# Approximation and localized polynomial frame on conic domains

Yuan Xu  Department of Mathematics, University of Oregon, Eugene, OR
97403–1222, USA<EMAIL_ADDRESS>
###### Abstract.

Highly localized kernels constructed by orthogonal polynomials have been
fundamental in the recent development of approximation and computational
analysis on the unit sphere, unit ball, and several other regular domains. In
this work, we first study homogeneous spaces that are assumed to contain
highly localized kernels and establish a framework for approximation and
localized tight frames in such spaces, which extends recent works on bounded
regular domains. We then show that the framework is applicable to homogeneous
spaces defined on bounded conic domains, which consists of conic surfaces and
the solid domains bounded by such surfaces and hyperplanes. The highly
localized kernels on conic domains require precise estimates that rely on
recently discovered addition formulas for orthogonal polynomials with respect
to special weight functions on each domain and an intrinsic distance that
takes into account the boundary of the domain, the latter is not comparable to
the Euclidean distance at around the apex of the cone.

The main results provide construction of semi-discrete localized tight frame
in weighted $L^{2}$ norm and characterization of best approximation by
polynomials on conic domains. The latter is achieved by using a
$K$-functional, defined via the differential operator that has orthogonal
polynomials as eigenfunctions, as well as a modulus of smoothness defined via
a multiplier operator that is equivalent to the $K$-functional. Several
intermediate results are of interest in their own right, including the
Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel
functions, and Bernstein type inequalities. Moreover, although the highly
localizable kernels hold only for special families of weight functions on each
domain, many intermediate results are shown to hold for doubling weights
defined via the intrinsic distance on the domain.

###### Key words and phrases:

Approximation, conic domain, Fourier orthogonal series, homogeneous spaces,
localized kernel, orthogonal polynomials, polynomial frame

###### 2010 Mathematics Subject Classification:

41A10, 41A63, 42C10, 42C40

The author is partially supported by Simons Foundation Grant #849676.

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 New contributions
      1. 1.2.1 Localizable homogeneous space
      2. 1.2.2 Best approximation by polynomials
      3. 1.2.3 Conic surface ${\mathbb{V}}_{0}^{d+1}$ and cone ${\mathbb{V}}^{d+1}$
    3. 1.3 Organization and convention
  2. 2 Homogeneous spaces with highly localized kernels
    1. 2.1 Localizable homogeneous spaces
      1. 2.1.1 Homogeneous spaces
      2. 2.1.2 Orthogonal polynomials
      3. 2.1.3 Localizable homogeneous spaces
    2. 2.2 Examples of localizable homogeneous spaces
      1. 2.2.1 The interval $[-1,1]$ with the Jacobi weight
      2. 2.2.2 The unit sphere ${\mathbb{S}^{d-1}}$
      3. 2.2.3 The unit ball ${\mathbb{B}}^{d}$ with classical weight function
    3. 2.3 An auxiliary maximal function
    4. 2.4 Marcinkiewicz-Zygmund inequality
    5. 2.5 Christoffel function for doubling weight
    6. 2.6 Positive cubature rules
    7. 2.7 Tight polynomial frames
  3. 3 Polynomial Approximation on homogeneous spaces
    1. 3.1 Addition formula and differential operator
    2. 3.2 Convolution structure
    3. 3.3 Characterization of best approximation
    4. 3.4 Near best approximation operator
    5. 3.5 Bernstein and Nikolskii inequalities
    6. 3.6 Proof of the main results
  4. 4 Homogeneous space on conic surfaces
    1. 4.1 Distance on the surface of the cone
    2. 4.2 A family of doubling weights
    3. 4.3 Orthogonal polynomials on the conic surface
    4. 4.4 Highly localized kernels
    5. 4.5 Maximal ${\varepsilon}$-separated sets and MZ inequality
    6. 4.6 Cubature rules and localized tight frames
    7. 4.7 Characterization of best approximation
  5. 5 Homogeneous space on solid cones
    1. 5.1 Distance on solid cone
    2. 5.2 A family of doubling weights
    3. 5.3 Orthogonal polynomials on the solid cone
    4. 5.4 Highly localized kernels
    5. 5.5 Maximal ${\varepsilon}$-separated sets and MZ inequality
    6. 5.6 Cubature rules and localized tight frames
    7. 5.7 Characterization of best approximation

##  1\. Introduction

In recent years highly localized kernels constructed via orthogonal
polynomials have become important tools in approximation theory, both
computational and theoretical harmonic analysis, and functional analysis. We
start with an introduction on the background of this development in the first
subsection, describe our main results in the second subsection and state the
organization of the work in the third subsection.

###  1.1. Background

Let $\Omega$ be a set in ${\mathbb{R}}^{d}$, either an algebraic surface or a
domain with non-empty interior, and let $\varpi$ be a nonnegative weight
function defined on $\Omega$, normalized with unit integral with respect to
the Lebesgue measure $\mathrm{d}x$ on $\Omega$, so that the bilinear form

${\langle}f,g{\rangle}_{\varpi}=\int_{\Omega}f(x)g(x)\varpi(x)\mathrm{d}x$

is a well defined inner product on the space of polynomials restricted to
$\Omega$. Let ${\mathcal{V}}_{n}^{d}(\varpi)$ be the space of orthogonal
polynomials of degree $n$ with respect to this inner product. Assume that
$\varpi$ is regular so that the orthogonal decomposition

$L^{2}(\Omega,\varpi)=\bigoplus_{n=0}^{\infty}{\mathcal{V}}_{n}^{d}(\varpi)$

holds. Let $P_{n}(\varpi;\cdot,\cdot)$ be the reproducing kernel of the space
${\mathcal{V}}_{n}^{d}(\varpi)$. The projection operator
$\operatorname{proj}_{n}:L^{2}(\Omega,\varpi)\mapsto{\mathcal{V}}_{n}^{d}(\varpi)$
can be written as

$(\operatorname{proj}_{n}f)(x)=\int_{\Omega}P_{n}(\varpi;x,y)f(y)\varpi(y)\mathrm{d}y,\qquad
f\in L^{2}(\Omega,\varpi).$

If $\widehat{a}$ is a cut-off function, defined as a compactly supported
function in $C^{\infty}({\mathbb{R}}_{+})$ and $\widehat{a}(t)=1$ for $t$ near
zero, then our highly localized kernels are of the form

$L_{n}(\varpi;x,y):=\sum_{j=0}^{\infty}\widehat{a}\Big{(}\frac{j}{n}\Big{)}P_{j}(\varpi;x,y).$

For orthogonal systems on several regular domains, the kernel
$L_{n}(\varpi;x,y)$ for a suitable $\widehat{a}$ satisfies a localization
principle, meaning that the kernel decays at rates faster than any inverse
polynomial rate away from the main diagonal $y=x$ in $\Omega\times\Omega$ with
respect to an intrinsic distance in $\Omega$.

To get a sense of the highly localized estimate, we specify to two cases. The
first one is the unit sphere ${\mathbb{S}^{d-1}}$ of ${\mathbb{R}}^{d}$ with
$\varpi(x)\mathrm{d}x=\mathrm{d}\sigma$ is the surface measure, for which the
kernel $L_{n}(\cdot,\cdot)=L_{n}(\varpi;\cdot,\cdot)$ is highly localized in
the sense that [34], for every ${\kappa}>0$,

(1.1) $|L_{n}(x,y)|\leq
c_{\kappa}\frac{n^{d-1}}{(1+n\,{\mathsf{d}}_{\SS}(x,y))^{\kappa}},\qquad
x,y\in{\mathbb{S}^{d-1}},$

where ${\mathsf{d}}_{\SS}$ denotes the geodesic distance

(1.2) ${\mathsf{d}}_{\SS}(x,y)=\arccos{\langle}x,y{\rangle},\qquad
x,y\in{\mathbb{S}^{d-1}}.$

For the unit ball, the classical weight function is
$\varpi_{\mu}(x)=(1-\|x\|^{2})^{\mu-\frac{1}{2}}$, $\mu>-\frac{1}{2}$. For
$\mu\geq 0$, the kernel $L_{n}(\varpi_{\mu};\cdot,\cdot)$ is highly localized
in the sense that [38]

(1.3) $\left|L_{n}(\varpi_{\mu};x,y)\right|\leq
c_{\kappa}\frac{n^{d}}{\sqrt{\varpi_{\mu}(n;x)}\sqrt{\varpi_{\mu}(n;y)}(1+n\,{\mathsf{d}}_{\mathbb{B}}(x,y))^{\kappa}},\qquad
x,y\in{\mathbb{B}}^{d},$

where $\varpi_{\mu}(n;x)=(1-\|x\|^{2}+n^{-2})^{\mu}$ and
${\mathsf{d}}_{\mathbb{B}}$ is the distance on ${\mathbb{B}}^{d}$ defined by

(1.4)
${\mathsf{d}}_{\mathbb{B}}(x,y)=\arccos\left({\langle}x,y{\rangle}+\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\right).$

By choosing ${\kappa}$ large, the kernel decays away form $y=x$ by (1.1) and
(1.3) faster than any polynomial rate. Furthermore, under a technical
restriction on the cut-off function, both estimates can be improved to sub-
exponential rate [28]. The proof of these estimates relies on closed-form
formula of the reproducing kernel $P_{n}(\varpi;\cdot,\cdot)$ on the unit
sphere and the unit ball; see the next subsection.

Let us denote the integral operator with $L_{n}(\varpi)$ as its kernel by
$L_{n}*f$; that is,

(1.5) $L_{n}*f(x):=\int_{\Omega}f(y)L_{n}(\varpi;x,y)\varpi(y)\mathrm{d}y.$

If $\widehat{a}$ has the support $[0,2]$ and satisfies $\widehat{a}(t)=1$ for
$0\leq t\leq 1$, then the operator $L_{n}*f$ is a polynomial of degree $2n$
and it reproduces polynomials of degree $n$. Furthermore, in many cases, it
provides the near best polynomial approximation to $f\in L^{p}(\Omega,\varpi)$
in the sense that

$\|L_{n}*f-f\|_{L^{p}(\Omega,\varpi)}\leq c\inf_{\deg g\leq
n}\|f-g\|_{L^{p}(\Omega,\varpi)},\quad 1\leq p\leq\infty.$

The rapid decay of the kernel makes it a powerful tool for studying polynomial
approximation on $\Omega$. Moreover, the operator $L_{n}*f$ plays an important
role in the characterization of best approximation, which is a central problem
in approximation theory. For the unit sphere, for example, it is used in the
classical result that characterizes the error of best polynomial approximation
by a modulus of smoothness defied via spherical means or by its equivalent
$K$-functional defined via the Laplace-Beltrami operator on the sphere. The
theory has a long history that spans from [4, 36] to [42] with contributions
from many researchers in between. It has also been extended, more recently, to
weight approximation on the sphere, the unit ball, and the simplex in [52].

The highly localized kernels can also be used to construct localized frames
via a semi-continuous Calderón type decomposition based on the decomposition,

$f=\sum_{j=0}^{\infty}L_{2^{j}}\ast L_{2^{j}}\ast f,\qquad f\in
L^{2}(\Omega,\varpi),$

which holds if the cut-off function $\widehat{a}$ satisfies
$\sum_{k=0}^{\infty}|\widehat{a}(2^{-k}t)|^{2}=1$. To define the frame
elements, the integrals in the right-hand side of the above expansion need to
be appropriately discretized by a cubature rule, established with the help of
the highly localized kernels, which states that

$\int_{\Omega}f(x)\mathrm{d}x=\sum_{\xi\in\Xi_{j}}{\lambda}_{\xi}f(\xi),\qquad
f\in\Pi_{2^{j}}(\Omega),$

where ${\lambda}_{\xi}>0$, $\Xi_{j}$ is a finite subset in $\Omega$, and
$\Pi_{n}(\Omega)$ denotes the space of polynomials, restricted on $\Omega$, of
degree at most $n$. The discretization leads to the tight frame
$\\{\psi_{\xi}\\}_{\xi\in\Xi}$ of $L^{2}(\Omega,\varpi)$, where $\Xi$ is a
discrete set of well separated, with respect to the intrinsic distance, points
in $\Omega$, and the frame elements $\psi_{\xi}$, also called needlets, are of
the form

$\psi_{\xi}(x)=\sqrt{{\lambda}_{\xi}}L_{2^{j}}(x,\xi),\qquad\xi\in\Xi,$

which is highly localized with its center at $\xi$ since ${\lambda}_{\xi}$ can
be quantized and $L_{2^{j}}(x,\xi)$ is highly localized. The tight frame means
that, for all $f\in L^{2}(\Omega,\varpi)$,

$f=\sum_{\xi\in\Xi}{\langle}f,\psi_{\xi}{\rangle}_{\varpi}\psi_{\xi}\quad\hbox{and}\quad\int_{\Omega}|f(x)|^{2}\varpi(x)\mathrm{d}x=\sum_{\xi\in\Xi}|{\langle}f,\psi_{\xi}{\rangle}|^{2}.$

Because of their rapid decay, needlets can be used for decomposition of spaces
of functions and distributions in various settings, including $L^{p}$,
Sobolev, Besov, and Triebel- Lizorkin spaces.

The above narrative has been carried out in the setting of the regular
domains. The localized kernels are most heavily studied and used when $\Omega$
is the unit sphere ${\mathbb{S}^{d-1}}$ of ${\mathbb{R}}^{d}$, for which our
outline of the localized frames is initiated in [34, 35]. They are further
explored in, for example, [12, 28], on the sphere, and also studied on the
interval $[-1,1]$ with the Jacobi weight [5, 24, 37], the unit ball with the
Gegenbauer weight [25, 38], and the simplex with the multivariable Jacobi
weight [28] as well as for ${\mathbb{R}}_{+}^{d}$ with the product Laguerre
weight and ${\mathbb{R}}^{d}$ with the Hermite weight [15, 18, 20, 39] and for
product domains [29]. The subexponential decay of the kernel was established
in [28], following earlier work in [16]. The needlets on the sphere and the
ball have found applications in computational harmonics analysis, mathematical
physics and statistics; see, for example, [2, 3, 26, 27, 31, 49] and
references in them.

###  1.2. New contributions

Our starting point is the recent study of orthogonal structures on conic
domains of ${\mathbb{R}}^{d+1}$ in [53, 54], which shows that many properties
of spherical harmonics and classical orthogonal polynomials on the unit ball
have analogs on conic domains. Most importantly, there are families of
orthogonal polynomials on conic domains that possess properties akin to the
addition formula and the Laplace-Beltrami operator for the spherical
harmonics, the former provides a closed-form formula for the reproducing
kernels and the latter is a second order differential operator that has
orthogonal polynomials as eigenfunctions. This paves the way for carrying out
the narrative outlined in the previous subsection on conic domains, which has
been a topic largely untouched hitherto. In the present paper, we will deal
with two types of conic domains and they are standardized as:

  1. (1)

conic surface ${\mathbb{V}}_{0}^{d+1}$ defined by

${\mathbb{V}}_{0}^{d+1}:=\left\\{(x,t)\in{\mathbb{R}}^{d+1}:\|x\|=t,\,0\leq
t\leq 1,\,x\in{\mathbb{R}}^{d}\right\\};$

  2. (2)

solid cone ${\mathbb{V}}^{d+1}$ bounded by ${\mathbb{V}}_{0}^{d+1}$ and the
hyperplane $t=1$.

Two other conic domains, double hyperbolic surface and solid hyperboloid, will
be dealt with elsewhere because of the page limitation. On each domain, a
closed-form formula of the reproducing kernels is uncovered for a family of
weight functions, which allows us to study highly localized kernels and
carries out the program described in Subsection 1.1, and a differential
operator is defined that has orthogonal polynomials as eigenfunctions, which
leads to a $K$-functional that is used to characterize the best approximation
by polynomials on the domain.

Instead of dealing with each case individually, we will develop a unified
theory for all homogeneous spaces that contain highly localized kernels that
we describe in the first two subsections below. The unified theory is
applicable to conic domains, which is described in the third and the fourth
subsections.

####  1.2.1. Localizable homogeneous space

A homogeneous space is a measure space $(\Omega,\mathrm{d}\mu,{\mathsf{d}})$,
where $\mathrm{d}\mu$ is a nonnegative doubling measure with respect to the
metric ${\mathsf{d}}(\cdot,\cdot)$ on $\Omega$. When
$\mathrm{d}\mu={\mathsf{w}}(x)\mathrm{d}x$, where ${\mathsf{w}}$ is a doubling
weight function on $\Omega$, we denote the homogenous space by
$(\Omega,{\mathsf{w}},{\mathsf{d}})$. For example, the space
$({\mathbb{S}^{d-1}},\mathrm{d}\sigma,{\mathsf{d}}_{\SS})$ is a homogeneous
space, so is $({\mathbb{B}}^{d},\varpi_{\mu},{\mathsf{d}}_{\mathbb{B}})$ for
$\mu>-\frac{1}{2}$.

We call a homogeneous space $(\Omega,\varpi,{\mathsf{d}})$ localizable if the
orthogonal polynomials with respect to $\varpi$ on $\Omega$ admit highly
localized kernels, which requires a fast decay estimate on
$|L_{n}(\varpi;x,y)|$ as well as on
$|L_{n}(\varpi;x_{1},y)-L_{n}(\varpi;x_{2},y)|$; see Subsection 2.1 for
precise definition. Our first main result is to show that the narrative on the
localized tight frame in the previous subsection can be established in a
localizable homogenous space. This provides a unified theory that is
applicable to conical domains to be studied in later sections. Furthermore, it
allows us to establish several intermediate results that are of interest in
their own right for doubling weight functions, including Marcinkiewicz-Zygmund
inequalities and positive cubature rules.

The development generalizes and follows closely recent studies on regular
domains, especially those on the unit sphere and the unit ball. We follow the
approach on the unit sphere in [7, 12], which is formulated in the homogeneous
space on the sphere. While some of the results for the unit sphere can be
extended easily, others need to be dealt with more carefully. Amongst various
reasons, we mention that, on the unit sphere, it is the Lebesgue measure
$\mathrm{d}\sigma$ that admits the highly localized kernels. It is, however,
not the case in general. For example, the highly localized kernels on the
conic surface ${\mathbb{V}}_{0}^{d+1}$ are established for the weight function
${\mathsf{w}}_{-1,{\gamma}}(t)=t^{-1}(1-t)^{\gamma}$, which does not include
the Lebesgue measure. While spherical caps of the same radius on
${\mathbb{S}^{d-1}}$ have the same constant surface area, this is not the case
on ${\mathbb{V}}_{0}^{d+1}$, where we need to deal with conical caps defined
via the intrinsic distance ${\mathsf{d}}_{{\mathbb{V}}_{0}}$, defined in (1.6)
below, and their volumes are measured by integrals over the caps against the
weight function ${\mathsf{w}}_{-1,{\gamma}}$.

####  1.2.2. Best approximation by polynomials

The characterization of best approximation is at the core of approximation
theory (cf. [17]). The classical approach on the unit sphere relies
essentially on multiplier operators of the Fourier harmonic series and the
convolution structure arising from the addition formula. We provide a general
framework on localized homogeneous space by assuming that the reproducing
kernel $P_{n}(\varpi;\cdot,\cdot)$ satisfies an addition formula of a specific
form satisfied by all known cases, including those on conic domains, and that
there is a second order differential operator that has orthogonal polynomials
as eigenfunctions. The former allows us to introduce a convolution structure
that can be used to define a modulus of smoothness via a multiplier operator,
and the latter is used to define a $K$-functional. The two quantities are then
shown to be equivalent and either one can be used to characterize the error of
the best polynomial approximation on the domain. The framework generalizes the
classical result on the unit sphere and is applicable to other regular domains
satisfying our assumptions, which are met by all four types of conic domains.

####  1.2.3. Conic surface ${\mathbb{V}}_{0}^{d+1}$ and cone
${\mathbb{V}}^{d+1}$

On the conic surface ${\mathbb{V}}_{0}^{d+1}$, the orthogonal structure is
studied in [53] for the weight function
${\mathsf{w}}_{{\beta},{\gamma}}(t)=t^{\beta}(1-t)^{\gamma}$ for ${\beta}>-d$
and ${\gamma}>-1$. For our study we need an intrinsic metric on the conic
surface. In contrast to the unit sphere with its geodesic distance
${\mathsf{d}}_{\SS}(\cdot,\cdot)$ that depend only on relative positions of
the points, the conic surface ${\mathbb{V}}_{0}^{d+1}$ has a boundary at $t=1$
and an apex point at $t=0$ that its metric needs to take into account. The
distance that we will use is defined by

(1.6)
${\mathsf{d}}_{{\mathbb{V}}_{0}}((x,t),(y,s))=\arccos\left[\sqrt{\frac{{\langle}x,y{\rangle}+ts}{2}}+\sqrt{1-t}\sqrt{1-s}\right].$

Using this distance function and the closed-form formula of the reproducing
kernel proved in [53], we shall show that ${\mathsf{w}}_{{\beta},{\gamma}}$ is
a doubling weight and that the space
$(\Omega,{\mathsf{w}}_{-1,{\gamma}},{\mathsf{d}}_{{\mathbb{V}}_{0}})$ is a
localizable homogeneous space. These results are established through delicate
estimates, partly because the distance
${\mathsf{d}}_{{\mathbb{V}}_{0}}(\cdot,\cdot)$ is not comparable to the
Euclidean distance at around the apex of ${\mathbb{V}}_{0}^{d+1}$. With the
highly localized kernel, we can then carry out our program and establish
positive cubature rules, and use them to establish the highly localized frame.
Furthermore, using the second order differential operator that has orthogonal
polynomials with respect to ${\mathsf{w}}_{-1,{\gamma}}$ as eigenfunctions,
which is an analog of the Laplace-Beltrami operator in the unit sphere, we can
define a $K$-functional and use it to establish a characterization of the best
approximation by polynomials in the weighted $L^{p}$ norm on
${\mathbb{V}}_{0}^{d+1}$.

Our study on the cone ${\mathbb{V}}^{d+1}$ follows along the similar line. The
distance function is defined by identifying ${\mathbb{V}}^{d+1}$ with a subset
of ${\mathbb{V}}_{0}^{d+2}$. The orthogonal structure is studied in [53] for
the weight function
$W_{{\beta},{\gamma},\mu}(x,t)=t^{\beta}(1-t)^{\gamma}(t^{2}-\|x\|^{2})^{\mu-\frac{1}{2}}$,
which has a more involved close form formula for its reproducing kernels.
While part of the estimate can be carried out for the kernel associated with
$W_{0,{\gamma},\mu}$, the localizable homogeneous space is established for
$W_{0,{\gamma},0}$, which nevertheless allows us to complete our program on
the solid cone ${\mathbb{V}}^{d+1}$.

Several remarks are in order over the above description of our main results.
First of all, analysis on conical domains have not been seriously studied in
the literature as far as we are aware, and it does pose new phenomena and new
challenges. In particular, while the framework for localizable homogeneous
space is applicable to conic domains, the verification of assertions under
which the framework holds possess considerable difficulty in each case.
Second, there are many intermediate results that are of interests. For
example, to quantize the coefficients in positive cubature rule, we need
bounds on the Christoeffel functions

$\displaystyle{\lambda}_{n}({\mathsf{w}};x)=\inf_{\begin{subarray}{c}g(x)=1\\\
g\in\Pi_{n}(\Omega)\end{subarray}}\int_{\Omega}|g(x)|^{2}{\mathsf{w}}(x)\mathrm{d}x,$

which are of interest in their own right. See, for example, [6, 14, 22, 23,
40, 41, 50] and their references for recent works on Christoffel functions in
several variables. While the lower bound of the Christoeffel function on a
homogenous space can be derived using highly localized kernels when
${\mathsf{w}}$ admits such kernels, we will establish the upper bound for all
doubling weight under the assumption that certain fast decaying polynomials
exist on $\Omega$. The latter requires the construction of such polynomials in
each conic domain. Third, there have been several recent works that deal with
multivariate approximation by polynomials or Marcinkiewicz-Zygmund
inequalities and positive cubature rules on either polyhedra domain or $C^{2}$
domain [8, 9, 47, 48]. They do not cover conical domains and our approach is
different and relies on a specific orthogonal structure on the domain, akin to
those on the unit sphere and the unit ball.

Finally, we should mention the important book Analysis on Symmetric Cones
[19] and the literature around it. While the topic works in the theory of
Euclidean Jordan algebras and lies in a more abstract setting, our study
requires very specific structures of Fourier orthogonal series and is far less
abstract. We have not been able to discern a connection between the two topics
duo to our lack of background in Jordan algebras. It would be of great
interest if a connection could be identified.

###  1.3. Organization and convention

The paper is organized as follows. The localizable homogeneous space will be
defined and studied in the next section, which consists of results discussed
in Subsection 1.2.1. The best approximation by polynomials in the homogeneous
space is discussed in the third section, which consists of the results
described in Subsection 1.2.2. The results on the conic surface and solid cone
will be discussed in Sections 4 and 5, respectively. Each section will contain
several subsections and its organization will be described in the preamble of
the section.

Throughout this paper, we will denote by $L^{p}(\Omega,{\mathsf{w}})$ the
weighted $L^{p}$ space with respect to the weight function ${\mathsf{w}}$
defined on the domain $\Omega$ for $1\leq p\leq\infty$. Its norm will be
denote by $\|\cdot\|_{p,{\mathsf{w}}}$ for $1\leq p\leq\infty$ with the
understanding that the space is $C(\Omega)$ with the uniform norm when
$p=\infty$.

For conical domains, in an attempt to distinguish conic surfaces and solid
conic bodies, we shall denote the operator on the surface in sans serif font,
such as ${\mathsf{P}}_{n}$ and ${\mathsf{L}}_{n}$, and the operator on the
solid domains in bold font, such as ${\mathbf{P}}_{n}$ and ${\mathbf{L}}_{n}$.

Finally, we shall use $c$, $c^{\prime}$, $c_{1}$, $c_{2}$ etc. to denote
positive constants that depend on fixed parameters and their values may change
from line to line. Furthermore, we shall write $A\sim B$ if $c^{\prime}A\leq
B\leq cA$.

##  2\. Homogeneous spaces with highly localized kernels

Much of recent work on regular domains, such as the sphere, the ball, and the
regular simplex falls in the framework of homogeneous space. What
distinguishes these regular domains is the existence of highly localized
kernels constructed via orthogonal polynomials. From the prior work on these
regular domains and what we will prove for the conic domains latter sections,
it becomes clear that the desired estimates for the highly localized kernels
share a common formation. In this section, we work with homogeneous spaces
that are assumed to contain highly localized kernels and carry out our
analysis on such spaces.

The first subsection contains basics on homogeneous space and orthogonal
polynomials, and it contains three assertions that define highly localized
kernels precisely. Three examples are given in the second subsection, which
also serves as a review of the Jacobi polynomials, spherical harmonics, and
orthogonal polynomials on the ball that will be used later. An auxiliary
maximal function is introduced in the third subsection and, with the help of
highly localized kernels for $\varpi$, is shown to be bounded in weighted
$L^{p}$ space with respect to a doubling weight. Using this maximal function,
Marcinkiewicz-Zygmund inequalities are established in the fourth subsection
for all doubling weights. The fifth subsection contains the fourth assertion
that is used to obtain an upper bound for the Christoeffel function. The
latter is needed in the sixth subsection, where the positive cubature rules
are established. With these preparations, the localized tight frame is defined
and studied in the seventh subsection.

The development of this section follows and generalizes the study on the unit
sphere. Some parts of the work follow those on the sphere with little extra
effort but by no means all. Since the geodesic distance on the sphere and the
surface measure on the unit sphere will be replaced by a distance function and
a weight function that offer no specific geometric information, we will
provide complete proofs in the most part and will be brief only when the proof
follows that on the unit sphere fairly straightforwardly.

###  2.1. Localizable homogeneous spaces

####  2.1.1. Homogeneous spaces

A homogeneous space is a measure space $(\Omega,\mu,{\mathsf{d}})$ with a
positive measure $\mu$ and a metric ${\mathsf{d}}$ such that all open balls

$B(x,r)=\\{y\in\Omega:{\mathsf{d}}(x,y)<r\\},$

are measurable and $\mu$ is a regular measure satisfying the doubling property

$\mu(B(x,2r))\leq c\mu(B(x,r)),\qquad\forall x\in\Omega,\quad\forall r>0,$

where $c$ is independent of $x$ and $r$. Such a measure $\mu$ is called a
doubling measure.

Let $\Omega$ be a domain in ${\mathbb{R}}^{d}$. We assume that $\Omega$ is
equipped with an intrinsic distance ${\mathsf{d}}$, so that the Lebesuge
measure ${\mathsf{m}}$ on $\Omega$ is a doubling measure and, for each
${\varepsilon}>0$, $\min_{x\in\Omega}{\mathsf{m}}(B(x,{\varepsilon}))\geq
c_{\varepsilon}>0$. Let ${\mathsf{w}}$ be a nonnegative integrable function
defined on $\Omega$. For a given set $E\subset\Omega$, we define

${\mathsf{w}}(E)=\int_{E}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x).$

The weight function ${\mathsf{w}}$ is called a doubling weight if its
satisfies the doubling condition: there exists a constant $L>0$ such that

${\mathsf{w}}(B(x,2r))\leq L{\mathsf{w}}(B(x,r)),\qquad\forall
x\in\Omega,\quad r\in(0,r_{0}),$

where $r_{0}$ is the largest positive number such that $B(x,r)\subset\Omega$.
If ${\mathsf{w}}$ is a doubling weight, then
${\mathsf{d}}\mu={\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}$ is a doubling measure.
We are particularly interested in the homogeneous space with its measure so
defined, which we denote by $(\Omega,{\mathsf{w}},{\mathsf{d}})$.

For a doubling weight ${\mathsf{w}}$ on $\Omega$ with respect to
${\mathsf{d}}$, we denote by $L({\mathsf{w}})$ the least constant $L$ for the
doubling condition ${\mathsf{w}}(B(x,2r))\leq L{\mathsf{w}}(B(x,r))$. Let
${\alpha}({\mathsf{w}})$ be a positive number such that

(2.1)
$\sup_{B(x,r)\subset\Omega}\frac{{\mathsf{w}}(B(x,2^{m}r))}{{\mathsf{w}}(B(x,r))}\leq
c_{L({\mathsf{w}})}2^{{\alpha}({\mathsf{w}})m},\quad m=1,2,\ldots.$

Then ${\alpha}({\mathsf{w}})\leq\log_{2}L({\mathsf{w}})$, as can be seen by
iteration. We call $L({\mathsf{w}})$ the doubling constant and
${\alpha}({\mathsf{w}})$ the doubling index.

######  Lemma 2.1.

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$.

  1. (i)

If $0<r<t$ and $x\in\Omega$, then

${\mathsf{w}}(B(x,t))\leq
c_{L({\mathsf{w}})}\left(\frac{t}{r}\right)^{{\alpha}({\mathsf{w}})}{\mathsf{w}}(B(x,r)).$

  2. (ii)

For $x,y\in\Omega$ and $n=1,2,\ldots$,

${\mathsf{w}}(B(x,n^{-1}))\leq
c_{L({\mathsf{w}})}(1+n{\mathsf{d}}(x,y))^{\alpha({\mathsf{w}})}{\mathsf{w}}(B(y,n^{-1})).$

###### Proof.

Assume $2^{m-1}\leq t/r\leq 2^{m}$, (i) follows immediately from the
definition of ${\alpha}({\mathsf{w}})$. By the triangle inequality of the
distance function,
${\mathsf{w}}(B(x,n^{-1}))\leq{\mathsf{w}}(B(y,{\mathsf{d}}(x,y)+n^{-1}))$,
applying (i) to bound it by ${\mathsf{w}}(B(y,n^{-1}))$ proves (ii). ∎

Many fundamental results that hold for the Euclidean space can be extended to
homogeneous spaces [44]. For example, the Hardy-Littlewood maximal function is
defined, for a locally integrable function $f$, by

(2.2)
$M_{\mathsf{w}}f(x):=\sup_{r>0}\frac{1}{{\mathsf{w}}(B(x,r))}\int_{B(x,r)}|f(y)|{\mathsf{w}}(y)\mathrm{d}{\mathsf{m}},\qquad
x\in\Omega,$

which possesses the usual properties of the maximal function [44, p. 13]. For
example, for every $f\in L^{p}(\Omega,{\mathsf{w}})$,

(2.3) $\|M_{\mathsf{m}}f\|_{p,{\mathsf{w}}}\leq
c_{p}\|f\|_{p,{\mathsf{w}}},\qquad 1<p\leq\infty,$

where $\|f\|_{p,{\mathsf{w}}}$ denotes the $L^{p}$ norm with respect to the
measure ${\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}$.

We study homogeneous spaces that contain highly localized kernels, to be
defined below. Such spaces are known to hold for certain weight functions on
several regular domains. We adopt the convention of denoting the weight
function that admits highly localized kernels by $\varpi$ and we also assume
that $\varpi$ is normalized by
$\int_{\Omega}\varpi(x)\mathrm{d}{\mathsf{m}}=1$. The kernels are defined via
orthogonal polynomials with respect to the inner product
${\langle}\cdot,\cdot{\rangle}_{\varpi}$ defined in terms of $\varpi$ in
(2.4).

####  2.1.2. Orthogonal polynomials

Let $\Pi(\Omega)$ denote the space of polynomials restricted to $\Omega$ and
let $\Pi_{n}(\Omega)$ denote its subspace of polynomials of degree at most
$n$. If the interior of $\Omega$ is open, then $\Pi(\Omega)$ contains all
polynomials of degree $n$ in $d$ variables and we write
$\Pi_{n}=\Pi_{n}(\Omega)$. If $\Omega$ is an algebraic surface, such as the
sphere ${\mathbb{S}^{d-1}}$, then $\Pi_{n}(\Omega)$ contains all polynomials
restricted to the surface.

Let $\varpi$ be a normalized weight function on $\Omega$. Then the bilinear
form ${\langle}\cdot,\cdot{\rangle}_{\varpi}$

(2.4)
${\langle}f,g{\rangle}_{\varpi}:=\int_{\Omega}f(x)g(x)\varpi(x)\mathrm{d}x$

is a well defined inner product on $\Pi_{n}(\Omega)$. Let
${\mathcal{V}}_{n}^{d}(\Omega,\varpi)$ be the space of orthogonal polynomials
of degree $n$ with respect to this inner product. We are particularly
interested in the case when $\Omega$ is either a quadratic surface, such as
the unit sphere ${\mathbb{S}^{d-1}}$ or the conical surface
${\mathbb{V}}_{0}^{d}$, for which

(2.5)
$\dim{\mathcal{V}}_{n}^{d}(\Omega,\varpi)=\binom{n+d-2}{n}+\binom{n+d-3}{n-1},\quad
n=1,2,3,\ldots,$

where we assume $\binom{n}{k}=0$ if $k<0$, or a solid domain bounded by a
quadratic surface, such as the unit ball ${\mathbb{B}}^{d}$ and the solid cone
${\mathbb{V}}^{d}$, for which

(2.6) $\dim{\mathcal{V}}_{n}^{d}(\Omega,\varpi)=\binom{n+d-1}{n},\quad
n=0,1,2,\ldots,$

The reproducing kernel of ${\mathcal{V}}_{n}^{d}(\Omega,\varpi)$, denoted by
$P_{n}(\varpi;\cdot,\cdot)$, is uniquely determined by

$\int_{\Omega}f(y)P_{n}(\varpi;x,y)\varpi(y)\mathrm{d}{\mathsf{m}}(y)=f(x),\qquad\forall
f\in{\mathcal{V}}_{n}^{d}(\Omega,\varpi).$

If $\\{P_{\nu,n}:1\leq\nu\leq\dim{\mathcal{V}}_{n}^{d}(\Omega,\varpi)\\}$ is
an orthogonal basis of ${\mathcal{V}}_{n}^{d}(\Omega,\varpi)$, then

(2.7)
$P_{n}(\varpi;x,y)=\sum_{\nu=1}^{\dim{\mathcal{V}}_{n}^{d}(\Omega,\varpi)}\frac{P_{\nu,n}(x)P_{\nu,n}(y)}{{\langle}P_{\nu,n},P_{\nu,n}{\rangle}_{\varpi}}.$

The kernel plays an essential role in the study of the Fourier orthogonal
series, since it is the kernel of the orthogonal projection operator
$\operatorname{proj}_{n}:L^{2}(\Omega,\varpi)\mapsto{\mathcal{V}}_{n}^{d}(\Omega,\varpi)$;
more precisely,

(2.8)
$\operatorname{proj}_{n}(\varpi;f,x)=\int_{\Omega}f(y)P_{n}(\varpi;x,y)\varpi(y)\mathrm{d}{\mathsf{m}}(y),\quad
f\in L^{2}(\Omega,\varpi).$

The Fourier orthogonal series of $f\in L^{2}(\Omega,\varpi)$ is defined by

(2.9)
$f=\sum_{n=0}^{\infty}\operatorname{proj}_{n}(\varpi;f)=\sum_{n=0}^{\infty}\sum_{\nu=1}^{\dim{\mathcal{V}}_{n}^{d}(\Omega,\varpi)}\widehat{f}_{\nu,n}P_{\nu,n},\quad\widehat{f}_{\nu,n}=\frac{{\langle}f,P_{\nu,n}{\rangle}_{\varpi}}{{\langle}P_{\nu,n},P_{\nu,n}{\rangle}_{\varpi}}.$

For these definitions and orthogonal polynomials of several variables in
general, we refer to [14].

Our highly localized kernels are defined by a sampling of the kernels of the
Fourier series through a smooth cut-off function, which we define first.

######  Definition 2.2.

A nonnegative function $\widehat{a}\in C^{\infty}({\mathbb{R}})$ is said to be
admissible if it obeys either one of the conditions

  1. $(a)$

$\mathrm{supp}\,\widehat{a}\subset[0,2]$ and $\widehat{a}(t)=1$, $t\in[0,1]$;
or

  2. $(b)$

$\mathrm{supp}\,\widehat{a}\subset[1/2,2]$.

Let $\widehat{a}$ be an admissible cut-off function. For $n=0,1,2,\ldots$, we
define the kernel function $L_{n}(\varpi)$ on $\Omega\times\Omega$ by

(2.10)
$L_{n}(\varpi;x,y)=\sum_{k=0}^{\infty}\widehat{a}\left(\frac{k}{n}\right)P_{n}(\varpi;x,y),\qquad
x,y\in\Omega,$

Since $\widehat{a}$ is supported on $[0,2]$, this is a kernel of polynomials
of degree at most $2n$ in either $x$ or $y$ variable.

####  2.1.3. Localizable homogeneous spaces

The main purpose of this section is to study homogeneous spaces
$(\Omega,\varpi,{\mathsf{d}})$ for which the kernels
$L_{n}(\varpi;\cdot,\cdot)$ are assumed to satisfy the following assertions.

######  Definition 2.3.

The kernels $L_{n}(\varpi;\cdot,\cdot)$, $n=1,2,\ldots$, are called highly
localized if they satisfy the following assertions:

  1. Assertion 1. For ${\kappa}>0$ and $x,y\in\Omega$,

$|L_{n}(\varpi;x,y)|\leq
c_{\kappa}\frac{1}{\sqrt{\varpi\\!\left(B(x,n^{-1})\right)}\sqrt{\varpi\\!\left(B(y,n^{-1})\right)}\left(1+n{\mathsf{d}}(x,y)\right)^{\kappa}}.$

  2. Assertion 2. For $0<\delta\leq\delta_{0}$ with some $\delta_{0}<1$ and $x_{1}\in B(x_{2},\frac{\delta}{n})$,

$\left|L_{n}(\varpi;x_{1},y)-L_{n}(\varpi;x_{2},y)\right|\leq
c_{\kappa}\frac{n{\mathsf{d}}(x_{1},x_{2})}{\sqrt{\varpi\\!\left(B(x_{1},n^{-1})\right)}\sqrt{\varpi\\!\left(B(x_{2},n^{-1})\right)}\left(1+n{\mathsf{d}}(x_{2},y)\right)^{\kappa}}.$

  3. Assertion 3. For sufficient large ${\kappa}>0$,

$\displaystyle\int_{\Omega}\frac{\varpi(y)}{\varpi\\!\left(B(y,n^{-1})\right)\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{{\kappa}}}\mathrm{d}{\mathsf{m}}(y)\leq
c.$

The third assertion affirms the sharpness of the first two assertions. It
implies the following more general inequality.

######  Lemma 2.4.

Let $\varpi$ be a doubling weight that satisfies Assertion 3 with
${\kappa}>0$. For $0<p<\infty$, let
$\tau={\kappa}-\frac{p}{2}{\alpha}(\varpi)|1-\frac{p}{2}|>0$. Then, for
$x\in\Omega$,

(2.11)
$\int_{\Omega}\frac{\varpi(y)}{\varpi\\!\left(B\left(y,n^{-1}\right)\right)^{\frac{p}{2}}\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{\tau}}\mathrm{d}{\mathsf{m}}(y)\leq
c\,\varpi\\!\left(B\left(x,n^{-1}\right)\right)^{1-\frac{p}{2}}.$

###### Proof.

Denote the left-hand side of (2.11) by $J_{p,\tau}$. When $p=2$, the
inequality (2.11) is precisely the Assertion 3. The case $p\neq 2$ can be
deduced from the case $d=2$. Indeed, by Lemma 2.1, it is easy to see that,

(2.12)
$\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{-{\alpha}(\varpi)}\leq\frac{\varpi\\!\left(B\left(y,n^{-1}\right)\right)}{\varpi\\!\left(B\left(x,n^{-1}\right)\right)}\leq\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{{\alpha}(\varpi)}.$

For $p\neq 2$, using the right-hand side inequality if $0<p<2$ and the left-
hand side inequality if $p>2$, we obtain

$\varpi\\!\left(B\left(y,n^{-1}\right)\right)^{\frac{p}{2}}\geq\frac{\varpi\\!\left(B\left(y,n^{-1}\right)\right)\varpi\\!\left(B\left(x,n^{-1}\right)\right)^{\frac{p}{2}-1}}{\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{{\alpha}(w)|1-\frac{p}{2}|}},$

which gives
$J_{p,\tau}\leq\varpi\\!\left(B\left(x,n^{-1}\right)\right)^{1-\frac{p}{2}}J_{2,{\kappa}}$.
Hence, (2.11) for $p\neq 2$ follows from the case $p=2$. The proof is
completed. ∎

######  Corollary 2.5.

For $0<p<\infty$ and $x\in\Omega$, the highly localized kernel satisfies

(2.13)
$\int_{\Omega}\left|L_{n}(\varpi;x,y)\right|^{p}\varpi(y)\mathrm{d}{\mathsf{m}}(y)\leq
c\left[\varpi\\!\left(B\\!\left(x,n^{-1}\right)\right)\right]^{1-p}.$

In particular, for $p=1$, it ensures that the Assertion 1 is sharp in the
sense that the estimate guarantees the boundedness of
$\left|L_{n}(\varpi;x,y)\right|$ and it ensures that Assertion 2 is sharp in
the same sense.

######  Definition 2.6.

The homogeneous space $(\Omega,\varpi,{\mathsf{d}})$ is called localizable if
$\varpi$ is a doubling weight that admits highly localized kernels. For
convenience, we also say that the domain $\Omega$, or the weight $\varpi$,
admits a localizable homogeneous space without specifying $\varpi$, or
$\Omega$.

Throughout the rest of this section, we assume that the weight function
$\varpi$ admits a localizable homogeneous space on $\Omega$.

###  2.2. Examples of localizable homogeneous spaces

In this subsection, we give three examples of localizable homogeneous spaces:
the interval $[-1,1]$ with the Jacobi weight, the unit sphere
${\mathbb{S}^{d-1}}$ with the surface measure, the unit ball
${\mathbb{B}}^{d}$ with its classical weight function. These cases have been
thoroughly studied fairly recently in the literature, and they motivate our
definition of the localizable homogeneous spaces. Moreover, their
corresponding orthogonal polynomials serve as the building blocks for
orthogonal polynomials on the conic domains.

####  2.2.1. The interval $[-1,1]$ with the Jacobi weight

For ${\alpha},{\beta}>-1$, the Jacobi weight function is defined by

$w_{{\alpha},{\beta}}(t):=(1-t)^{\alpha}(1+t)^{\beta},\qquad-1<x<1.$

Its normalization constant $c^{\prime}_{{\alpha},{\beta}}$, defined by
$c^{\prime}_{{\alpha},{\beta}}\int_{-1}^{1}w_{{\alpha},{\beta}}(x)dx=1$, is
given by

(2.14)
$c^{\prime}_{{\alpha},{\beta}}=\frac{1}{2^{{\alpha}+{\beta}+1}}c_{{\alpha},{\beta}}\quad\hbox{with}\quad
c_{{\alpha},{\beta}}:=\frac{\Gamma({\alpha}+{\beta}+2)}{\Gamma({\alpha}+1)\Gamma({\beta}+1)}.$

The intrinsic distance of the interval $[-1,1]$ is defined by

(2.15)
${\mathsf{d}}_{[-1,1]}(t,s)=\arccos\left(ts+\sqrt{1-t^{2}}\sqrt{1-s^{2}}\right),$

which is the projection of the distance $|{\theta}-\phi|$ on the upper half of
the unit circle if we set $t=\cos{\theta}$ and $s=\cos\phi$. The space
$([-1,1],w_{{\alpha},{\beta}},{\mathsf{d}}_{[-1,1]})$ is a localizable
homogeneous space if ${\alpha},{\beta}\geq-\frac{1}{2}$.

The orthogonal polynomials with respect to $w_{{\alpha},{\beta}}$ are the
Jacobi polynomials $P_{n}^{({\alpha},{\beta})}$, which are given by the
hypergeometric function as [46, (4.21.2)]

$P_{n}^{({\alpha},{\beta})}(x)=\frac{({\alpha}+1)_{n}}{n!}{}_{2}F_{1}\left(\begin{matrix}-n,n+{\alpha}+{\beta}+1\\\
{\alpha}+1\end{matrix};\frac{1-x}{2}\right).$

These polynomials satisfy the orthogonal relations

$c_{{\alpha},{\beta}}^{\prime}\int_{-1}^{1}P_{n}^{({\alpha},{\beta})}(x)P_{m}^{({\alpha},{\beta})}(x)w_{{\alpha},{\beta}}(x)\mathrm{d}x=h_{n}^{({\alpha},{\beta})}\delta_{m,n},$

where $h_{n}^{({\alpha},{\beta})}$ is the square of the $L^{2}$ norm that
satisfies

$h_{n}^{({\alpha},{\beta})}=\frac{({\alpha}+1)_{n}({\beta}+1)_{n}({\alpha}+{\beta}+n+1)}{n!({\alpha}+{\beta}+2)_{n}({\alpha}+{\beta}+2n+1)}.$

The Jacobi polynomials are eigenfunctions of a second order differential
operator:

$\left[(1-x)\partial^{2}-({\alpha}-{\beta}+({\alpha}+{\beta}+2)x)\partial\right]u=-n(n+{\alpha}+{\beta}+1)u,$

where $\partial f=f^{\prime}$ and $u=P_{n}^{({\alpha},{\beta})}$ for
$n=0,1,2,\ldots$. The highly localized kernel for the Jacobi polynomials will
be denoted by

(2.16)
$L_{n}^{({\alpha},{\beta})}(x,y)=\sum_{j=0}^{\infty}\widehat{a}\Big{(}\frac{j}{n}\Big{)}\frac{P_{j}^{({\alpha},{\beta})}(x)P_{j}^{({\alpha},{\beta})}(y)}{h_{j}^{({\alpha},{\beta})}},$

where $\widehat{a}$ is an admissible cutoff function. This kernel decays
rapidly in terms of the distance ${\mathsf{d}}_{[-1,1]}(\cdot,\cdot)$. Let the
ball $B(x,r)$ be defined with respect to this distance. Then the Jacobi weight
is a doubling weight and satisfies,

$w_{{\alpha},{\beta}}(B(x,n^{-1}))\sim n^{-1}w_{{\alpha},{\beta}}(n;t),$

where $w_{{\alpha},{\beta}}(n;t)$ is defined by

$w_{{\alpha},{\beta}}(n;t)=(1-t+n^{-2})^{{\alpha}+\frac{1}{2}}(1+t+n^{-2})^{{\beta}+\frac{1}{2}}.$

The following estimate of the localized kernel can be found in [37].

######  Theorem 2.7.

Let $\alpha,\beta\geq-1/2$ and $0<{\varepsilon}\leq 1$. Then, for any
${\kappa}>0$ and $n\geq 1$,

$|L_{n}^{({\alpha},{\beta})}(t,s)|\leq\frac{c\,n}{\sqrt{w_{{\alpha},{\beta}}(n;t)}\sqrt{w_{{\alpha},{\beta}}(n;s)}}\left(1+n{\mathsf{d}}_{[-1,1]}(t,s)\right)^{-{\kappa}},$

This verifies Assertion 1 of the Definition 2.3. The Assertions 2 and 3 can be
found in [37, 24], respectively. The proof of the estimate relies on the
estimate for the special case

$L_{n}^{({\alpha},{\beta})}(t)=L_{n}^{({\alpha},{\beta})}(t,1),\qquad-1\leq
t\leq 1.$

If $s=1$ and $t=\cos{\theta}$, then
${\mathsf{d}}_{[-1,1]}(t,1)={\theta}\sim\sin\theta/2\sim\sqrt{1-t}$ and we
obtain [5],

(2.17) $\left|L_{n}^{({\alpha},{\beta})}(t)\right|\leq
c\,\frac{n^{2{\alpha}+2}}{\left(1+n\sqrt{1-t}\right)^{\kappa}},\quad-1\leq
t\leq 1,$

which is used to establish the estimate in Theorem 2.7. There is also a more
general estimate stated under weaker assumption on the cut-off function [5]
and [12, Theorem 2.6.7], which will be useful in later development.

######  Theorem 2.8.

Let $\ell$ be a positive integer and let $\eta$ be a function that satisfy,
$\eta\in C^{3\ell-1}({\mathbb{R}})$, $\mathrm{supp}\,\eta\subset[0,2]$ and
$\eta^{(j)}(0)=0$ for $j=0,1,2,\ldots,3\ell-2$. Then, for
${\alpha}\geq{\beta}\geq-\frac{1}{2}$, $t\in[-1,1]$ and $n\in{\mathbb{N}}$,

(2.18) $\left|\frac{d^{m}}{dt^{m}}L_{n}^{({\alpha},{\beta})}(t)\right|\leq
c_{\ell,m,{\alpha}}\left\|\eta^{(3\ell-1)}\right\|_{\infty}\frac{n^{2{\alpha}+2m+2}}{(1+n\sqrt{1-t})^{\ell}},\quad
m=0,1,2,\ldots.$

The Jacobi polynomials with equal parameters are the Gegenbauer polynomials
$C_{n}^{\lambda}$, which are of particular interest for our study. These
polynomials are orthogonal with respect to the the weight function

$w_{\lambda}(x)=(1-x^{2})^{{\lambda}-\frac{1}{2}},\quad-1<x<1,\quad{\lambda}>-\tfrac{1}{2}.$

More precisely, the polynomials $C_{n}^{\lambda}$ satisfy the orthogonal
relation

(2.19)
$c_{{\lambda}}\int_{-1}^{1}C_{n}^{{\lambda}}(x)C_{m}^{{\lambda}}(x)w_{\lambda}(x)\mathrm{d}x=h_{n}^{{\lambda}}\delta_{n,m},$

where the normalization constant $c_{\lambda}$ of $w_{\lambda}$ and the norm
square $h_{n}^{\lambda}$ are given by

(2.20)
$c_{{\lambda}}=\frac{\Gamma({\lambda}+1)}{\Gamma(\frac{1}{2})\Gamma({\lambda}+\frac{1}{2})}\qquad\hbox{and}\qquad
h_{n}^{\lambda}=\frac{{\lambda}}{n+{\lambda}}C_{n}^{\lambda}(1)=\frac{{\lambda}}{n+{\lambda}}\frac{(2{\lambda})_{n}}{n!}.$

The polynomial $C_{n}^{\lambda}$ is a constant multiple of the Jacobi
polynomial $P_{n}^{({\lambda}-\frac{1}{2},{\lambda}-\frac{1}{2})}$ and it is
also related to the Jacobi polynomial by a quadratic transform [46, (4.7.1)]

(2.21)
$C_{2n}^{\lambda}(x)=\frac{({\lambda})_{n}}{(\frac{1}{2})_{n}}P_{n}^{({\lambda}-\frac{1}{2},{\lambda}-\frac{1}{2})}(2x^{2}-1).$

####  2.2.2. The unit sphere ${\mathbb{S}^{d-1}}$

With the unit weight function, or the Lebesgue measure $\mathrm{d}\sigma$, the
space $({\mathbb{S}^{d-1}},\mathrm{d}\sigma,\mathrm{d}_{\SS})$ is a
localizable homogeneous space, where the metric
$\mathrm{d}_{\SS}=\mathrm{d}_{{\mathbb{S}^{d-1}}}$ is the geodesic distance
defined in (1.2). The orthogonal polynomials are spherical harmonics. Let
${\mathcal{P}}_{n}^{d}$ denote the space of homogeneous polynomials of degree
$n$ in $d$ variables. A spherical harmonics $Y$ of degree $n$ is an element of
${\mathcal{P}}_{n}^{d}$ that satisfies $\Delta Y=0$, where $\Delta$ is the
Laplace operator of ${\mathbb{R}}^{d}$. If $Y\in{\mathcal{P}}_{n}^{d}$, then
$Y(x)=\|x\|^{n}Y(x^{\prime})$, $x^{\prime}=x/\|x\|\in{\mathbb{S}^{d-1}}$, so
that $Y$ is determined by its restriction on the unit sphere.

Let ${\mathcal{H}}_{n}({\mathbb{S}^{d-1}})$ denote the space of spherical
harmonics of degree $n$. Its dimension
$\dim{\mathcal{H}}_{n}({\mathbb{S}^{d-1}})$ is given by (2.5). Spherical
harmonics of different degrees are orthogonal on the sphere. For
$n\in{\mathbb{N}}_{0}$ let
$\\{Y_{\ell}^{n}:1\leq\ell\leq\dim{\mathcal{H}}_{n}({\mathbb{S}^{d-1}})\\}$ be
an orthonormal basis of ${\mathcal{H}}_{n}({\mathbb{S}^{d-1}})$; then

$\frac{1}{{\omega}_{d}}\int_{\mathbb{S}^{d-1}}Y_{\ell}^{n}(\xi)Y_{\ell^{\prime}}^{m}(\xi)\mathrm{d}\sigma(\xi)=\delta_{\ell,\ell^{\prime}}\delta_{m,n},$

where ${\omega}_{d}$ denotes the surface area
${\omega}_{d}=2\pi^{\frac{d}{2}}/\Gamma(\frac{d}{2})$ of ${\mathbb{S}^{d-1}}$.
Let ${\mathsf{P}}_{n}(\cdot,\cdot)$ denote the reproducing kernel of the space
${\mathcal{H}}_{n}({\mathbb{S}^{d-1}})$. Among many properties of the
spherical harmonics (see, for example, [12, 45]), one characteristic property
is a closed-formula for this kernel [12, (1.2.3) and (1.2.7)],

(2.22)
${\mathsf{P}}_{n}(x,y)=\sum_{\ell=1}^{\dim{\mathcal{H}}_{n}({\mathbb{S}^{d-1}})}Y_{\ell}^{n}(x)Y_{\ell}^{n}(y)=Z_{n}^{\frac{d-2}{2}}({\langle}x,y{\rangle}),\quad
x,y\in{\mathbb{S}^{d-1}},$

where $Z_{n}^{\lambda}$ is defined in terms of the Gegenbauer polynomial by

(2.23)
$Z_{n}^{\lambda}(t)=\frac{n+{\lambda}}{{\lambda}}C_{n}^{\lambda}(t),\qquad{\lambda}=\frac{d-2}{2}.$

Because of the second equal sign, the identity (2.22) is often called the
addition formula of spherical harmonics. The function in its right-hand side
is often called zonal harmonic, since it depends only on
${\langle}x,y{\rangle}$. This formula allows us to use (2.17) to derive the
estimate (1.1), which is Assertion 1, for the highly localized kernel

${\mathsf{L}}_{n}(x,y)=\sum_{k=0}^{\infty}\widehat{a}\left(\frac{k}{n}\right){\mathsf{P}}_{k}(x,y),\qquad
x,y\in{\mathbb{S}^{d-1}}.$

The other two assertions also hold; see, for example, [12, Thm. 6.5] and [12,
Cor. 2.6.6].

Another characteristic property of the spherical harmonics is that they are
eigenfunctions of the Laplace-Beltrami operator $\Delta_{0}$ on the sphere,
which is the restriction of the Laplace operator $\Delta$ on the unit sphere;
see, for example, [12, Section 1.4] for its explicit expression. More
precisely, it is known [12, (1.4.9)] that

(2.24) $\Delta_{0}Y=-n(n+d-2)Y,\qquad Y\in{\mathcal{H}}_{n}^{d},$

so that the eigenvalues depend only on $n$. This relation, as well as the
operator $\Delta_{0}$, plays an important role in the approximation theory on
the unit sphere.

####  2.2.3. The unit ball ${\mathbb{B}}^{d}$ with classical weight function

The classical weight function on the unit ball ${\mathbb{B}}^{d}$ of
${\mathbb{R}}^{d}$ is defined by

(2.25) $W_{\mu}(x)=(1-\|x\|)^{\mu-\frac{1}{2}},\qquad
x\in{\mathbb{B}}^{d},\quad\mu>-\tfrac{1}{2}.$

Its normalization constant is
$b_{\mu}^{\mathbb{B}}=\Gamma(\mu+\frac{d+1}{2})/(\pi^{\frac{d}{2}}\Gamma(\mu+\frac{1}{2}))$.
For the ball $B(x,r)$ defined via the distance function
${\mathsf{d}}_{\mathbb{B}}$ in (1.4) on the unit ball [38, Lemma 5.1],

$W_{\mu}(B(x,r))\sim r^{d}\left(\sqrt{1-\|x\|^{2}}+r\right)^{2\mu},\qquad
0<r\leq 1,$

so that $W_{\mu}$ is a doubling weight. The space
$({\mathbb{B}}^{d},W_{\mu},{\mathsf{d}}_{\mathbb{B}})$ is a localizable
homogeneous space for $\mu\geq 0$, where
${\mathsf{d}}_{\mathbb{B}}(\cdot,\cdot)$ is the distance defined in (1.4) on
the unit ball.

Let ${\mathcal{V}}_{n}^{d}({\mathbb{B}}^{d},W_{\mu})$ be the space of
orthogonal polynomials of degree $n$ with respect to $W_{\mu}$. An orthogonal
basis of ${\mathcal{V}}_{n}^{d}({\mathbb{B}}^{d},W_{\mu})$ can be given
explicitly in terms of the Jacobi polynomials and spherical harmonics. For
$0\leq m\leq n/2$, let
$\\{Y_{\ell}^{n-2m}:1\leq\ell\leq\dim{\mathcal{H}}_{n-2m}({\mathbb{S}^{d-1}})\\}$
be an orthonormal basis of ${\mathcal{H}}_{n-2m}^{d}$. Define [14, (5.2.4)]

(2.26)
$P_{\ell,m}^{n}(x)=P_{m}^{(\mu-\frac{1}{2},n-2m+\frac{d-2}{2})}\left(2\|x\|^{2}-1\right)Y_{\ell}^{n-2m}(x).$

Then $\\{P_{\ell,m}^{n}:0\leq m\leq
n/2,1\leq\ell\leq\dim{\mathcal{H}}_{n-2m}({\mathbb{S}^{d-1}})\\}$ is an
orthogonal basis of ${\mathcal{V}}_{n}^{d}(W_{\mu})$. Let
${\mathbf{P}}_{n}(W_{\mu};\cdot,\cdot)$ denote the reproducing kernel of the
space ${\mathcal{V}}_{n}({\mathbb{B}}^{d},W_{\mu})$. This kernel satisfies an
analog of the addition formula [51]: for $x,y\in{\mathbb{B}}^{d}$,

(2.27)
$\displaystyle{\mathbf{P}}_{n}(W_{\mu};x,y)=c_{\mu-\frac{1}{2}}\int_{-1}^{1}Z_{n}^{\mu+\frac{d-1}{2}}$
$\displaystyle\Big{(}{\langle}x,y{\rangle}+u\sqrt{1-\|x\|^{2}}\sqrt{1-\|y\|^{2}}\Big{)}$
$\displaystyle\times(1-u^{2})^{\mu-1}\mathrm{d}u,$

where $\mu>0$ and it holds for $\mu=0$ under the limit

(2.28) $\lim_{\mu\to
0+}c_{\mu-\frac{1}{2}}\int_{-1}^{1}f(t)(1-t^{2})^{\mu-1}\mathrm{d}u=\frac{f(1)+f(-1)}{2}.$

This is an analog of the addition formula for the unit ball and it allows us
to use (2.17) to derive the estimate (1.3), which is Assertion 1, for the
highly localized kernel now denoted by

${\mathbf{L}}_{n}(W_{\mu};x,y)=\sum_{k=0}^{\infty}\widehat{a}\left(\frac{k}{n}\right){\mathbf{P}}_{k}(W_{\mu};x,y).$

All three assertions were proved in [38]. For the first one, see also [12,
Thm. 11.5.3].

The orthogonal polynomials with respect to $W_{\mu}$ on the unit ball are
eigenfunctions of a second order differential operator [14, (5.2.3)]:

(2.29)
$\left(\Delta-{\langle}x,\nabla{\rangle}^{2}-(2\mu+d-1){\langle}x,\nabla{\rangle}\right)u=-n(n+2\mu+d-1)u$

for all $u\in{\mathcal{V}}_{n}({\mathbb{B}}^{d},W_{\mu})$. The differential
operator in the left-hand side is the analog of the Laplace-Beltrami operator
on the sphere.

###  2.3. An auxiliary maximal function

Some of the polynomial inequalities in the latter sections will be established
for norms defined via a doubling weight. The study of approximation and
polynomial inequality with doubling weights was pioneered in [32] for the
interval $[-1,1]$ and developed subsequently in [7] for the unit sphere
${\mathbb{S}^{d-1}}$. The main tool for the latter is an auxiliary maximal
function. We define its analog on the domain $\Omega$ that admits a
localizable homogeneous space.

######  Definition 2.9.

For ${\beta}>0$, $n\in{\mathbb{N}}$ and $f\in C(\Omega)$, a maximal function
$f_{{\beta},n}^{\ast}$ is defined by

$f_{{\beta},n}^{\ast}(x):=\max_{y\in\Omega}|f(y)|\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{-{\beta}},\quad
x\in\Omega.$

The development below follows closely [12, Sect. 5.2 and 5.3] but with a
somewhat more streamlined proof.

We clearly have $|f(x)|\leq f_{{\beta},n}^{\ast}(x)$ for all $x\in\Omega$. The
maximal function $f_{{\beta},n}^{\ast}$ has an upper bound in the Hardy-
Littlewood maximum function $M_{\mu}$ defined in (2.2). To see this, we first
prove a lemma. For $f\in C(\Omega)$ and $r>0$, we define

${\rm osc}(f)(x,r):=\sup_{x_{1},x_{2}\in B(x,r)}|f(x_{1})-f(x_{2})|,\qquad
x\in\Omega.$

######  Lemma 2.10.

If $f\in\Pi_{n}(\Omega)$ and $0<\delta\leq 1$, then for any ${\beta}>0$,

${\rm osc}(f)\left(x,\tfrac{\delta}{n}\right)\leq c_{\beta}\delta
f_{{\beta},n}^{\ast}(x),\qquad x\in\Omega,$

where the constant $c_{\beta}$ depends only on $d$ and ${\beta}$ when
${\beta}$ is large.

###### Proof.

We use the highly localized kernel $L_{n}$ with an admissible cut-off function
of type (a). For $\delta>0$ and $x,y\in\Omega$, set

$A_{n,\delta}(x,y):=\sup_{z\in
B(x,\frac{\delta}{n})}\left|L_{n}(\varpi;x,y)-L_{n}(\varpi;z,y)\right|.$

Let $L_{n}(\varpi)*f$ be defined as in (1.5). Then $L_{n}(\varpi)*f=f$ for
$f\in\Pi_{n}(\Omega)$. If $x_{1},x_{2}\in B(x,r)$, then by the triangle
inequality

$\left|L_{n}(\varpi;x_{1},y)-L_{n}(\varpi;x_{2},y)\right|\leq
2A_{n,\delta}(x,y).$

Hence, it follows by (1.5) that

$\displaystyle{\rm osc}(f)\left(x,\tfrac{\delta}{n}\right)$ $\displaystyle\leq
2\int_{\Omega}|f(y)|A_{n,\delta}(x,y)\varpi(y)\mathrm{d}{\mathsf{m}}(y)$
$\displaystyle\leq
2f_{{\beta},n}^{\ast}(x)\int_{\Omega}(1+n{\mathsf{d}}(x,y))^{{\beta}}A_{n,\delta}(x,y)\,\mathrm{d}{\mathsf{m}}(y)$
$\displaystyle\leq c_{\beta}\delta f_{{\beta},n}^{\ast}(x),$

where the last step follows from Assertion 1 and Assertion 3. ∎

######  Theorem 2.11.

Let ${\mathsf{w}}$ be a doubling weight. For $f\in\Pi_{n}(\Omega)$,
${\beta}>0$ and setting ${\gamma}={\alpha}({\mathsf{w}})/{\beta}$, then

(2.30) $f_{{\beta},n}^{\ast}(x)\leq
c_{{\beta},L({\mathsf{w}})}\left(M_{\mathsf{w}}\big{(}|f|^{\gamma}\big{)}(x)\right)^{1/{\gamma}},\qquad
x\in\Omega.$

###### Proof.

We apply Lemma 2.10 with $c_{\beta}\delta=\frac{1}{4}$ to obtain, by
$(a+b)^{\gamma}\leq 2^{\gamma}(a^{\gamma}+b^{\gamma})$,

$\displaystyle\frac{|f(y)|^{{\gamma}}}{(1+n{\mathsf{d}}(x,y))^{{\alpha}({\mathsf{w}})}}\leq
2^{{\gamma}}\frac{\min_{z\in
B(y,\frac{\delta}{n})}|f(z)|^{{\gamma}}}{(1+n{\mathsf{d}}(x,y))^{{\alpha}({\mathsf{w}})}}+\frac{1}{2^{\gamma}}\left(f_{{\beta},n}^{\ast}(x)\right)^{\gamma},$

Taking maximum over $x\in\Omega$, the left-hand side becomes
$\big{(}f_{{\beta},n}^{\ast}(x)\big{)}^{\gamma}$ using
${\alpha}({\mathsf{w}})={\beta}{\gamma}$, so that the inequality can be
rearranged to give

(2.31) $\left(f_{{\beta},n}^{\ast}(x)\right)^{\gamma}\leq
c(1+n{\mathsf{d}}(x,y))^{-{\alpha}({\mathsf{w}})}\min_{z\in
B(y,\frac{\delta}{n})}|f(z)|^{\gamma},$

where $c=2^{\gamma}/(1-2^{-{\gamma}})$. We now estimate $\min_{z\in
B(y,\frac{\delta}{n})}|f(z)|^{\gamma}$, following the proof in the case of the
unit sphere [12, Thm. 5.2.2].

Let ${\theta}=\max\\{\frac{\delta}{n},{\mathsf{d}}(x,y)\\}$. Then
$B(y,\frac{\delta}{n})\subset B(x,2{\theta})\subset B(y,3{\theta})$. By the
doubling condition for ${\mathsf{w}}$ and (ii) of Lemma 2.1,

$\displaystyle{\mathsf{w}}\big{(}B(y,\tfrac{\delta}{n})\big{)}\geq\frac{1}{c_{L({\mathsf{w}})}}\left(\frac{3{\theta}n}{\delta}\right)^{-{\alpha}({\mathsf{w}})}{\mathsf{w}}(B(y,3{\theta}))\geq\frac{1}{c_{L({\mathsf{w}})}}\left(\frac{3{\theta}n}{\delta}\right)^{-{\alpha}({\mathsf{w}})}{\mathsf{w}}(B(x,2{\theta})),$

which implies, together with $n{\theta}\leq 1+n{\mathsf{d}}(x,y)$, the
estimate

$\displaystyle\min_{z\in B(y,\frac{\delta}{n})}|f(z)|^{\gamma}$
$\displaystyle\leq\frac{1}{{\mathsf{w}}(B(y,\frac{\delta}{n}))}\int_{B(y,\frac{\delta}{n})}|f(z)|^{{\gamma}}{\mathsf{w}}(z)\,\mathrm{d}{\mathsf{m}}(z)$
$\displaystyle\leq
c_{L({\mathsf{w}})}\left(\frac{3{\theta}n}{\delta}\right)^{{\alpha}({\mathsf{w}})}\frac{1}{{\mathsf{w}}(B(x,2{\theta}))}\int_{B(x,2{\theta})}|f(z)|^{\gamma}{\mathsf{w}}(z)\,\mathrm{d}{\mathsf{m}}(z)$
$\displaystyle\leq
c_{L({\mathsf{w}})}\left(\frac{3}{\delta}\right)^{{\alpha}({\mathsf{w}})}(1+n{\mathsf{d}}(x,y))^{{\alpha}({\mathsf{w}})}M_{{\mathsf{w}}}\left(|f|^{\gamma}\right)(x).$

Together with (2.31), this completes the proof. ∎

Let $\|\cdot\|_{p,{\mathsf{w}}}$ denote the norm of
$L^{p}(\Omega,{\mathsf{w}})$ and we adopt this notation for $0<p<1$, even
though it is no longer a norm. The $L^{p}$ boundedness of
$f_{{\beta},n}^{\ast}$ follows from that of $M_{\mu}(|f|^{\gamma})$ for
${\gamma}p>1$ or ${\beta}>{\alpha}({\mathsf{w}})/p$.

######  Corollary 2.12.

If $0<p\leq\infty$, $f\in\Pi_{n}(\Omega)$ and
${\beta}>{\alpha}({\mathsf{w}})/p$, then

$\|f\|_{p,{\mathsf{w}}}\leq\|f_{{\beta},n}^{\ast}\|_{p,{\mathsf{w}}}\leq
c\|f\|_{p,{\mathsf{w}}},$

where $c$ depends also on $L({\mathsf{w}})$ and ${\beta}$ when ${\beta}$ is
either large or close to ${\alpha}({\mathsf{w}})/p$.

###  2.4. Marcinkiewicz-Zygmund inequality

These inequalities are between the $L^{p}$ norm and a discrete $L^{p}$ norm
defined with respect to a well-distributed set of points. We start with the
definition of well-separated sets of points.

######  Definition 2.13.

Let $\Xi$ be a discrete set in $\Omega$.

  1. (a)

A finite collection of subsets $\\{S_{z}:z\in\Xi\\}$ is called a partition of
$\Omega$ if $S_{z}^{\circ}\cap S_{y}^{\circ}=\emptyset$ when $z\neq y$ and
$\Omega=\bigcup_{z\in\Xi}S_{z}$.

  2. (b)

Let ${\varepsilon}>0$. A discrete subset $\Xi$ of $\Omega$ is called
${\varepsilon}$-separated if ${\mathsf{d}}(x,y)\geq{\varepsilon}$ for every
two distinct points $x,y\in\Xi$.

  3. (c)

$\Xi$ is called maximal if there is a constant $c_{d}>1$ such that

(2.32) $1\leq\sum_{z\in\Xi}\chi_{B(z,{\varepsilon})}(x)\leq
c_{d},\qquad\forall x\in\Omega,$

where $\chi_{E}$ denotes the characteristic function of the set $E$.

Evidently (2.32) implies $\Omega=\bigcup_{z\in\Xi}B(z,{\varepsilon})$.
Moreover, it implies that the cardinality $\\#\Xi$ of $\Xi$ satisfies

$c_{1}\frac{{\mathsf{m}}(\Omega)}{\max_{z\in\Xi}{\mathsf{m}}(B(z,{\varepsilon}))}\leq\\#\Xi\leq
c_{2}\frac{{\mathsf{m}}(\Omega)}{\min_{z\in\Xi}{\mathsf{m}}(B(z,{\varepsilon}))}.$

For the unit sphere, all balls of the same radius have the same volume; that
is, $\sigma(B(x,r))=\sigma(B(0,1))r^{d-1}$ for all $x\in{\mathbb{S}^{d-1}}$,
where $\sigma$ is the surface measure on ${\mathbb{S}^{d-1}}$. This allows us
to deduce (2.32) from $\Omega=\bigcup_{x\in\Xi}B(x,{\varepsilon})$ and the
cardinality of the maximal ${\varepsilon}$-separated set $\Xi_{\SS}$ on the
sphere ${\mathbb{S}^{d-1}}$ satisfies
$c_{d}^{\prime}{\varepsilon}^{-d+1}\leq\\#\Xi_{\SS}\leq
c_{d}{\varepsilon}^{-d+1}$; see, for example, [12, p. 114].

As we shall show in the next section that the volume of the ball $B(x,r)$ on
the conic surface depends on the position of $x$. In particular, we no longer
have $\varpi(B(x,{\varepsilon}))\sim\varpi(B(y,{\varepsilon}))$ for all $x,y$
in the domain in general.

######  Theorem 2.14.

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. Let
${\varepsilon}=\frac{\delta}{n}$ for $n=1,2,\ldots$ and $\delta\in(0,1)$. If
$\Xi$ is a maximal ${\varepsilon}$-separated subset of $\Omega$, then for all
$f\in\Pi_{n}(\Omega)$ and $0<p<\infty$,

(2.33) $\bigg{(}\sum_{y\in\Xi}\left|{\rm
osc}(f)\left(y,\tfrac{\delta}{n}\right)\right|^{p}{\mathsf{w}}\\!\left(B(y,\tfrac{\delta}{n})\right)\bigg{)}^{\frac{1}{p}}\leq
c_{p}\delta\|f\|_{p,{\mathsf{w}}},$

where $c_{p}$ depends on $p$, when $p$ is close to $0$, and on $d$ and
$L({\mathsf{w}})$.

###### Proof.

If $y\in B(x,\frac{\delta}{n})$, then
$f_{2{\alpha}({\mathsf{w}})/p,n}^{\ast}(y)\sim
f_{2{\alpha}({\mathsf{w}})/p,n}^{\ast}(x)$. Hence, it follows from Lemma 2.10
that

$\displaystyle\sum_{y\in\Xi}\left|{\rm
osc}(f)\left(y,\tfrac{\delta}{n}\right)\right|^{p}{\mathsf{w}}\\!\left(B(y,\tfrac{\delta}{n})\right)$
$\displaystyle\leq
c(c_{p}\delta)^{p}\sum_{y\in\Xi}\int_{B(y,\frac{\delta}{n})}\left(f_{2{\alpha}({\mathsf{w}})/p,n}^{\ast}(x)\right)^{p}{\mathsf{w}}(x)\,\mathrm{d}{\mathsf{m}}(x)$
$\displaystyle\leq(c_{p}\delta)^{p}\int_{\Omega}\left(f_{2{\alpha}({\mathsf{w}})/p,n}^{\ast}(x)\right)^{p}{\mathsf{w}}(x)\,\mathrm{d}{\mathsf{m}}(x).$

The last integral is bounded by $\|f\|_{p,{\mathsf{w}}}^{p}$ by Corollary
2.12. ∎

We now prove the Marcinkiewicz-Zygmund inequality for polynomials.

######  Theorem 2.15.

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. Let $\Xi$ be a maximal
$\frac{\delta}{n}$-separated subset of $\Omega$ and let $\delta>0$ and
$\delta\leq 1$.

  1. $(i)$

For all $0<p<\infty$ and $f\in\Pi_{m}(\Omega)$ with $n\leq m\leq cn$,

(2.34) $\Bigg{(}\sum_{z\in\Xi}\Big{(}\max_{x\in
B(z,\frac{\delta}{n})}|f(x)|^{p}\Big{)}{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{n})\right)\Bigg{)}^{\frac{1}{p}}\leq
c_{{\mathsf{w}},p}\|f\|_{p,{\mathsf{w}}},$

where $c_{{\mathsf{w}},p}$ depends on $L({\mathsf{w}})$ and on $p$ when $p$ is
close to $0$.

  2. $(ii)$

For $0<r<1$, there is a $\delta_{r}>0$ such that for $\delta\leq\delta_{r}$,
$r\leq p<\infty$ and $f\in\Pi_{n}(\Omega)$,

(2.35) $\displaystyle\|f\|_{p,{\mathsf{w}}}\leq
c_{{\mathsf{w}},r}\bigg{(}\sum_{z\in\Xi}\bigg{(}\min_{x\in
B\bigl{(}z,\tfrac{\delta}{n}\bigr{)}}|f(x)|^{p}\bigg{)}{\mathsf{w}}\bigl{(}B(z,\tfrac{\delta}{n})\bigr{)}\Bigg{)}^{\frac{1}{p}},$

where $c_{{\mathsf{w}},r}$ depends only on $L({\mathsf{w}})$ and on $r$ when
$r$ is close to $0$. Furthermore,

(2.36) $\|f\|_{\infty}\leq c\max_{z\in\Xi}|f(z)|.$

###### Proof.

For every $y\in\Xi$, choose $z_{y}\in B(y,\frac{\delta}{n})$ such that
$|f(z_{y})|=\sup_{x\in B(y,\frac{\delta}{n})}|f(x)|$. For
$f\in\Pi_{m}(\Omega)$, we have $|f(z_{y})|\leq f_{{\beta},m}^{\ast}(z_{y})\leq
cf_{{\beta},n}^{\ast}(y)$. Hence,

$\displaystyle\sum_{z\in\Xi}\Big{(}\max_{x\in
B(z,\frac{\delta}{n})}|f(x)|^{p}\Big{)}{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{n})\right)$
$\displaystyle\leq
c\sum_{z\in\Xi}\big{(}f_{{\beta},n}^{\ast}(y)\big{)}^{p}{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{n})\right)$
$\displaystyle\leq
c\int_{\Omega}\big{(}f_{{\beta},n}^{\ast}(x)\big{)}^{p}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x),$

from which (i) follows from the bound in Corollary 2.12. We now prove (ii).
Since $\Xi$ is maximal $\frac{\delta}{n}$-separated,

$\displaystyle\|f\|_{p,w}^{p}$
$\displaystyle\leq\sum_{z\in\Xi}\int_{B(z,\frac{\delta}{n})}|f(x)|^{p}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)$
$\displaystyle\leq
2^{p}\sum_{z\in\Xi}|\mathrm{osc}(f)\left(z,\tfrac{\delta}{n}\right)|^{p}{\mathsf{w}}\left(B\big{(}z,\tfrac{\delta}{n}\big{)}\right)+2^{p}\sum_{z\in\Xi}\bigg{(}\min_{x\in
B\bigl{(}z,\tfrac{\delta}{n}\bigr{)}}|f(x)|^{p}\bigg{)}{\mathsf{w}}\left(B\big{(}z,\tfrac{\delta}{n}\big{)}\right).$

The first term in the right-hand side is bounded by
$(2c_{r}\delta)^{p}\|f\|_{p,w}^{p}$ by Lemma 2.10. Hence, choosing
$\delta_{r}=1/(4c_{r})$ so that $(2c_{r}\delta)^{p}\leq 2^{-p}$ for
$\delta\leq\delta_{r}$, we conclude that

$(1-2^{-p})\|f\|_{p,{\mathsf{w}}}^{p}\leq
2^{p}\sum_{z\in\Xi}\bigg{(}\min_{x\in
B\bigl{(}z,\tfrac{\delta}{n}\big{)}}|f(x)|^{p}\bigg{)}{\mathsf{w}}\big{(}B(z,\tfrac{\delta}{n})\big{)}.$

Taking the power of $1/p$ proves (2.35). In particular, since the constant in
(2.35) is independent of $p$, it readily implies (2.36). This completes the
proof. ∎

For the unit sphere without weight and the unit ball with the classical
weight, the MZ inequality was established in [5, 33, 34, 38]. For the unit
sphere with the doubling weight, it was established in [7].

###  2.5. Christoffel function for doubling weight

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. The Christoffel function
${\lambda}_{n}({\mathsf{w}};\cdot)$ is defined by

(2.37)
$\displaystyle{\lambda}_{n}({\mathsf{w}};x):=\inf_{\begin{subarray}{c}g(x)=1\\\
g\in\Pi_{n}(\Omega)\end{subarray}}\int_{\Omega}|g(x)|^{2}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x).$

It is known that the Christoffel function is closely related to the kernel
$K_{n}({\mathsf{w}};\cdot,\cdot)$ of the partial sum operator

$K_{n}({\mathsf{w}};x,y)=\sum_{k=0}^{n}P_{k}({\mathsf{w}};x,y),\qquad
x,y\in\Omega.$

More precisely, it is known [14, Theorem 3.6.6.] that, for $n=0,1,2,\ldots$,

(2.38)
$\displaystyle{\lambda}_{n}({\mathsf{w}};x)=\frac{1}{K_{n}({\mathsf{w}};x,x)},\qquad
x\in\Omega.$

The Christoffel function encodes essential information on weighted
approximation by polynomials on the domain $\Omega$. It will also be useful in
the study of cubature rules in the next subsection, for which we need an upper
bound of ${\lambda}_{n}({\mathsf{w}};x)$. We establish this bound for all
doubling weights on the localizable homogeneous space under the assumption
that there exist fast decaying polynomials on $\Omega$. More precisely, we
make the following assertion.

Assertion 4. Let $\Omega$ be compact. For each $x\in\Omega$, there is a
nonnegative polynomial $T_{x}$ of degree at most $n$ that satisfies

  1. (1)

$T_{x}(x)=1$, $T_{x}(y)\geq\delta>0$ for $y\in B(x,\frac{1}{n})$ for some
$\delta$ independent of $n$, and, for each ${\gamma}>1$,

$0\leq T_{x}(y)\leq c_{\gamma}(1+n{\mathsf{d}}(x,y))^{-{\gamma}},\qquad
y\in\Omega;$

  2. (2)

there is a polynomial $q_{n}$ such that $q_{n}(x)T_{x}(y)$ is a polynomial of
degree at most $rn$, for some positive integer $r$, in $x$-variable and
$c_{1}\leq q_{n}(x)\leq c_{2}$ for $x\in\Omega$ for some positive numbers
$c_{1}$ and $c_{2}$.

The fast decaying polynomials of one variable are studied in [30], see also
[43], which leads to fast decaying polynomials on the unit sphere by the
addition formula (2.22). For the unit ball, such polynomials are constructed
in [38]. We use these polynomials to derive an upper bound for the Christoffel
function. First, we need a lemma.

######  Lemma 2.16.

Assume Assertion 4. Let ${\alpha}>0$ be a positive number and let $x\in\Omega$
be fixed. For a doubling weight ${\mathsf{w}}$ on $\Omega$, there is a
polynomial $Q_{x}$ of degree $n$ such that, for all $y\in\Omega$,

(2.39)
$c_{1}(1+n{\mathsf{d}}(x,y))^{{\alpha}}{\mathsf{w}}(B(y,\tfrac{1}{n}))\leq
Q_{x}(y)\leq
c_{2}(1+n{\mathsf{d}}(x,y))^{{\alpha}}{\mathsf{w}}(B(y,\tfrac{1}{n})),$

where $c_{1}$ and $c_{2}$ are positive constant independent of $n$, $x$ and
$y$.

###### Proof.

Let $T_{x}$ and $q$ be polynomials in Assertion 4 of degree at most
$\lfloor\frac{n}{2}\rfloor$. For fixed $x$, we define $Q_{x}$ by

$Q_{x}(y)=q_{n}(y)\int_{\Omega}T_{y}(v)(1+n{\mathsf{d}}(x,v))^{{\alpha}}{\mathsf{w}}(v)\mathrm{d}{\mathsf{m}}(v),\qquad
y\in\Omega.$

Since $q_{n}(y)T_{y}(v)$ is a polynomial of degree at most $n$ in $y$, it is
easy to see that $Q_{x}$ is a polynomial of degree at most $n$. Using
Assertion 4 with ${\gamma}>{\alpha}+1$ and using the triangle inequality
$1+n{\mathsf{d}}(x,v)\leq(1+n{\mathsf{d}}(x,y))(1+n{\mathsf{d}}(y,v))$, we
obtain

$\displaystyle Q_{x}(y)\,$ $\displaystyle\leq
c(1+n{\mathsf{d}}(x,y))^{\alpha}\int_{\Omega}\frac{1}{(1+n{\mathsf{d}}(y,v))^{{\gamma}-{\alpha}}}{\mathsf{w}}(v)\mathrm{d}{\mathsf{m}}(v)$
$\displaystyle\leq
c(1+n{\mathsf{d}}(x,y))^{\alpha}{\mathsf{w}}\big{(}B(y,\tfrac{1}{n})\big{)},$

since, by the doubling property of ${\mathsf{m}}$ and choosing ${\gamma}$
sufficiently large,

(2.40)
$\displaystyle\int_{\Omega}\frac{{\mathsf{w}}(v)\mathrm{d}{\mathsf{m}}(v)}{(1+n{\mathsf{d}}(y,v))^{{\gamma}-{\alpha}}}\,$
$\displaystyle\leq\sum_{k=1}^{\infty}\int_{B(y,\frac{2^{k}}{n})\setminus
B(y,\frac{2^{k-1}}{n})}\frac{{\mathsf{w}}(v)\mathrm{d}{\mathsf{m}}(v)}{(1+n{\mathsf{d}}(y,v))^{{\gamma}-{\alpha}}}$
$\displaystyle\leq
c\sum_{k=1}^{\infty}\frac{1}{(1+2^{k-1})^{{\gamma}-{\alpha}}}{\mathsf{w}}\\!\left(B\big{(}y,\tfrac{2^{k}}{n}\big{)}\right)$
$\displaystyle\leq
c\,{\mathsf{w}}\\!\left(B\big{(}y,\tfrac{1}{n}\big{)}\right)\sum_{k=1}^{\infty}\frac{L({\mathsf{w}})^{k-1}}{(1+2^{k-1})^{{\gamma}-{\alpha}}}$
$\displaystyle\leq
c\,{\mathsf{w}}\\!\left(B\big{(}y,\tfrac{1}{n}\big{)}\right).$

This establishes the upper bound of $Q_{x}$. In the other direction, we use
$q_{n}(y)\geq c_{1}$, $T_{y}(v)\geq\delta>0$ for $v\in B(y,\frac{1}{n})$ and
$T_{y}(v)\geq 0$ to obtain

$\displaystyle Q_{x}(y)\,$
$\displaystyle\geq(1+n{\mathsf{d}}(x,y))^{\alpha}\int_{\Omega}T_{y}(v)(1+n{\mathsf{d}}(y,v))^{-{\alpha}}{\mathsf{w}}(v)\mathrm{d}{\mathsf{m}}(v)$
$\displaystyle\geq
c\delta(1+n{\mathsf{d}}(x,y))^{\alpha}\int_{B(y,\frac{1}{n})}(1+n{\mathsf{d}}(y,v))^{-{\alpha}}{\mathsf{w}}(v)\mathrm{d}{\mathsf{m}}(v)$
$\displaystyle\geq
c\delta(1+n{\mathsf{d}}(x,y))^{\alpha}{\mathsf{w}}\\!\left(B\big{(}y,\tfrac{1}{n}\big{)}\right),$

since the last integral is trivially bounded below by
$c{\mathsf{w}}\\!\left(B\big{(}y,\tfrac{1}{n}\big{)}\right)$. This completes
the proof. ∎

######  Proposition 2.17.

Assume Assertion 4 holds. Let ${\mathsf{w}}$ be a doubling weight on the
domain $\Omega$. Then

${\lambda}_{n}({\mathsf{w}};x)\leq
c\,{\mathsf{w}}\\!\left(B(x,\tfrac{1}{n})\right),\qquad x\in\Omega.$

###### Proof.

Let $m=\lfloor\frac{n}{3}\rfloor$. Let $T_{x}(y)$ be the polynomial of degree
$m$ in Assertion 4. Let $\Xi$ be a maximal $\frac{\delta}{n}$-separated set in
$\Omega$ with $\delta\leq\delta_{0}$ as in Theorem 2.15. By the inequality
(2.35) with $p=1$ and (ii) of Lemma 2.1 for both ${\mathsf{m}}$ and
${\mathsf{w}}$, we obtain

$\displaystyle\int_{\Omega}[T_{x}(y)]^{2}{\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y)\,$
$\displaystyle\leq
c\sum_{z\in\Xi}[T_{x}(z)]^{2}{\mathsf{w}}\big{(}B(z,\tfrac{\delta}{n})\big{)}$
$\displaystyle\leq
c\frac{{\mathsf{w}}\big{(}B(x,\tfrac{\delta}{n})\big{)}}{[{\mathsf{m}}\big{(}B(x,\tfrac{\delta}{n})\big{)}]^{2}}\sum_{z\in\Xi}[T_{x}(z)]^{2}(1+n{\mathsf{d}}(x,z))^{{\alpha}}[{\mathsf{m}}\big{(}B(z,\tfrac{\delta}{n})\big{)}]^{2},$

where ${\alpha}={\alpha}({\mathsf{w}})+2{\alpha}({\mathsf{m}})$. Let $Q_{x}$
be the polynomial of degree $m$ that satisfies (2.39) with ${\mathsf{m}}$ in
place of ${\mathsf{w}}$. Using the lower bound of $Q_{x}$ in (2.39) and
applying the inequality (2.34) with $p=1$ on the polynomial
$[T_{x}(z)]^{2}Q_{y}(z)$ of degree $n$ with respect to the measure
$\mathrm{d}{\mathsf{m}}$, we obtain

$\displaystyle\int_{\Omega}[T_{x}(y)]^{2}{\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y)$
$\displaystyle\leq
c\frac{{\mathsf{w}}\big{(}B(x,\tfrac{\delta}{n})\big{)}}{[{\mathsf{m}}\big{(}B(x,\tfrac{\delta}{n})\big{)}]^{2}}\sum_{z\in\Xi}[T_{x}(z)]^{2}Q_{x}(z){\mathsf{m}}(B(z,\tfrac{\delta}{n}))$
$\displaystyle\leq
c\,\frac{{\mathsf{w}}\big{(}B(x,\tfrac{\delta}{n})\big{)}}{[{\mathsf{m}}\big{(}B(x,\tfrac{\delta}{n})\big{)}]^{2}}\int_{\Omega}[T_{x}(v)]^{2}Q_{x}(v)\mathrm{d}{\mathsf{m}}(v).$

Using now the upper bound of $Q_{x}$ in (2.39), again with ${\mathsf{m}}$ in
place of ${\mathsf{w}}$, and the upper bound of $T_{x}(v)$ in Assertion 4, and
(ii) of Lemma 2.1 for the measure $\mathrm{d}{\mathsf{m}}$, we further deduce
that the last integral is bounded by

$\displaystyle\int_{\Omega}[T_{x}(v)]^{2}Q_{x}(v)\mathrm{d}{\mathsf{m}}(v)$
$\displaystyle\leq
c\int_{\Omega}\frac{{\mathsf{m}}(B(v,\tfrac{\delta}{n}))}{(1+n{\mathsf{d}}(x,v))^{2{\gamma}-{\alpha}}}\mathrm{d}{\mathsf{m}}(v)$
$\displaystyle\leq
c{\mathsf{m}}(B(x,\tfrac{\delta}{n}))\int_{\Omega}\frac{1}{(1+n{\mathsf{d}}(x,z))^{2{\gamma}-{\alpha}-{\alpha}({\mathsf{m}})}}\mathrm{d}{\mathsf{m}}(v)$
$\displaystyle\leq
c\big{[}{\mathsf{m}}\big{(}B(x,\tfrac{\delta}{n})\big{)}\big{]}^{2},$

where the last step follows from (2.40). Together, from the last two displayed
inequalities follows that

$\int_{\Omega}[T_{x}(y)]^{2}{\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y)\leq
c{\mathsf{w}}\big{(}B(x,\tfrac{\delta}{n})\big{)}.$

Since $T_{x}(x)=1$, this completes the proof by (2.37). ∎

######  Proposition 2.18.

If ${\mathsf{w}}$ is a weight function for which Assertion 1 holds, then

${\lambda}_{n}({\mathsf{w}};x)\geq
c\,{\mathsf{w}}\\!\left(B(x,\tfrac{1}{n})\right),\qquad x\in\Omega.$

###### Proof.

Let $L_{n}({\mathsf{w}};\cdot,\cdot)$ be the localized kernel defined with a
cut-off function of type (a). It follows immediately that
$K_{n}({\mathsf{w}};x,x)\leq L_{n}({\mathsf{w}};x,x)$. Hence, if
${\mathsf{w}}$ admits Assertion 1, then

$K_{n}({\mathsf{w}};x,x)\leq
L_{n}({\mathsf{w}};x,x)\leq\frac{c}{{\mathsf{w}}(B(x,\tfrac{1}{n}))},$

which is equivalent to the lower bound ${\lambda}_{n}({\mathsf{w}};x)$ by
(2.38). ∎

Together, the last two propositions give both upper and lower bounds for the
Christoffel functions. Such bounds are known for regular domains, see for
example [14, 23, 50] and the references therein, and it has been established
recently for fairly general convex domains [22, 40] and planar domains with
piecewise boundary [41]. None of the previous results, however, apply to conic
domains that will come out as special cases of the above theorem.

###  2.6. Positive cubature rules

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. A cubature rule of degree
$n$ for ${\mathsf{w}}$ is a finite linear combination of point evaluations
that satisfies

$\int_{\Omega}f(x){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\sum_{k=1}^{N}{\lambda}_{k}f(x_{k}),\qquad\forall
f\in\Pi_{n}(\Omega),$

where ${\lambda}_{k}\in{\mathbb{R}}$ and $x_{k}\in\Omega$. When
${\lambda}_{k}>0$ for all $k$, the cubature rule is called positive. Given a
maximal ${\varepsilon}$-separated subset $\Xi$ on $\Omega$, we establish the
existence of a positive cubature rule based on the points in $\Xi$, which
plays an essential role for discretizing our near best approximation
$L_{n}*f$.

The proof will use the Farkas lemma (cf. [12, Lemma 6.3.2]) stated below.

######  Lemma 2.19.

Let $V$ be a finite dimensional real Hilbert space with inner product
${\langle}\cdot,\cdot{\rangle}$. Then for any points $a^{1},a^{2},\dots,a^{m}$
and $\zeta\in V$, exactly of the following two systems has a solution:

  1. (1)

$\displaystyle{\sum_{j=1}^{m}\mu_{j}a^{j}=\zeta},\qquad
0\leq\mu_{1},\mu_{2},\ldots,\mu_{m}\in{\mathbb{R}}$.

  2. (2)

${\langle}a^{j},x{\rangle}\geq 0$, $j=1,2,\ldots,m$ and
${\langle}\zeta,x{\rangle}<0$ for some $x\in V$.

We use the lemma to prove the existence of the positive cubature rules. For
the unit sphere and the unit ball, this was established in [33] and later in
[5, 34, 38]. We follow the proof in [12, Theorem 6.3.3] for the doubling
weight on the unit sphere.

######  Theorem 2.20.

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. Let $\Xi$ be a maximum
$\frac{\delta}{n}$-separated subset of $\Omega$. There is a $\delta_{0}>0$
such that for $0<\delta<\delta_{0}$ there exist positive numbers
${\lambda}_{z}$, $z\in\Xi$, so that

(2.41)
$\int_{\Omega}f(x){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\sum_{z\in\Xi}{\lambda}_{z}f(z),\qquad\forall
f\in\Pi_{n}(\Omega).$

Moreover, ${\lambda}_{z}\geq
c_{1}{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{n})\right)$ and, if Assertion 4
holds, then
${\lambda}_{z}\sim{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{n})\right)$.

###### Proof.

Let $\varpi$ be the weight function that admits highly localized kernels on
$\Omega$. We use Lemma 2.19 with the inner product defined by
${\langle}\cdot,\cdot{\rangle}_{\varpi}$. Recall that
$K_{n}(\varpi;\cdot,\cdot)$ denotes the reproducing kernel of
$\Pi_{n}(\Omega)$. Let $\\{z_{k}:1\leq k\leq N\\}$ be an enumeration of $\Xi$.
We define the functions $a^{j}$ and $\zeta$ by

$\displaystyle a^{j}(x)\,$
$\displaystyle=K_{n}\big{(}\varpi;x,z_{j}\big{)},\qquad 1\leq j\leq N,$
$\displaystyle\zeta(x)$
$\displaystyle=\int_{\Omega}K_{n}\big{(}\varpi;x,y\big{)}{\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y)-\frac{1}{2c_{d}}\sum_{j=1}^{N}K_{n}\big{(}\varpi;x,z_{j}\big{)}{\mathsf{w}}\big{(}B(z_{j},\tfrac{\delta}{n})\big{)},$

where $c_{d}$ is the constant in (2.32). For every $f\in\Pi_{n}(\Omega)$, the
reproducing property of $K_{n}(\varpi;\cdot,\cdot)$ implies that

$\displaystyle{\langle}f,a^{j}{\rangle}_{\varpi}$
$\displaystyle=f(z_{j}),\qquad 1\leq j\leq N,$
$\displaystyle{\langle}f,\zeta{\rangle}_{\varpi}$
$\displaystyle=\int_{\Omega}f(y){\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y)-\frac{1}{2c_{0}}\sum_{j=1}^{N}f(z_{j}){\mathsf{w}}\\!\big{(}B(z_{j}),\tfrac{\delta}{n}\big{)}.$

Assume ${\langle}f,a^{j}{\rangle}_{\varpi}\geq 0$ for all $j$, so that $f$ is
nonnegative on $\Xi$. Then, by (2.32),

$\displaystyle\int_{\Omega}f(x)$
$\displaystyle{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\sum_{j=1}^{N}\int_{B(z_{j},\frac{\delta}{n})}f(x)\frac{{\mathsf{w}}(x)}{\sum_{z\in\Xi}\chi_{B(y,\frac{\delta}{n})}(x)}\mathrm{d}{\mathsf{m}}(x)$
$\displaystyle\geq\frac{1}{c_{d}}\sum_{j=1}^{N}f(z_{j})\int_{B(z_{j},\frac{\delta}{n})}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)-\sum_{j=1}^{N}\int_{B(z_{j},\frac{\delta}{n})}\mathrm{osc}(f)(x,\tfrac{\delta}{n}){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)$
$\displaystyle\geq\frac{1}{c_{d}}\sum_{j=1}^{N}f(z_{j}){\mathsf{w}}\big{(}B(z_{j},\tfrac{\delta}{n})\big{)}-c_{1}\delta\int_{\Omega}|f(x)|{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)$

by (2.33). Hence, using the Marcinkiewicz-Zygmund inequality in (ii) of
Theorem 2.15 with $p=1$, we conclude that

$\displaystyle{\langle}f,\zeta{\rangle}_{\varpi}\geq\left((2c_{d})^{-1}-c_{{\mathsf{w}},1}c_{1}\delta\right)\sum_{j=1}^{N}f(z_{j}){\mathsf{w}}\big{(}B(z_{j},\tfrac{\delta}{n})\big{)}\geq
0$

if we choose $\delta\leq(2c_{d})^{-1}/(c_{{\mathsf{w}},1}c_{1})$.
Consequently, (2) of Farkas lemma does not hold so that (1) must hold, which
shows that there exist $\mu_{j}\geq 0$ such that

$\int_{\Omega}f(x){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)-\frac{1}{2c_{0}}\sum_{j=1}^{N}f(z_{j}){\mathsf{w}}\\!\big{(}B(z_{j},\tfrac{\delta}{n})\big{)}=\sum_{j=1}^{N}\mu_{j}f(z_{j}).$

This establishes the cubature rule (2.41) with
${\lambda}_{z_{j}}=(2c_{0})^{-1}{\mathsf{w}}(B(z_{j},\tfrac{\delta}{n}))+\mu_{j}$.

It follows immediately that ${\lambda}_{z}\geq
c{\mathsf{w}}(B(z,\tfrac{\delta}{n}))$ for all $z\in\Xi$. To prove the upper
bound of this inequality, we set $m=\lfloor\frac{n}{2}\rfloor$ and use the
reproducing kernel $K_{m}({\mathsf{w}};\cdot,\cdot)$ of $\Pi_{n}(\Omega)$ with
respect to ${\mathsf{w}}$. Since $K_{m}({\mathsf{w}};z,z)>0$, we define a
polynomial

$q_{z}(x)=\frac{K_{m}({\mathsf{w}};x,z)}{K_{m}({\mathsf{w}};z,z)},\qquad
z\in\Xi.$

For each $z$, the polynomial $q_{z}^{2}$ is a polynomial of degree at most $n$
and $q_{z}(z)=1$. Hence, by (2.41),

${\lambda}_{z}\leq\sum_{y\in\Xi}{\lambda}_{y}[q_{z}(y)]^{2}=\int_{\Omega}[q_{z}(x)]^{2}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\int_{\Omega}\frac{K_{m}({\mathsf{w}};x,z)^{2}}{K_{m}({\mathsf{w}};z,z)^{2}}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x).$

Hence, using the reproducing property of the kernel
$K_{m}({\mathsf{w}};\cdot,\cdot)$,

$\int_{\Omega}K_{m}({\mathsf{w}};x,z)^{2}{\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=K_{m}({\mathsf{w}};z,z),$

we conclude that ${\lambda}_{z}\leq
c[K_{m}({\mathsf{w}};z,z)]^{-1}={\lambda}_{n}({\mathsf{w}};z)$. Hence,
${\lambda}_{z}\leq c\,{\mathsf{w}}(B(z,\tfrac{\delta}{n}))$ by Proposition
2.17. ∎

###  2.7. Tight polynomial frames

The highly localized kernels are powerful tools when they exist. We use them
to study approximation behavior of the integral operator $L_{n}(\varpi)*f$
that has $L_{n}\big{(}\varpi;\cdot,\cdot)$ as its kernel, or more precisely,

(2.42)
$L_{n}(\varpi)*f(x)=\int_{\Omega}f(y)L_{n}\big{(}\varpi;x,y)\varpi(y)\mathrm{d}{\mathsf{m}}(y).$

The fast decaying of the kernel indicates that $L_{n}(\varpi)*f$ provides a
good approximation to the function $f$. Its approximation property will be
studied in the next section. For now, we observe that $L_{n}(\varpi)*f$ is a
polynomial of degree at most $2n$ by the property of the cut-off function.

We now use the operator $L_{n}(\varpi)*f$ and the positive cubature rules
based on the discrete set $\Xi$ to construct tight polynomial frames in
$L^{2}(\Omega,\varpi)$. The construction is fairly standard by now, so is the
proof of the tight frame. See, for example, [12, 34, 37, 38] for the case of
the interval, the unit sphere and the unit ball. We recall the procedure and
will be brief in proof.

Let $(\Omega,\varpi,{\mathsf{d}})$ be a localizable homogeneous space. For the
highly localized kernels $L_{n}(\varpi;x,y)$, we assume its cut-off function
$\widehat{a}$ is of type (b) and satisfies

(2.43) $\displaystyle\begin{split}\widehat{a}(t)\geq\rho>0,\qquad&\mbox{if
$t\in[3/5,5/3]$},\\\
[\widehat{a}(t)]^{2}+[\widehat{a}(2t)]^{2}=1,\qquad&\mbox{if
$t\in[1/2,1]$.}\end{split}$

If $g$ is a nonnegative even function in $C^{\infty}({\mathbb{R}})$, so that
$\mathrm{supp}\,g=[-1,1]$, $g(0)=1$ and $|g(t)|^{2}+|g(t+1)|^{2}=1$ on
$[-1,0]$, then $\widehat{a}(t):=g(\log_{2}t)$ has the desired properties. The
last assumption on $\widehat{a}$ implies

(2.44)
$\sum_{j=0}^{\infty}\left[\widehat{a}\left(\frac{t}{2^{j}}\right)\right]^{2}=1,\quad
t\in[1,\infty).$

Let ${\mathsf{w}}$ be the doubling weight that admits Assertion 4. Then the
cubature rules in Theorem 2.20 are well established for ${\mathsf{w}}$. For
$j=0,1,\ldots,$ let ${\varepsilon}_{j}=\frac{\delta}{2^{j}}$ and let $\Xi_{j}$
be a maximal ${\varepsilon}_{j}$-separated subset in $\Omega$, where $\delta$
is chosen so that the cubature rule in Theorem 2.20 holds; that is, there are
${\lambda}_{z,j}>0$ for $z\in\Xi_{j}$ such that

(2.45)
$\int_{\Omega}f(x){\mathsf{w}}(x)\mathrm{d}{\mathsf{m}}(x)=\sum_{z\in\Xi_{j}}\Xi_{z,j}f(z),\qquad
f\in\Pi_{2^{j}}(\Omega);$

moreover, ${\lambda}_{z,j}\sim{\mathsf{w}}(B(z,2^{-j}))$ since we assume that
Assertion 4 holds for ${\mathsf{w}}$. In terms of the kernel
$L_{n}({\mathsf{w}};\cdot,\cdot)$ defined via the cut-off function
$\widehat{a}$, we define

$F_{0}(x,z):=1,\quad\hbox{and}\quad
F_{j}(x,z):=L_{2^{j-1}}({\mathsf{w}};x,z),\quad j=1,2,3,\ldots.$

Accordingly, we define $F_{j}*f=L_{2^{j-1}}*f$, that is,

$F_{j}*f(x):=\int_{\Omega}f(y)F_{j}(x,y){\mathsf{w}}(y)\mathrm{d}{\mathsf{m}}(y),\qquad
j=0,1,2,\ldots.$

Now, for $z\in\Xi_{j}$ and $j=1,2,\ldots$, we define our frame elements by

$\psi_{z,j}(x):=\sqrt{{\lambda}_{z,j}}F_{j}(x,z).$

Then $\Psi:=\big{\\{}\psi_{z,j}:z\in\Xi_{j},\quad 1\leq j\leq\infty\big{\\}}$
is a frame system.

By the orthogonality of the reproducing kernel, it follows readily that

$\displaystyle
F_{j}*F_{j}*f=\sum_{k=1}^{2^{j}}\left|\widehat{a}\left(\frac{k}{2^{j-1}}\right)\right|^{2}\operatorname{proj}_{k}({\mathsf{w}};f).$

Hence, the following semi-discrete Calderón type decomposition follows from
(2.44),

(2.46)
$f=\sum_{k=0}^{\infty}\operatorname{proj}_{k}({\mathsf{w}};f)=\sum_{j=0}^{\infty}F_{j}*F_{j}*f,\qquad
f\in L^{2}(\Omega,{\mathsf{w}}).$

Furthermore, the system $\Psi$ is a tight frame in
$L^{2}(\Omega,{\mathsf{w}})$ norm.

######  Theorem 2.21.

Assume that $\Omega$ admits a localizable homogenous space. Let ${\mathsf{w}}$
be a doubling weight satisfying Assertion 4. If $f\in
L^{2}(\Omega,{\mathsf{w}})$, then

(2.47) $f=\sum_{j=0}^{\infty}\sum_{z\in\Xi_{j}}\langle
f,\psi_{z,j}\rangle_{\mathsf{w}}\psi_{z,j}\qquad\mbox{in
$L^{2}(\Omega,{\mathsf{w}})$}$

and

(2.48)
$\|f\|_{2,{\mathsf{w}}}=\Big{(}\sum_{j=0}^{\infty}\sum_{z\in\Xi_{j}}|\langle
f,\psi_{z,j}\rangle_{\mathsf{w}}|^{2}\Big{)}^{1/2}.$

###### Proof.

Since $F_{j}(x,\cdot)F_{j}(\cdot,y)$ is a polynomial of degree $2^{j}$, we can
apply the cubature rule (2.45) to discretize $F_{j}*F_{j}$, which gives

$F_{j}*F_{j}(x,y)=\int_{\Omega}F_{j}(x,z)F_{j}(z,y){\mathsf{w}}(z)\mathrm{d}{\mathsf{m}}(z)=\sum_{z\in\Xi_{j}}\psi_{z,j}(x)\psi_{z,j}(y),$

from which the identity (2.47) follows from (2.46) right away. Furthermore,
let $S_{R}f$ be the sum $S_{R}f=\sum_{j=0}^{R}\sum_{z\in\Xi_{j}}\langle
f,\psi_{z,j}\rangle_{\mathsf{w}}\psi_{z,j}$. Then, it follows immediately

${\langle}f,S_{R}f{\rangle}_{\mathsf{w}}=\sum_{j=0}^{R}\sum_{z\in\Xi_{j}}\left|\langle
f,\psi_{z,j}\rangle_{\mathsf{w}}\right|^{2}.$

Taking the limit $R\to\infty$ we obtain the tight frame identity (2.48). ∎

For the weight function $\varpi$ that admits the highly localized kernels, the
frame element $\psi_{z,j}$ has near exponential rate of decay away from its
center with respect to the distance ${\mathsf{d}}(\cdot,\cdot)$ on $\Omega$.

######  Theorem 2.22.

Let $(\Omega,\varpi,{\mathsf{d}})$ be a localizable homogeneous space. There,
there is a constant $c_{\kappa}>0$ depending only on ${\kappa}$, $d$, $\varpi$
and $\widehat{a}$ such that for $z\in\Xi_{j}$, $j=0,1,\dots$,

(2.49) $|\psi_{z,j}(x)|\leq
c_{\kappa}\frac{1}{\sqrt{\varpi(B(z,2^{-j}))}(1+2^{j}{\mathsf{d}}(x,z))^{\kappa}},\quad
x\in\Omega.$

This follows readily from the fast decaying of the kernel and
${\lambda}_{z,j}\sim{\mathsf{w}}(B(z,2^{-j}))$. It shows that the frame $\Psi$
is highly localized, which makes it a powerful tool for various applications,
such as decompositions of functions spaces and computational harmonic
analysis, on more specific domains.

##  3\. Polynomial Approximation on homogeneous spaces

We consider approximation by polynomials in the space $L^{p}(\Omega,\varpi)$
when the weight function $\varpi$ posses two additional properties that are
analogs of two characteristics of spherical harmonics: addition formula for
the reproducing kernels and a differential operator that has orthogonal
polynomials as eigenfunctions. With appropriate weight functions, these
properties are shared by orthogonal polynomials on the unit sphere, on the
unit ball, and on the simplex. The addition formula premises that the
reproducing kernels possess a one-dimensional character, which leads to a
convolution structure that allows us to reduce much of the study of the
Fourier orthogonal series to that of the Fourier-Jacobi series of one
variable. The differential operator that has orthogonal polynomials as
eigenfunctions leads to a $K$-functional. Together they provide us with tools
for characterizing the best approximation by polynomials, which is a central
problem in approximation theory. The goal of this section is to develop this
framework, which extends the results on the unit sphere and the unit ball and
it is applicable on the conic domains.

The two characteristic properties are defined and discussed in the first
subsection. The first one is used to define the convolution structure that
leads to the definition of the modulus of smoothness in the second subsection.
The second one is used to define a $K$-functional in the third subsection,
where the main results on the characterization of the best polynomial
approximation are stated and discussed. The near-best approximation operator,
used to provide a direct estimate, is studied in the fourth subsection with
the help of the differential operator, where some of the results are
established with norms defined via a doubling weight. The Bernstein
inequality, essential for proving the inverse estimate, is proved in the fifth
subsection, together with the Nikolskii inequality. Finally, the proof of the
main results is given in the seventh subsection.

###  3.1. Addition formula and differential operator

Let $\varpi$ be a weight function defined on $\Omega$. For studying
approximation by polynomials, we require two more properties on the orthogonal
structure of $L^{2}(\Omega,\varpi)$. These properties are analogs to the two
characteristics of spherical harmonics on the unit sphere: the Laplace-
Beltrami operator (2.24) and the addition formula (2.22), which serve as the
quintessential examples in the study [53, 54] as well as in this subsection.
These properties are shared by several other domains, including the unit ball,
the simplex, and conic domains. Below we assume these properties and use them
to carry out our subsequent study.

The first property that we assume is an analog of the Laplace-Beltrami
operator. Recall that ${\mathcal{V}}_{n}(\Omega,\varpi)$ denote the space of
orthogonal polynomials of degree $n$ with respect to $\varpi$ on $\Omega$.

######  Definition 3.1.

Let $\varpi$ be a weight function on $\Omega$. We denote by
${\mathfrak{D}}_{\varpi}$ the second order derivation operator that has
orthogonal polynomials with respect to $\varpi$ as eigenfunctions; more
precisely,

(3.1)
${\mathfrak{D}}_{\varpi}Y=-\mu(n)Y,\qquad\forall\,Y\in{\mathcal{V}}_{n}(\Omega,\varpi),$

where $\mu$ is a nonnegative quadratic polynomial.

For the unit sphere ${\mathbb{S}^{d-1}}$, the operator
${\mathfrak{D}}_{\varpi}$ is the Laplace-Beltrami operator given by (2.24).
For the unit ball ${\mathbb{B}}^{d}$, ${\mathfrak{D}}_{\varpi}$ is the
operator given by (2.29). In both cases, the operator is a differential
operator. For the unit sphere and the unit ball with reflection invariant
weight functions, ${\mathfrak{D}}_{\varpi}$ is a differential-difference
operator, called Dunkl Laplacian [13, 14]. For $d=2$ and $\Omega$ is a domain
with non-empty interior, the existence of a second order differential operator
that satisfies (3.1) was characterized in [21]. There are essentially five
cases with positive weight functions on regular domains but only two are
compact domains: disk and triangle. The characterization for $d>2$ remains an
open problem. The operators for the conic domains are the recent additions,
discovered in [53, 54] and will be recalled in later sections.

The second property that we assume is the addition formula, which gives a
closed-form formula for the reproducing kernel $P_{n}(\varpi;\cdot,\cdot)$ of
${\mathcal{V}}_{n}(\Omega,\varpi)$.

######  Definition 3.2.

Let $\varpi$ be a weight function on $\Omega$. The reproducing kernel
$P_{n}(\varpi;\cdot,\cdot)$ is said to satisfy an addition formula if, for
some ${\alpha}\geq{\beta}\geq-\frac{1}{2}$,

$P_{n}(\varpi;x,y)=\int_{[-1,1]^{m}}Z_{n}^{({\alpha},{\beta})}\big{(}\xi(x,y;u)\big{)}\mathrm{d}\tau(u),\qquad
Z_{n}^{({\alpha},{\beta})}(t)=\frac{P_{n}^{({\alpha},{\beta})}(1)P_{n}^{({\alpha},{\beta})}(t)}{h_{n}^{({\alpha},{\beta})}},$

where $m$ is a positive integer; $\xi(x,y;u)$ is a function of
$u\in[-1,1]^{m}$, symmetric in $x$ and $y$, and $\xi(x,y;u)\in[-1,1]$;
moreover, $\mathrm{d}\tau$ is a probability measure on $[-1,1]^{m}$, which can
degenerate to have a finite support.

The classical addition formula is the one for the unit sphere
${\mathbb{S}^{d-1}}$, given in (2.22), which is the degenerate case, with
${\alpha}={\beta}=\frac{d-3}{2}$. The one for the classical weight
$\varpi_{\mu}$ on the unit ball ${\mathbb{B}}^{d}$ is given by (2.27), which
has $m=1$ and ${\alpha}={\beta}=\mu+\frac{d-2}{2}$. Similar formula holds for
reflection invariant weight functions on the unit sphere and on the unit ball
(cf. [14, p. 221 and p. 265]). Furthermore, for the simplex in
${\mathbb{R}}^{d}$ (cf. [14, p. 275]) and the conic domains
${\mathbb{V}}_{0}^{d+1}$ and ${\mathbb{V}}^{d+1}$, the polynomial
$Z_{n}^{({\alpha},{\beta})}$ is given by $Z_{2n}^{\lambda}$ for some
${\lambda}\geq 0$, so that, by the quadratic transform (2.21), it is given by
the Jacobi polynomial with ${\alpha}={\lambda}-\frac{1}{2}$ and
${\beta}=-\frac{1}{2}$.

The addition formula premises a one-dimensional structure for the reproducing
kernel. In all known cases, the estimates of the highly localized kernel
$L_{n}(\varpi;\cdot,\cdot)$ are established with the help of this formula,
since the addition formula implies

(3.2)
$L_{n}(\varpi;x,y)=\int_{[-1,1]^{m}}L_{n}^{({\alpha},{\beta})}\big{(}\xi(x,y;u)\big{)}\mathrm{d}\tau(u),$

where the function $t\mapsto
L_{n}^{({\alpha},{\beta})}(t)=L_{n}^{({\alpha},{\beta})}(t,1)$ is the kernel
defined in (2.16) for the Jacobi polynomials.

In the rest of this section, we assume that the orthogonal polynomials with
respect to $\varpi$ posses both the derivation operator and the addition
formula. We do not, however, require that $\varpi$ admits highly localized
kernels that satisfy Assertions 1–3. In fact, some of our results hold if
$\varpi$ satisfies Assertions 1 and 3, but need not satisfy Assertion 2.

###  3.2. Convolution structure

Making use of the one-dimensional structure premised by the addition formula,
we define a convolution operator on $\Omega$.

######  Definition 3.3.

Assume $\varpi$ admits the addition formula. For $f\in L^{1}(\Omega,\varpi)$
and $g\in L^{1}([-1,1],w_{{\alpha},{\beta}})$, we define the convolution of
$f$ and $g$ by

$(f\ast_{\varpi}g)(x):=\int_{\Omega}f(y)T^{({\alpha},{\beta})}g(x,y)\varpi(y)\mathrm{d}{\mathsf{m}}(y),$

where the operator $g\mapsto T^{({\alpha},{\beta})}g$ is defined by

$\displaystyle
T^{({\alpha},{\beta})}g(x,y):=\int_{[-1,1]^{m}}g\big{(}\xi(x,y;u)\big{)}\mathrm{d}\tau(u).$

The addition formula shows that the projection operator
$\operatorname{proj}_{n}(\varpi;f)$ and the reproducing kernel
$P_{n}(\varpi;\cdot,\cdot)$ satisfy

(3.3)
$\operatorname{proj}_{n}(\varpi;f,x)=f*_{\varpi}Z_{n}^{({\alpha},{\beta})}\quad\hbox{and}\quad
P_{n}(\varpi;x,y)=T^{({\alpha},{\beta})}\left(Z_{n}^{({\alpha},{\beta})}\right).$

The operator $T^{({\alpha},{\beta})}$ is defined in [54] with a more generic
orthogonal polynomial in place of $P_{n}^{({\alpha},{\beta})}$, where the
following boundedness of the operator is established [54, Lemma 6.3].

######  Lemma 3.4.

Let $g\in L^{1}([-1,1],w_{{\alpha},{\beta}})$. Then, for each
$Q_{n}\in{\mathcal{V}}_{n}(\Omega,\varpi)$,

(3.4)
$\int_{\Omega}T^{({\alpha},{\beta})}g(x,y)Q_{n}(y)\varpi(y)\mathrm{d}y=\Lambda_{n}^{({\alpha},{\beta})}(g)Q_{n}(x),$

where

$\Lambda_{n}^{({\alpha},{\beta})}(g)=c_{{\alpha},{\beta}}\int_{-1}^{1}g(t)R_{n}^{({\alpha},{\beta})}(t)w_{{\alpha},{\beta}}(t)\mathrm{d}t\quad\hbox{with}\quad
R_{n}^{({\alpha},{\beta})}(t):=\frac{P_{n}^{({\alpha},{\beta})}(t)}{P_{n}^{({\alpha},{\beta})}(1)}$

Furthermore, for $1\leq p\leq\infty$ and $x\in\Omega$,

(3.5)
$\left\|T^{({\alpha},{\beta})}g(x,\cdot)\right\|_{L^{p}(\Omega,\varpi)}\leq\|g\|_{L^{p}([-1,1],w_{{\alpha},{\beta}})}.$

The convolution operator $f\ast_{\varpi}g$ is also defined in [54] and shown
to satisfy the usual Young’s inequality.

######  Theorem 3.5.

Let $p,q,r\geq 1$ and $p^{-1}=r^{-1}+q^{-1}-1$. For $f\in
L^{q}(\Omega,\varpi)$ and $g\in L^{r}([-1,1];w_{{\alpha},{\beta}})$,

(3.6)
$\|f\ast_{\varpi}g\|_{L^{p}(\Omega,\varpi)}\leq\|f\|_{L^{q}(\Omega,\varpi)}\|g\|_{L^{r}([-1,1];\varpi)}.$

As an application of these results, we consider the Cesàro $(C,\delta)$ means
$S_{n}^{\delta}(\varpi;f)$ of the Fourier orthogonal series with respect to
$\varpi$. For $\delta>0$, the operator $S_{n}^{\delta}(\varpi)$ is defined by

$S_{n}^{\delta}(\varpi;f):=\frac{1}{\binom{n+\delta}{n}}\sum_{k=0}^{n}\binom{n-k+\delta}{n-k}\operatorname{proj}_{k}(\varpi;f).$

It can be written as an integral operator with the kernel
$K_{n}^{\delta}(\varpi;\cdot,\cdot)$ being the $(C,\delta)$ mean of the
reproducing kernel $P_{n}(\varpi;\cdot,\cdot)$. By the addition formula, the
kernel $K_{n}^{\delta}(\varpi;\cdot,\cdot)$ can be written as

$K_{n}^{\delta}(\varpi;x,y)=\int_{[-1,1]^{m}}k_{n}^{({\alpha},{\beta}),\delta}(\xi(x,y;u),1)\mathrm{d}\tau(u),$

where $k_{n}^{({\alpha},{\beta}),\delta}(s,t)$ denotes the kernel of the
$(C,\delta)$ mean of the Fourier-Jacobi series. In particular, using the
positivity of the kernel $k_{n}^{({\alpha},{\beta}),\delta}$ [1] and the
boundedness of the kernel [46, Theorem 9.1.3], we obtain the following
theorem.

######  Theorem 3.6.

Let $\varpi$ be a weight function that admits the addition formula. The Cesàro
$(C,\delta)$ means of the Fourier orthogonal series with respect to $\varpi$
satisfy

  1. (1)

If $\delta\geq{\alpha}+{\beta}+2$, then $S_{n}^{\delta}(\varpi;f)$ is a
nonnegative operator;

  2. (2)

If $\delta>{\alpha}+\frac{1}{2}$, then for $n=0,1,2,\ldots$,

$\left\|S_{n}^{\delta}(\varpi;f)\right\|_{p,\varpi}\leq\|f\|_{p,\varpi},\qquad
1\leq p\leq\infty.$

To further explore the one-dimensional structure that leads to the definition
of the convolution, we define an operator $S_{{\theta},\varpi}f$ as follows.

######  Definition 3.7.

Assume the addition formula holds for the weight function $\varpi$. For
$0\leq{\theta}\leq\pi$, the translation operator $S_{{\theta},\varpi}$ is
defined by

(3.7)
$\operatorname{proj}_{n}(\varpi;S_{{\theta},\varpi}f)=R_{n}^{({\alpha},{\beta})}(\cos{\theta})\operatorname{proj}_{n}(\varpi;f),\quad
n=0,1,2,\ldots.$

As a corollary of Theorem 3.6, a function $f\in L^{1}(\Omega,\varpi)$ is
uniquely determined by its orthogonal projections
$\operatorname{proj}_{n}(\varpi;f)$, $n\geq 0$. Hence, the operator
$S_{{\theta},\varpi}$ is well defined for all $f\in L^{1}(\Omega,\varpi)$. For
the unit sphere with the Lebesgue measure $\mathrm{d}\sigma$, the operator
$S_{\theta}$ is an integral operator given by [12, (2.16)]

$S_{{\theta},\mathrm{d}\sigma}f(x)=\frac{1}{{\omega}_{d-1}(\sin{\theta})^{d-1}}\int_{c(x,{\theta})}f(y)\mathrm{d}\ell(y),$

where $c(x,{\theta})=\\{y:{\langle}x,y{\rangle}=\cos{\theta}\\}$ is the
spherical cap and $\mathrm{d}\ell$ denotes the Lebesgue measure on the cap.
For the unit ball ${\mathbb{B}}^{d}$ with the classical weight $W_{\mu}$, the
operator $S_{{\theta},W_{\mu}}$ can also be written as an explicit integral
operator [12, Theorem 11.2.6].

######  Proposition 3.8.

The operator $S_{{\theta},\varpi}$ satisfies the following properties:

  1. (i)

For $f\in L^{2}(\Omega,\varpi)$ and $g\in L^{1}([-1,1],w_{{\alpha},{\beta}})$,

$(f*_{\varpi}g)(x)=c_{{\alpha},{\beta}}\int_{0}^{\pi}S_{{\theta},\varpi}f(x)g(\cos{\theta})w_{{\alpha},{\beta}}(\cos{\theta})\sin{\theta}\mathrm{d}{\theta}.$

  2. (ii)

$S_{{\theta},\varpi}f$ preserves positivity; that is,
$S_{{\theta},\varpi}f\geq 0$ if $f\geq 0$.

  3. (iii)

For $f\in L^{p}(\varpi;\Omega)$, if $1\leq p\leq\infty$, or $f\in C(\Omega)$
if $p=\infty$,

$\|S_{{\theta},\varpi}f\|_{p,\varpi}\leq\|f\|_{p,\varpi}\quad\hbox{and}\quad\lim_{{\theta}\to
0}\|S_{{\theta},\varpi}f-f\|_{\varpi,p}=0.$

###### Proof.

From the Fubini theorem and (3.4), it is easy to see that

$\displaystyle\operatorname{proj}_{n}(\varpi;f*_{\varpi}g)\,$
$\displaystyle=\Lambda_{n}^{({\alpha},{\beta})}(g)\operatorname{proj}_{n}(\varpi;f)$
$\displaystyle=c_{{\alpha},{\beta}}\int_{0}^{\pi}\operatorname{proj}_{n}(\varpi;S_{{\theta},\varpi}f)g(\cos{\theta})w_{{\alpha},{\beta}}(\cos{\theta})\sin{\theta}\mathrm{d}{\theta},$

from which (i) follows. To prove (ii), we let $g_{n}$ be a non-negative
function such that $g_{n}(\cos{\theta})$ is supported on
$[-\frac{1}{n},\frac{1}{n}]$ and
$\int_{0}^{\pi}g_{n}(\cos{\theta})w_{{\alpha},{\beta}}({\theta})\sin{\theta}\mathrm{d}{\theta}=1$.
Then, using the expression in (i), $f\ast_{\varpi}g_{n}$ converges to
$S_{{\theta},\varpi}f$, which proves (ii). Moreover, by (3.6),
$\|f\ast_{\varpi}g_{n}\|_{p,\varpi}\leq\|f\|_{p,\varpi}$, so that by the Fatou
lemma, $\|S_{{\theta},\varpi}f\|_{p,\varpi}\leq\|f\|_{p,\varpi}$. Finally, if
$f_{n}=S_{n}^{\delta}(\varpi;f)$ with $\delta\geq{\alpha}+{\beta}+2$, then
$f_{n}\geq 0$ and $f_{n}\to f$ in $L^{p}(\Omega,\varpi)$; furthermore, by
(3.7), $S_{{\theta},\varpi}f_{n}$ converges to $f_{n}$ for ${\theta}\to 0$;
consequently, by the triangle inequality,
$\|S_{{\theta},\varpi}f-f\|_{p,\varpi}$ converges to $0$ when ${\theta}\to 0$.
This completes the proof. ∎

The operator $S_{{\theta},\varpi}$ can be used to define a modulus of
smoothness. For $r>0$, we defined the $r$-th difference operator

$\triangle_{{\theta},\varpi}^{r}=\left(I-S_{{\theta},\varpi}\right)^{r/2}=\sum_{n=0}^{\infty}(-1)^{n}\binom{r/2}{n}(S_{{\theta},\varpi})^{n},$

where $I$ denote the identity operator, in the distribution sense, by

(3.8)
$\operatorname{proj}_{n}\left(\varpi;\triangle_{{\theta},\varpi}^{r}f\right)=\left(1-R_{n}^{({\alpha},{\beta})}(\cos\theta)\right)^{r/2}\operatorname{proj}_{n}(\varpi;f),\quad
n=0,1,2,\cdots.$

######  Definition 3.9.

Let $r>0$ and $0<{\theta}<\pi$. For $f\in L^{p}(\Omega,\varpi)$ and $1\leq
p<\infty$ or $f\in C(\Omega)$ and $p=\infty$, the weighted $r$th order modulus
of smoothness is defined by

(3.9) $\omega_{r}(f,t)_{p,\varpi}:=\sup_{0<\theta\leq
t}\left\|\triangle_{{\theta},\varpi}^{r}f\right\|_{p,\varpi},\quad 0<t<\pi.$

For the unit sphere with the Lebesuge measure, the definition of this modulus
of smoothness is classical. For weighed approximation on the unit sphere, the
unit ball, and the simplex, it is given in [52]. The proof of the following
lemma is standard, see [12, Proposition 10.1.2] for example, and will be
omitted.

######  Proposition 3.10.

Let $f\in L^{p}(\Omega,\varpi)$ if $1\leq p<\infty$ and $f\in C(\Omega)$ if
$p=\infty$. Then

  1. (1)

${\omega}_{r}(f,t)_{p,\varpi}\leq 2^{r+2}\|f\|_{p,\varpi}$;

  2. (2)

$\omega_{r}(f,t)_{p,\varpi}\to 0$ if $t\to 0+$;

  3. (3)

$\omega_{r}(f,t)_{p,\varpi}$ is monotone nondecreasing on $(0,\pi)$;

  4. (4)

$\omega_{r}(f+g,t)_{p,\varpi}\leq\omega_{r}(f,t)_{\kappa,p}+\omega_{r}(g,t)_{p,\varpi}$;

  5. (5)

For $0<s<r$,

$\omega_{r}(f,t)_{p,\varpi}\leq 2^{(r-s)+2}\omega_{s}(f,t)_{p,\varpi}.$

###  3.3. Characterization of best approximation

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. For $f\in
L^{p}(\Omega,{\mathsf{w}})$, we denote by $E_{n}(f)_{p,{\mathsf{w}}}$ the best
approximation to $f$ from $\Pi_{n}(\Omega)$ in the norm
$\|\cdot\|_{p,{\mathsf{w}}}$ of $L^{p}(\Omega,{\mathsf{w}})$; that is,

$E_{n}(f)_{p,{\mathsf{w}}}:=\inf_{g\in\Pi_{n}(\Omega)}\|f-g\|_{p,{\mathsf{w}}},\qquad
1\leq p\leq\infty.$

A central problem of approximation theory is to character this quantity by the
smoothness of the functions, usually in terms of a modulus of smoothness or a
$K$-functional, which are often equivalent. In this subsection, we state our
main results on the characterization of the best approximation.

The modulus of smoothness is already defined. We now define the $K$-functional
via the differential operator ${\mathfrak{D}}_{\varpi}$ in (3.1). Since the
operator $-{\mathfrak{D}}_{\varpi}$ has nonnegative eigenvalues, it is a non-
negative operator. A function $f\in L^{p}(\Omega;\varpi)$ belongs to the
Sobolev space $W_{p}^{r}(\Omega;\varpi)$ if there is a function $g\in
L^{p}(\Omega;\varpi)$, which we denote by
$(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}f$, such that

(3.10)
$\operatorname{proj}_{n}\left(\varpi;(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}f\right)=\mu(n)^{\frac{r}{2}}\operatorname{proj}_{n}(\varpi;f),$

where we assume that $f,g\in C(\Omega)$ when $p=\infty$. The fractional
differential operator $(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}f$ is a linear
operator on the space $W_{p}^{r}(\Omega;\varpi)$ defined by (3.1).

Let ${\mathsf{w}}$ be a doubling weight. We denote by
$W_{p}^{r}(\Omega,{\mathsf{w}})$ the Sobolev space that consists of $f\in
L^{p}(\Omega,{\mathsf{w}})$ such that
$(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}f\in L^{p}(\Omega,{\mathsf{w}})$. The
space is well defined as we shall see later in Theorem 3.17. We define our
$K$-functional as follows.

######  Definition 3.11.

Let $r>0$ and $1\leq p\leq\infty$. The $r$-th $K$-functional of $f\in
L^{p}(\Omega,{\mathsf{w}})$ is defined by

$K_{r}(f,t)_{p,{\mathsf{w}}}:=\inf_{g\in
W_{p}^{r}(\Omega,{\mathsf{w}})}\left\\{\|f-g\|_{p,{\mathsf{w}}}+t^{r}\left\|(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}g\right\|_{p,{\mathsf{w}}}\right\\}.$

We first state of our characterization theorem in terms of the $K$-functional,
which contains two parts, the first part is the direct estimate or the Jackson
inequality, and the second part is the inverse estimate. For the first part,
we need another assertion to ensure that a Bernstein inequality holds. This is
Assertion 5 and it will be stated in Subsection 3.5.

######  Theorem 3.12.

Let $\varpi$ be a weight function that admits Assertions 1, 3 and 5. Let $f\in
L^{p}(\Omega,\varpi)$ if $1\leq p<\infty$ and $f\in C(\Omega)$ if $p=\infty$.
Then for $r>0$ and $n=1,2,\ldots$,

  1. (i)

the direct estimate

$E_{n}(f)_{p,\varpi}\leq cK_{r}(f;n^{-1})_{p,\varpi};$

  2. (ii)

the inverse estimate

$K_{r}(f;n^{-1})_{p,\varpi}\leq
cn^{-r}\sum_{k=0}^{n}(k+1)^{r-1}E_{k}(f)_{p,\varpi}.$

It is worth mentioning that the direct estimate holds for the weight norm of
$\varpi$, whereas the inverse estimate holds for all doubling weight. We can
also replace the above characterization by the modulus of smoothness
$\varpi_{r}(f;t)_{p,\varpi}$, since the two quantities will be shown to be
equivalent.

######  Theorem 3.13.

Let $\varpi$ be a weight function that admits Assertions 1, 3 and 5. Let $f\in
L^{p}(\Omega,\varpi)$ if $1\leq p<\infty$ and $f\in C(\Omega)$ if $p=\infty$.
If $0<{\theta}\leq\pi/2$ and $r>0$, then

(3.11)
$c_{1}K_{r}(f;{\theta})_{p,\varpi}\leq{\omega}_{r}(f;{\theta})_{p,\varpi}\leq
c_{2}K_{r}(f;{\theta})_{p,\varpi}.$

For the unit sphere and the unit ball, these theorems are known to hold when
$\varpi$ being reflection invariant weight functions. For the unit sphere with
the Lebesgue measure, the characterization of best approximation has a long
history. Starting from [4, 36] for $r=2$, the problem was studied by a number
of authors and finally completed in [42]; we refer to [12, p. 102 and p. 263]
for historical notes. These results are extended to the weighted case, with
reflection invariant weight functions, on the unit sphere, the unit ball, and
the simplex, in [52].

Another pair of $K$-functional and modulus of smoothness was used to
characterize the best approximation for the unit sphere with the Lebesgue
measure in [11]. The approach appears to be very much domain-specific and
relies on the geometric and the differential structure on the unit sphere.

We will give the proof of these two theorems in Subsection 3.6, after proving
several preliminary results that are of interest in their own right.

###  3.4. Near best approximation operator

In this subsection we study the approximation property of the operator
$L_{n}(\varpi)*f$, defined in (2.42), that has the highly localized kernel as
its kernel. By (3.2), the operator can be written in terms of the convolution
operator by

$L_{n}(\varpi)*f=f\ast_{\varpi}L_{n}^{({\alpha},{\beta})},\qquad
n=0,1,2,\ldots.$

We now show that $L_{n}(\varpi)*f$ provides near-best approximation to $f$.

######  Theorem 3.14.

Let $\widehat{a}$ be admissible of type $(a)$. Assume $\varpi$ admits
Assertions 1 and 3. Then the operator $L_{n}*f$ satisfies

  1. (a)

$L_{n}(\varpi)*f$ is a polynomial of degree at most $2n$;

  2. (b)

$L_{n}(\varpi)*g=g$ for any polynomial $g\in\Pi_{n}(\Omega)$ of degree $n$;

  3. (c)

For $f\in L^{p}(\Omega,\varpi)$, $1\leq p\leq\infty$,

$\|L_{n}(\varpi)*f\|_{p,\varpi}\leq
c\|f\|_{p,\varpi}\quad\hbox{and}\quad\|L_{n}(\varpi)*f-f\|_{p,\varpi}\leq
c\,E_{n}(f)_{p,\varpi}.$

###### Proof.

The first two properties are immediate consequences of the definition. For the
third property, in the case $p=1$ we obtain

$\|L_{n}(\varpi)*f\|_{1,\varpi}\leq
c\|f\|_{1,\varpi}\max_{x\in\Omega}\int_{\Omega}\left|L_{n}\big{(}\varpi;x,y)\right|\varpi(y)\mathrm{d}{\mathsf{m}}(y).$

By Assertion 1 and (2.11) with $p=1$ and a large $\tau$, the integral in the
right-hand side is bound by

$\displaystyle\max_{x\in\Omega}\int_{\Omega}\frac{\varpi(y)\mathrm{d}{\mathsf{m}}(y)}{\sqrt{\varpi\left(B\left(y,n^{-1}\right)\right)}\sqrt{\varpi\left(B\left(x,n^{-1}\right)\right)}\big{(}1+n{\mathsf{d}}(x,y)\big{)}^{\tau}}\leq
c,$

which shows that $\|L_{n}(\varpi)*f\|_{p,{\mathsf{w}}}$ is bounded for $p=1$.
The same inequality also shows that the boundedness holds for $p=\infty$. The
case $1<p<\infty$ follows from the Riesz-Thorin interpolation theorem. By item
(b), the boundedness of $\|L_{n}(\varpi)*f\|_{p,\varpi}$ implies, by the
triangle inequality,

$\|L_{n}(\varpi)*f-f\|_{p,\varpi}\leq\|L_{n}(\varpi)*(f-g)\|_{p,\varpi}+\|f-g\|_{p,\varpi}\leq
c\|f-g\|_{p,\varpi}$

for any polynomial $g\in\Pi_{n}(\Omega)$. Taking infimum over $g$ completes
the proof. ∎

Because of property (c), we call the operator $L_{n}(\varpi)*f$ near best
approximation polynomial. Such operators have been used extensively in
approximation theory and computational harmonic analysis on the unit sphere
and the unit ball (see [12], for example), and in various other domains.

######  Theorem 3.15.

Let the assumption be the same as in the previous theorem. For $f\in
W_{p}^{r}(\Omega,\varpi)$, $1\leq p\leq\infty$,

(3.12) $\|f-L_{n}(\varpi)*f\|_{p,\varpi}\leq
cn^{-r}\left\|(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}f\right\|_{p,\varpi},\qquad
n=1,2,\ldots.$

###### Proof.

Let $\ell$ be a positive integer. Without loos of generality, we can assume
$n>\ell$. Since $L_{n}(\varpi)*f=f$ if $f$ is a polynomial of degree $n$, we
can write

$\displaystyle f-L_{n}(\varpi)*f\,$
$\displaystyle=\sum_{k=n+1}^{\infty}\left(1-\widehat{a}\left(\frac{k}{n}\right)\right)\operatorname{proj}_{k}(\varpi;f)$
$\displaystyle=\sum_{k=n+1}^{\infty}\\!\left(1-\widehat{a}\\!\left(\frac{k}{n}\right)\right)\mu(k)^{-\frac{r}{2}}\operatorname{proj}_{k}\\!\left(\varpi;F\right),$

where $F=\left(-{\mathfrak{D}}_{\varpi}\right)^{\frac{r}{2}}f$. Summation by
parts $\ell+1$ times, we obtain

$f-L_{n}(\varpi)*f=\sum_{k=n+1}^{\infty}b_{k}\operatorname{proj}_{k}\\!\left(\varpi;F\right)=\sum_{k=n+1}^{\infty}\left(\triangle^{\ell+1}b_{k}\right)A_{k}^{\ell}S_{k}^{\ell}\left(\varpi;F\right),$

where $b_{k}=(1-\widehat{a}(\frac{k}{n}))\mu(k)^{-\frac{r}{2}}$,
$A_{k}^{\ell}=\binom{k+\ell}{k}\sim k^{\ell}$ and
$S_{k}^{\ell}\left(\varpi;F\right)$ denotes the $k$-th Cesàro $(C,\ell)$ mean
of the Fourier orthogonal series with respect to $\varpi$. Since $\widehat{a}$
is in $C^{\infty}$ and its support is $[0,2]$, it is easy to see that
$|\triangle^{\ell+1}b_{k}|\leq ck^{-r-\ell-1}$. Hence, it follows that

$\sum_{k=n+1}^{\infty}\left|\triangle^{\ell+1}b_{k}\right|A_{k}^{\ell}\leq
c\sum_{k=n+1}^{\infty}k^{-r-\ell-1}k^{\ell}\leq cn^{r}.$

Since the choice of $\ell$ implies that
$\|S_{k}^{\ell}\left(\varpi;F\right)\|_{p,\varpi}\leq\|F\|_{p,\varpi}$ by the
convergence of the Cèsaro means, we obtain

$\|f-L_{n}(\varpi)*f\|_{p,\varpi}\leq cn^{-r}\|F\|_{p,\varpi},$

which is what we need to prove. ∎

###  3.5. Bernstein and Nikolskii inequalities

The Bernstein inequality is essential for studying approximation by
polynomials. We establish such an inequality for the deviation operator
${\mathfrak{D}}_{\varpi}$ defined by (3.1). First we prove a proposition that
is of independent interest.

######  Proposition 3.16.

Let $(\Omega,\varpi,{\mathsf{d}})$ be a localizable homogeneous space and
assume Assertion 4 holds. Let
$G_{n}(\cdot,\cdot):\Omega\times\Omega\mapsto{\mathbb{R}}$ be a kernel such
that $G_{n}(x,y)=G_{n}(y,x)$ for all $x,y\in\Omega$ and $G_{n}$ is a
polynomial of degree $n$ in either of its variables. Let $T:f\mapsto Tf$ be
the operator defined by

$Tf(x)=\int_{\Omega}f(y)G_{n}(x,y)\varpi(y)\mathrm{d}{\mathsf{m}}(y).$

If $f$ is a polynomial of degree $n$, then for $1\leq p\leq\infty$ and any
doubling weight ${\mathsf{w}}$,

(3.13) $\|Tf\|_{p,{\mathsf{w}}}\leq
c\|f\|_{p,{\mathsf{w}}}\max_{z\in\Omega}\int_{\Omega}|G_{n}(x,z)|(1+n{\mathsf{d}}(z,x))^{{\alpha}({\mathsf{w}})+{\alpha}(\varpi)}\varpi(x)\mathrm{d}{\mathsf{m}}(x).$

###### Proof.

Since $T$ is a linear integral operator, if $f$ is a polynomial of degree $n$
then $Tf$ is a polynomial of the same degree. Let $\Xi$ be an
${\varepsilon}$-separated set of $\Omega$ for
${\varepsilon}=\frac{\delta}{2n}$ so that the Marcinkiewicz-Zygmund
inequalities in Theorem 2.15 hold for all polynomials of degree $2n$. The
inequality is trivial for $p=\infty$ since it holds without
$(1+n{\mathsf{d}}(z,x))^{{\alpha}({\mathsf{w}})}$ term. For $1\leq p<\infty$,
by (2.35),

$\displaystyle\|Tf\|_{p,{\mathsf{w}}}^{p}\leq
c\sum_{z\in\Xi}|Tf(z)|^{p}{\mathsf{w}}\left(B(z,\tfrac{\delta}{2n})\right).$

Since $f(y)G_{n}(z,y)$ is a polynomial of degree $2n$ in $y$ variable,
applying (2.35) again, we obtain

$|Tf(z)|\leq
c\sum_{u\in\Xi}|f(u)||G_{n}(u,z)|\varpi\left(B(u,\tfrac{\delta}{2n})\right).$

Assume $1\leq p<\infty$. By the Hölder’s inequality,

$\displaystyle|Tf(z)|^{p}\,$ $\displaystyle\leq
c\sum_{u\in\Xi}|f(u)|^{p}|G_{n}(z,u)|\varpi\left(B(u,\tfrac{\delta}{2n})\right)\left(\sum_{u\in\Xi}|G_{n}(z,u)|\varpi\left(B(u,\tfrac{\delta}{2n})\right)\right)^{\frac{p}{q}}$
$\displaystyle\leq
c\sum_{u\in\Xi}|f(u)|^{p}|G_{n}(z,u)|\varpi\left(B(u,\tfrac{\delta}{2n})\right)\|G_{n}(z,\cdot)\|_{1}^{\frac{p}{q}},$

where we used (2.34) in the second step. Using (ii) of Lemma 2.1 for both
$\varpi$ and ${\mathsf{w}}$, we see that

$\displaystyle\sum_{z\in\Xi}|Tf(z)|^{p}{\mathsf{w}}\\!\left(B(z,\tfrac{\delta}{2n})\right)\leq
c\max_{z\in\Xi}\|G_{n}(z,\cdot)\|_{1}^{\frac{p}{q}}$
$\displaystyle\qquad\times\sum_{u\in\Xi}|f(u)|^{p}{\mathsf{w}}\\!\left(B(u,\tfrac{\delta}{2n})\right)\sum_{z\in\Xi}|G_{n}(z,u)|(1+n{\mathsf{d}}(z,u))^{{\alpha}({\mathsf{w}})+{\alpha}(\varpi)}\varpi\\!\left(B(z,\tfrac{\delta}{2n})\right).$

By Lemma 2.16, there is a nonnegative polynomial $Q_{u}$ of degree $n$ that
satisfies (2.39) with ${\alpha}={\alpha}({\mathsf{w}})+{\alpha}(\varpi)$,
which allows us to apply (2.34) on the polynomial $z\mapsto G(z,u)Q_{u}(z)$ to
show that the last sum in the right-hand side is bounded by

$\displaystyle
c\sum_{z\in\Xi}|G_{n}(z,u)|Q_{u}(z)\varpi\\!\left(B(z,\tfrac{\delta}{2n})\right)$
$\displaystyle\leq
c\int_{\Omega}|G_{n}(x,u)|Q_{u}(x)\varpi(x)\mathrm{d}{\mathsf{m}}(x)$
$\displaystyle\leq
c\int_{\Omega}|G_{n}(x,u)|(1+n{\mathsf{d}}(x,u))^{{\alpha}({\mathsf{w}})+{\alpha}(\varpi)}\varpi(x)\mathrm{d}{\mathsf{m}}(x).$

Putting these together and using (2.34) and (2.35), we have proved that

$\displaystyle\|Tf\|_{p,{\mathsf{w}}}^{p}\,$ $\displaystyle\leq
c\|f\|_{p,{\mathsf{w}}}^{p}\max_{z\in\Xi}\|G_{n}(z,\cdot)\|_{1}^{\frac{p}{q}}$
$\displaystyle\qquad\times\max_{u\in\Xi}\int_{\Omega}|G_{n}(x,u)|(1+n{\mathsf{d}}(x,u))^{{\alpha}({\mathsf{w}})+{\alpha}(\varpi)}\varpi(x)\mathrm{d}{\mathsf{m}}(x)$
$\displaystyle\leq
c\|f\|_{p,{\mathsf{w}}}^{p}\left(\max_{u\in\Xi}\int_{\Omega}|G_{n}(x,u)|(1+n{\mathsf{d}}(x,u))^{{\alpha}({\mathsf{w}})+{\alpha}(\varpi)}\varpi(x)\mathrm{d}{\mathsf{m}}(x)\right)^{1+\frac{p}{q}},$

where we have used the assumption that $G_{n}$ is symmetric in its variables.
Since $1+\frac{p}{q}=p$, this proves the stated inequality. ∎

For $r>0$, we denote by $L_{n}^{(r)}(\varpi;\cdot,\cdot)$ the kernel defined
by

(3.14)
$L_{n}^{(r)}(\varpi;x,y)=\sum_{n=0}^{\infty}\widehat{a}\left(\frac{k}{n}\right)[\mu(k)]^{r/2}P_{k}(\varpi;x,y),$

which is the kernel ${\mathfrak{D}}_{\varpi}^{r/2}L_{n}(x,y)$ with
${\mathfrak{D}}_{\varpi}^{r/2}$ applying on $x$ variable. Our Bernstein
inequality is proved under the following assumption on the decaying of this
kernel.

Assertion 5. For $r>0$ and ${\kappa}>0$, the kernel $L_{n}^{(r)}(\varpi)$
satisfies, for $x,y\in\Omega$,

$\left|L_{n}^{(r)}(\varpi;x,y)\right|\leq
c_{\kappa}\frac{n^{r}}{\sqrt{\varpi(B(x,n^{-1}))}\sqrt{\varpi(B(y,n^{-1}))}}(1+n{\mathsf{d}}(x,y))^{-{\kappa}}.$

######  Theorem 3.17.

Let $\varpi$ be a weight functions that admits Assertions 1–5. Let
${\mathsf{w}}$ be a doubling weight on $\Omega$. If $r>0$, $1\leq p\leq\infty$
and $f\in\Pi_{n}(\Omega)$, then

$\|(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}f\|_{p,{\mathsf{w}}}\leq
cn^{r}\|f\|_{p,{\mathsf{w}}}.$

###### Proof.

Since ${\mathsf{L}}_{n}(\varpi)*f$ reproduces polynomials of degree $n$, we
can write

$(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}f=(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}\left(L_{n}(\varpi)*f\right)=\int_{\Omega}f(y)L_{n}^{(r)}(\varpi;x,y)\varpi(y)\mathrm{d}{\mathsf{m}}(y),$

where $L_{n}^{(r)}(\varpi)$ is defined by (3.14). Applying Proposition 3.16,
we conclude that

$\displaystyle\left\|(-{\mathfrak{D}})^{\frac{r}{2}}f\right\|_{p,{\mathsf{w}}}\,$
$\displaystyle\leq
c\|f\|_{p,{\mathsf{w}}}\max_{x\in\Omega}\int_{\Omega}\left|L_{n}^{(r)}(\varpi;x,y)\right|(1+n{\mathsf{d}}(x,y))^{{\alpha}({\mathsf{w}})+{\alpha}(\varpi)}\varpi(y)\mathrm{d}{\mathsf{m}}(y).$

By Assertion 5, the integral in the right-hand side is bounded by

$c_{\kappa}n^{r}\int_{\Omega}\frac{\varpi(s)}{\sqrt{\varpi(B(x,n^{-1}))}\sqrt{\varpi(B(y,n^{-1}))}\left(1+n{\mathsf{d}}(x,y)\right)^{{\kappa}-{\alpha}({\mathsf{w}})-{\alpha}(\varpi)}}\mathrm{d}{\mathsf{m}}(y)\leq
c\,n^{r}$

using (2.11) with $p=1$ and
${\kappa}>{\alpha}({\mathsf{w}})+\frac{5}{4}{\alpha}(\varpi)$. ∎

For the Laplace-Beltrami operator on the unit sphere, this Bernestin
inequality is classical if ${\mathsf{w}}$ is the Lebesgue measure, whereas the
version with the doubling weight was proved in [7]. For the unit ball, the
inequality was established in [52] for the classical weight function with a
different proof that applies to reflection invariant weight functions on the
unit sphere and the unit ball.

We can also prove a Nikolskii type inequality for the doubling weight on
$\Omega$ that admits a localizable homogeneous space.

######  Theorem 3.18.

Let ${\mathsf{w}}$ be a doubling weight on $\Omega$. If $0<p<q\leq\infty$ and
$f\in\Pi_{n}(\Omega)$, then

(3.15) $\|f\|_{q,{\mathsf{w}}}\leq
cn^{(\frac{1}{p}-\frac{1}{q}){\alpha}({\mathsf{w}})}\|f\|_{p,{\mathsf{w}}}.$

###### Proof.

The proof is another one that follows as in the case of the unit sphere. The
main work lies in proving the case $q=\infty$, for which we choose a maximal
$\frac{\delta}{n}$-separated subset of $\Omega$, so that

$\displaystyle\|f\|_{\infty}\leq c\max_{z\in\Xi}|f(z)|\,$ $\displaystyle\leq
c\left(\min_{z\in\Xi}B\Big{(}z,\frac{\delta}{n}\Big{)}\right)^{-\frac{1}{p}}\Bigg{(}\sum_{z\in\Xi}{\mathsf{w}}\left(B\Big{(}z,\frac{\delta}{n}\Big{)}\right)|f(z)|^{p}\Bigg{)}^{\frac{1}{p}}$
$\displaystyle\leq
c\|f\|_{p,{\mathsf{w}}}\max_{z\in\Xi}\left(B\Big{(}z,\frac{1}{n}\Big{)}\right)^{-\frac{1}{p}}.$

Since $\Omega$ is compact, there is a positive number $\rho_{0}>0$ such that,
for each $z\in\Xi$, $1={\mathsf{w}}(\Omega)={\mathsf{w}}(B(z,\rho))$ for some
positive number $\rho\leq\rho_{0}$. Let $m$ be a positive integer such that
$2^{m-1}\leq\rho n\leq 2^{m}$. Then, by (i) of Lemma 2.1,

$1={\mathsf{w}}(B(z,\rho))\leq
c_{L({\mathsf{w}})}2^{m{\alpha}({\mathsf{w}})}{\mathsf{w}}(B(z,\tfrac{1}{n}))\leq
cn^{{\alpha}({\mathsf{w}})}{\mathsf{w}}(B(z,\tfrac{1}{n})),$

which implies that ${\mathsf{w}}(B(z,\tfrac{1}{n}))\leq
cn^{-{\alpha}({\mathsf{w}})}$. Combing the two inequalities, we conclude that
$\|f\|_{\infty}\leq
c\|f\|_{p,{\mathsf{w}}}n^{\frac{1}{p}{\alpha}({\mathsf{w}})}$. This proves
(3.15) for $q=\infty$.

The case $q<\infty$ reduces to that of $q=\infty$ since, using (3.15) for
$q=\infty$,

$\|f\|_{q,{\mathsf{w}}}^{q}\leq\|f\|_{\infty}^{q-p}\|f\|_{q,{\mathsf{w}}}^{p}\leq
c\left(\|f\|_{p,{\mathsf{w}}}n^{\frac{1}{p}{\alpha}({\mathsf{w}})}\right)^{q-p}\|f\|_{q,{\mathsf{w}}}^{p}=cn^{q(\frac{1}{p}-\frac{1}{q}){\alpha}({\mathsf{w}})}\|f\|_{p,\sigma}^{q},$

which is (3.15) for $q<\infty$. ∎

For the unit sphere, this inequality was established in [10]. The above proof
uses essentially the same argument as seen in [12, Theorem 5.5.1]. For the
unit ball with the classical weight, the inequality was proved in [25].

###  3.6. Proof of the main results

We are now ready to prove Theorems 3.12 and 3.13. The proof follows the same
procedure used in [42, 52], which is summarized in [12, section 10]. Since
most technical parts are essentially the same, we shall be brief.

Proof of Theorem 3.12.

To prove the direct estimate, we use Theorems 3.14 and 3.15 to obtain

$\displaystyle\left\|f-L_{n}\left(\varpi;f\right)\right\|_{p,\varpi}\,$
$\displaystyle\leq
2\|f-g\|_{p,\varpi}+\left\|g-L_{n}\left(\varpi;g\right)\right\|_{p,\varpi}$
$\displaystyle\leq
2\|f-g\|_{p,\varpi}+cn^{-r}\left\|(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}g\right\|_{p,\varpi}.$

Taking infimum over $g$ proves (i).

To prove the inverse estimate, we choose $m$ such that $2^{m-1}\leq n<2^{m}$.
Let us set $L_{2^{-1}}(\varpi)*f=0$. Then, choosing $g=L_{2^{m}}(\varpi)*f$,
we obtain by Theorem 3.14,

$\displaystyle K_{r}\left(f,n^{-1}\right)_{p,{\mathsf{w}}}$
$\displaystyle\,\leq\|f-L_{2^{m}}(\varpi)*f\|_{p,{\mathsf{w}}}+2^{-(m-1)r}\left\|(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}L_{2^{m}}(\varpi)*f\right\|_{p,{\mathsf{w}}}$
$\displaystyle\leq
cE_{n}(f)_{p,{\mathsf{w}}}+2^{-(m-1)r}\sum_{j=0}^{m}\left\|(-{\mathfrak{D}}_{\varpi})^{\frac{r}{2}}\left[L_{2^{j}}({\mathsf{w}})*f-L_{2^{j-1}}({\mathsf{w}})*f\right]\right\|_{p,{\mathsf{w}}}.$

Applying the Bernstein inequality Theorem 3.17 and using the triangle
inequality with Theorem 3.14, we conclude that

$\displaystyle K_{r}\left(f,n^{-1}\right)_{p,\varpi}$ $\displaystyle\,\leq
cE_{n}(f)_{p,\varpi}+2^{-mr}\sum_{j=0}^{m}2^{-jr}\left\|L_{2^{j}}(\varpi)*f-L_{2^{j-1}}(\varpi)*f\right\|_{p,{\mathsf{w}}}$
$\displaystyle\leq c\,2^{-mr}\sum_{j=0}^{m}2^{jr}E_{2^{j-1}}(f)_{p,\varpi}\leq
c\,n^{-r}\sum_{k=0}^{m}(k+1)^{r-1}E_{k}(f)_{p,\varpi}.$

This completes the proof. ∎

The proof of the equivalence of the $K$-functional and the modulus of
smoothness relies on two technical lemmas. Recall that $\widehat{a}$ is a cut-
off function and $\mu(j)$ is the eigenvalue in (3.1).

######  Lemma 3.19.

Let ${\alpha}\geq{\beta}\geq-\frac{1}{2}$. For $r>0$, $0\leq{\theta}\leq
3n^{-1}$, and any $\ell\in{\mathbb{N}}$,

(3.16)
$\sum_{j=0}^{2n}\Biggl{|}\triangle^{\ell+1}\biggl{[}\biggl{(}\frac{1-R^{({\alpha},{\beta})}_{j}(\cos{\theta})}{\mu(j){\theta}^{2}}\biggr{)}^{r}\widehat{a}\left(\frac{j}{n}\right)\biggr{]}\Biggr{|}(j+1)^{\ell}\leq
c,$

and

(3.17)
$\sum_{j=0}^{2n}\Biggl{|}\triangle^{\ell+1}\biggl{[}\biggl{(}\frac{\mu(j){\theta}^{2}}{1-R_{j}^{({\alpha},{\beta})}(\cos{\theta})}\biggr{)}^{r}\widehat{a}\left(\frac{j}{n}\right)\biggr{]}\Biggr{|}(j+1)^{\ell}\leq
c,$

where the difference $\triangle^{\ell+1}$ is acting on $j$ and $c$ depends
only on ${\alpha},{\beta},\ell,r$.

######  Lemma 3.20.

Let ${\alpha}\geq{\beta}\geq-\frac{1}{2}$ and let $\ell=\lceil{\alpha}\rceil$.
If $r>0$, $k^{-1}\leq{\theta}\leq\frac{\pi}{2}$, $j\in{\mathbb{N}}_{0}$ and
$0\leq j\leq\ell+1$, then for any $m\in{\mathbb{N}}$,

(3.18)
$\biggl{|}\triangle^{j}\biggl{(}\frac{(1-(1-R_{k}^{({\alpha},{\beta})}(\cos{\theta}))^{r})^{m+\ell+1}}{(1-R_{k}^{({\alpha},{\beta})}(\cos{\theta}))^{r}}\biggr{)}\biggr{|}\leq
c_{m,r}(k{\theta})^{-m{\alpha}}{\theta}^{j}.$

For ${\alpha}={\beta}={\lambda}$, these two lemma are Lemma 10.4.3 and Lemma
10.4.4 in [12]. The proof relies on properties of the Jacobi polynomials and
holds readily for ${\alpha}\geq{\beta}\geq-\frac{1}{2}$.

Proof of Theorem 3.13.

Choose $n$ such that ${\theta}^{-1}\leq n\leq 3{\theta}^{-1}$. Let $g$ be a
polynomial in $\Pi_{n}(\Omega)$. Then so is
$\left(I-S_{{\theta},\varpi}\right)^{r/2}g$ in $\Pi_{n}(\Omega)$ by (3.7). Let
$\widehat{a}$ be a cut-off function of type (a), so that $L_{n}(\varpi)$
reproduces polynomials of degree $n$. Using (3.10), it follows readily that

$\left(I-S_{{\theta},\varpi}\right)^{r/2}g={\theta}^{r}\sum_{j=0}^{2n}\biggl{(}\frac{1-R^{({\alpha},{\beta})}_{j}(\cos{\theta})}{\mu(j){\theta}^{2}}\biggr{)}^{r/2}\widehat{a}\Big{(}\frac{j}{n}\Big{)}\operatorname{proj}_{j}\left(\varpi;(-{\mathfrak{D}}_{\varpi})^{r/2}g\right).$

Following the proof of Theorem 3.15 to take a summation by parts and use the
boundedness of the Cesàro means, we can use (3.16) to obtain

(3.19)
$\displaystyle\left\|\left(I-S_{{\theta},\varpi}\right)^{r/2}g\right\|_{p,\varpi}\leq
cn^{-r}\left\|(-{\mathfrak{D}}_{\varpi})^{r/2}g\right\|_{p,\varpi}.$

Using this inequality with $g=L_{\lfloor n/2\rfloor}(\varpi)*f$ and the
Jackson estimate in Theorem 3.12, we obtain

$\displaystyle\left\|(I-S_{{\theta},\varpi})^{r/2}f\right\|_{p,\varpi}$
$\displaystyle\leq
c\left\|f-L_{\lfloor\frac{n}{2}\rfloor}(\varpi)*f\right\|_{p,\varpi}+\left\|(I-S_{{\theta},\varpi})^{r/2}L_{\lfloor\frac{n}{2}\rfloor}(\varpi)*f\right\|_{p,\varpi}$
$\displaystyle\leq
cK_{r}(f;n^{-1})_{p,\varpi}+cn^{-r}\left\|(-{\mathfrak{D}})^{r/2}L_{\lfloor\frac{n}{2}\rfloor}(\varpi)*f\right\|_{p,\varpi}.$

By the triangle inequality and applying the Bernstein inequality on
$L_{n}(\varpi)*(f-g)$, we obtain

$\displaystyle
n^{-r}\left\|(-{\mathfrak{D}})^{r/2}L_{\lfloor\frac{n}{2}\rfloor}(\varpi)*f\right\|_{p,\varpi}$
$\displaystyle\leq
c\left\|f-g\right\|_{p,\varpi}+n^{-r}\left\|(-{\mathfrak{D}})^{r/2}L_{\lfloor\frac{n}{2}\rfloor}(\varpi)*g\right\|_{p,\varpi}$
$\displaystyle\leq
c\left(\left\|f-g\right\|_{p,\varpi}+n^{-r}\left\|(-{\mathfrak{D}})^{r/2}g\right\|_{p,\varpi}\right),$

where we used, by (3.1),
$(-{\mathfrak{D}})^{r/2}L_{\lfloor\frac{n}{2}\rfloor}(\varpi)*g=L_{\lfloor\frac{n}{2}\rfloor}(\varpi)*(-{\mathfrak{D}})^{r/2}g$
in the second step. Hence, taking infimum over $g$, it follows from the above
two inequalities and ${\theta}\sim n^{-1}$ that

$\left\|(I-S_{{\theta},\varpi})^{r/2}f\right\|_{p,\varpi}\leq
cK_{r}(f,{\theta})_{p,\varpi},$

from which the right-hand inequality of (3.11) follows.

In the other direction, we can follow the proof of inequality (3.19) and use
(3.17) instead of (3.16), to establish

$\displaystyle
n^{-r}\left\|(-{\mathfrak{D}}_{\varpi})^{r/2}L_{n}(\varpi)*f\right\|_{p,\varpi}\leq
c\left\|\left(I-S_{{\theta},\varpi}\right)^{r/2}f\right\|_{p,\varpi}.$

Hence, in order to prove the left-hand inequality of (3.11), it suffices to
prove that

(3.20) $\displaystyle\left\|f-L_{n}(\varpi)*f\right\|_{p,\varpi}\leq
c\left\|\left(I-S_{{\theta},\varpi}\right)^{r/2}f\right\|_{p,\varpi}.$

Using $(1-r)^{-1}=\sum_{i=0}^{m+\ell}r^{i}+r^{m+\ell}(1-r)^{-1}$ with
$r=1-\big{(}1-R^{({\alpha},{\beta})}_{j}(\cos{\theta})\big{)}^{r/2}$, one can
write, as seen in [12, p. 253], that

$\displaystyle f-L_{n}(\varpi)*f$
$\displaystyle\,=\sum_{i=0}^{m+\ell}(I-L_{n}(\varpi))*\left(I-(I-S_{{\theta},\varpi})^{r/2}\right)^{i}F$
$\displaystyle+\sum_{j=n}^{\infty}\left(1-\widehat{a}\left(\frac{j}{n}\right)\right)\frac{\big{(}1-\big{(}1-R^{({\alpha},{\beta})}_{j}(\cos{\theta})\big{)}^{r/2}\big{)}^{m+\ell+1}F}{(1-R_{j}^{({\alpha},{\beta})}(\cos{\theta}))^{r/2}}\operatorname{proj}_{j}(\varpi;F),$

where $F=\left(I-S_{{\theta},\varpi}\right)^{r/2}f$. While the first term is
bounded by $\|F\|_{p,\varpi}$ by the boundedness of $L_{n}(\varpi)*f$ and
$(I-(I-S_{{\theta},\varpi})^{r/2})F$, the second one can be shown to be
bounded by $\|F\|_{p,\varpi}$ by using (3.18) with the summation by parts and
the boundedness of Cesàro means that we have used several times. This proves
(3.20) and completes the proof. ∎

##  4\. Homogeneous space on conic surfaces

In this section we work in the setting of homogeneous space on the conic
surface

${\mathbb{V}}_{0}^{d+1}=\\{(x,t):\|x\|=t,\,x\in{\mathbb{R}}^{d},\,0\leq t\leq
1\\}.$

We shall verify that the framework in the previous two sections is applicable
on this domain for the weight function $t^{-1}(1-t)^{\gamma}$, which has a
singularity at the apex. The verification is highly non-trivial because of new
phenomena and obstacles encountered. While the structure of orthogonal
polynomials on the conic surface shares characteristic features of spherical
harmonics, the conic surface is markedly different from that of the unit
sphere because of its apex and its boundary.

Our first task is to understand, in the first subsection, the intrinsic
distance function on the conic surface, which turns out to be incomparable to
the Euclidean distance around the apex. In the second subsection, we show that
the Jacobi weight function
${\mathsf{w}}_{{\beta},{\gamma}}(t)=t^{\beta}(1-t)^{\gamma}$ is a doubling
weight with respect to the intrinsic distance. The orthogonal structure with
respect to the Jacobi weight is reviewed in the third subsection, which is
used to verify Assertions 1-3 of the highly localized kernels for the weight
${\mathsf{w}}_{-1,{\gamma}}$ in the fourth subsection. Construction of
${\varepsilon}$-separated set of ${\mathbb{V}}_{0}^{d+1}$ is provided in the
fifth subsection and used to state the Marcinkiewicz-Zygmund inequality.
Assertion 4 is verified in the sixth subsection, which ensures that the
positive cubature rules and the tight localized frames can both be stated for
the conic surface. In the seventh subsection, Assertion 5 is verified and the
characterization of the best approximation by polynomials is stated.

###  4.1. Distance on the surface of the cone