# Sharp Constants of Approximation Theory. IV. Asymptotic Relations in General
Settings

Michael I. Ganzburg  Department of Mathematics
Hampton University
Hampton, VA 23668
USA<EMAIL_ADDRESS>
###### Abstract.

In this paper we first introduce the unified definition of the sharp constant
that includes constants in three major problems of approximation theory, such
as, inequalities for approximating elements, approximation of individual
elements, and approximation on classes of elements. Second, we find sufficient
conditions that imply limit relations between various sharp constants of
approximation theory in general settings. Third, a number of examples from
various areas of approximation theory illustrates the general approach.

###### Key words and phrases:

Error of approximation, asymptotic relations, sharp constants, V. A. Markov-
Bernstein inequality of different metrics, algebraic polynomials,
trigonometric polynomials, entire functions of exponential type

###### 2010 Mathematics Subject Classification:

Primary 41A44, 41A63; Secondary 41A17, 41A65

##  1\. Introduction

We continue the study of the sharp constants of approximation theory that
began in [31, 32, 44, 38, 41, 42, 43]. In this paper we discuss limit
relations between sharp constants of approximation theory in general settings.

Sharp constants of approximation theory have attracted much attention of
approximation analysts for more than a century (see monographs [74, 1, 52, 23,
60, 15, 53, 6] and references therein). Our unified definition of the sharp
constant (see (2.6)) includes constants in three major problems of
approximation theory, such as, inequalities for approximating elements
(Problem A), approximation of individual elements (Problem B), and
approximation on classes of elements (Problem C).

It was Bernstein who in 1913 and 1938 initiated the study of asymptotic
relations in Problem B for the errors of best polynomial approximation of
$f_{\lambda}(\tau):=|\tau|^{\lambda}$ by proving the following celebrated
result (see [7] for ${\lambda}=1$ and [8] for ${\lambda}>0$): for any
${\lambda}>0$, there is a constant $\mu_{\lambda}\in(0,{\infty})$ such that

$\lim_{n\to{\infty}}n^{\lambda}\min_{P_{n}\in{\mathcal{P}}_{n}}\max_{\tau\in[-1,1]}\left|f_{\lambda}(\tau)-P_{n}(\tau)\right|=\mu_{\lambda},$
(1.1)

where ${\mathcal{P}}_{n}$ is the class of all univariate algebraic polynomials
of degree at most $n$. Nikolskii [63] proved that an $L_{1}[-1,1]$-version of
(1.1) is valid as well. In addition to (1.1), Bernstein [8] also showed that
$\mu_{\lambda}{\sigma}^{-{\lambda}}$ coincides with the error of best
approximation of $f_{\lambda}$ by entire functions of exponential type
${\sigma},\,{\sigma}>0$, on the real axis, that is,

$\mu_{\lambda}={\sigma}^{\lambda}\inf_{g\in
B_{\sigma}}\sup_{\tau\in(-{\infty},{\infty})}\left|f_{\lambda}(\tau)-g(\tau)\right|.$
(1.2)

Later Bernstein [11, 13] (see also [74, Sect. 2.6.22] and [1, Appendix, Sect.
84]) extended the combination of relations (1.1) and (1.2) to any continuous
function $f$ of polynomial growth in the following form:

$\lim_{n\to{\infty}}\min_{P_{n}\in{\mathcal{P}}_{n}}\max_{\tau\in[-n/{\sigma},n/{\sigma}]}\left|f(\tau)-P_{n}(\tau)\right|=\inf_{g\in
B_{\sigma}}\sup_{\tau\in(-{\infty},{\infty})}\left|f(\tau)-g(\tau)\right|.$
(1.3)

This relation holds true for almost all ${\sigma}>0$, while for certain
functions (e.g., for $f_{\lambda}$) it is valid for all ${\sigma}>0$. In
particular, (1.3) implies (1.1) and (1.2). An $L_{p}$-version of (1.3) for
$p\in[1,{\infty})$ and all ${\sigma}>0$ was obtained by Raitsin [66]. Certain
periodic analogues of (1.3) were given in [10]. The corresponding multivariate
extensions of these results were established by the author [29, 30, 31, 32]
(see also Examples 7.8 and 7.9 in Section 7). Weighted $L_{p}$-versions of
(1.3) for $p\in(0,{\infty}]$ were obtained in [36] (see also Example 7.10).

In addition, Bernstein [9] discussed limit relations in Problem C between the
least upper bounds of the errors of best harmonic and polynomial approximation
on Hölder classes $H^{\lambda},\,{\lambda}\in(0,1]$. In particular, he proved
the following relation:

$\lim_{n\to{\infty}}n^{\lambda}\sup_{f\in
H^{\lambda}[-1,1]}\min_{P_{n}\in{\mathcal{P}}_{n}}\max_{\tau\in[-1,1]}\left|f(\tau)-P_{n}(\tau)\right|=\sup_{f\in
H^{\lambda}(-{\infty},{\infty})}\min_{g\in
B_{1}}\sup_{\tau\in(-{\infty},{\infty})}\left|f(\tau)-g(\tau)\right|.$ (1.4)

More general versions of (1.4) were discussed by Bernstein [12] and by the
author [31, 35] (see also Examples 7.15 and 7.16).

A general approach to limit (asymptotic) relations between sharp constants in
normed spaces in Problems B and C was developed by the author [31]. This
approach was later extended to ${\beta}$-normed spaces [32].

The first asymptotic relations between sharp constants in Nikolskii-type
inequalities of Problem A were established by Taikov [72, 73], Garsia et al
[46], and Gorbachev [47]. Levin and Lubinsky [57, 58] obtained more general
results. The latest asymptotics for the sharp constants in Nikolskii-Bernstein
type inequalities were given by the author and Tikhonov [44]; in particular,
the following relation was proved in [44]:

$\lim_{n\to{\infty}}n^{-s-1/p}\sup_{Q\in{\mathcal{T}}_{n}\setminus\\{0\\}}\frac{\left\|Q^{(s)}\right\|_{L_{\infty}([0,2\pi))}}{\left\|Q\right\|_{L_{p}([0,2\pi))}}={\sigma}^{-s-1/p}\sup_{g\in\left(B_{\sigma}\cap
L_{p}({\mathbb{R}})\right)\setminus\\{0\\}}\frac{\left\|g^{(s)}\right\|_{L_{\infty}((-{\infty},{\infty}))}}{\left\|g\right\|_{L_{p}((-{\infty},{\infty}))}},$
(1.5)

where ${\mathcal{T}}_{n}$ is the class of all univariate trigonometric
polynomials of degree at most $n$ and $p\in(0,{\infty}]$. A multivariate
generalization of (1.5) was obtained in [41]. Univariate and multivariate
versions of (1.5) for algebraic polynomials were given in [38, 21, 42, 43, 22]
(see also Examples 7.2, 7.3, and 7.4).

This paper was inspired by several publications [44, 38, 41, 42, 43] about
limit relations between sharp constants in univariate and multivariate Markov-
Bernstein type inequalities of different metrics for trigonometric and
algebraic polynomials and entire functions of exponential type. Analyzing
these publications, we noticed that the proofs of the corresponding limit
relations are based on verification of certain conditions that often coincide
with the conditions introduced in [31, 32]. As a result of this observation,
in this paper we develop a more general approach, compared with [31, 32], to
limit relations for sharp constants in vector spaces. In addition to Problems
B and C, this approach also includes inequalities for approximating elements.
However, in certain details the approach differs from the one in [31, 32] for
Problems B and C.

In Section 2 the definition of the sharp constant ${\mathcal{C}}$ is
introduced, and special cases of ${\mathcal{C}}$ in approximation theory and
analysis are discussed. In Section 3 we introduce conditions on sets, vector
spaces, and operators and study their properties. In Section 4 we prove
asymptotic relations between sharp constants in vector spaces that are based
on conditions from Section 3. The proofs are simple but we believe that this
general condition-based approach is useful in approximation theory. Special
cases of asymptotic relations in Problems A, B, and C are discussed in Section
5.

Function spaces, function classes, special operators, and their properties are
discussed in Section 6. In Section 7 for each of Problems A, B, and C, we
discuss limit relations between sharp constants in function spaces, including
three examples of Markov-Bernstein type inequalities of different metrics,
three examples of limit theorems for polynomial and harmonic approximation,
and two examples of limit relations between constants of approximation theory.
In the capacity of approximating elements in the examples, we use univariate
and multivariate trigonometric and algebraic polynomials and entire functions
of exponential type. These examples cover most of known results in this area
and Example 7.15 is new.

##  2\. Definitions and Properties of Sharp Constants in Vector Spaces

Here, we first describe major spaces that are discussed in this paper. Next,
we give the unified definition of a sharp constant in general settings and
study some of its properties. Finally, we discuss certain special cases of
sharp constants of approximation theory and analysis.

###  2.1. Spaces

Let $F$ be a vector space over the field of complex numbers ${\mathbb{C}}$ or
real numbers ${\mathbb{R}}$. Let $F$ be equipped with a functional
$\|\cdot\|_{F}:F\to[0,{\infty})$, which is not necessarily a norm though we
use the same notation. Occasionally, we will refer to $\|\cdot\|_{F}$ as a
”norm.” In particular, $\|\cdot\|_{F}$ could be a norm, a seminorm, or more
generally, a ${\beta}$-norm or a ${\beta}$-seminorm.

######  Definition 2.1.

[49, p. 1102] We say that $F$ is a _$\beta$ -normed space_, $\beta\in(0,1]$,
if there exists a ${\beta}$-_norm_ $\|\cdot\|_{F}$ on $F$ with the following
properties:

  * (a)

$\|f\|_{F}\geq 0$ for all $f\in F$ and $\|f\|_{F}=0$ if and only if $f=0$.

  * (b)

$\|\alpha f\|_{F}=|\alpha|\,\|f\|_{F}$ for all $\alpha\in{\mathbb{C}}$ (or
$\alpha\in{\mathbb{R}}$).

  * (c)

$\|f+g\|_{F}\leq\left(\|f\|_{F}^{\beta}+\|g\|_{F}^{\beta}\right)^{1/\beta}$
for all $f\in F,\,g\in F$.

Occasionally, we will refer to (b) and (c) as the homogeneity property and the
${\beta}$-triangle inequality, respectively. In case of
${\beta}=1,\,\|\cdot\|_{F}$ is a norm and $F$ is a normed space. Note that a
${\beta}$-norm is closely related to a ${\gamma}$-quasinorm $|||\cdot|||_{F}$
satisfying conditions (a), (b), and (c*)
$|||f+g|||_{F}\leq{\gamma}(|||f|||_{F}+|||g|||_{F}),\,{\gamma}\geq 1;\,f\in
F,\,g\in F.$ Obviously, a ${\beta}$-norm is a $2^{1/{\beta}-1}$-quasinorm.
Conversely, the Aoki-Rolewicz theorem [3, 68] states that if for some
$\beta\in(0,1],\,|||\cdot|||_{F}$ is a $2^{1/{\beta}-1}$-quasinorm, then there
exists a ${\beta}$-norm $\|\cdot\|_{F}$, which is equivalent to
$|||\cdot|||_{F}$.

If we replace property (a) with

  * (a*)

$\|f\|_{F}\geq 0$ for all $f\in F$ ,

then $\|\cdot\|_{F}$ and $F$ are called a ${\beta}$-_seminorm_ and a _$\beta$
-seminormed space_, respectively. For $\beta=1,\,\|\cdot\|_{F}$ is a seminorm
and $F$ is a seminormed space.

However, in this paper $\|\cdot\|_{F}$ is not necessarily a $\beta$-norm or a
$\beta$-seminorm. For example, the trivial functional

$\|f\|_{F,\mathrm{triv}}:=\left\\{\begin{array}[]{ll}0,&f=0,\\\ 1,&f\neq
0,\end{array}\right.\qquad f\in F,$ (2.1)

satisfies property (a) and also property (c) for any $\beta\in(0,1]$ but does
not satisfy property (b). Meantime, $\|\cdot\|_{F,\mathrm{triv}}$ satisfies
the symmetry property

  * (b*)

$\|-f\|_{F}=\|f\|_{F}$ for all $f\in F$ .

Note that a vector space $F$ can be equipped with two (or more) different
”norms;” we often denote them by $\|\cdot\|_{F}$ and $\|\cdot\|_{F}^{*}$. So
the equality $F=G$ for vector spaces $F$ and $G$ means that they are equal as
sets but can be equipped with different ”norms,” while the equality
$\|\cdot\|_{F}=\|\cdot\|_{G}$ also means that $F=G$.

In this paper we also discuss sequences of ”norms”
$\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ that possess either the
continuity (C) or generalized continuity (GC) properties, which can replace
the ${\beta}$-triangle inequality in certain problems.

######  Definition 2.2.

We say that $\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ possesses the
C-property if for any $h_{n}\in F_{n},\,g_{n}\in F_{n},\,n\in{\mathbb{N}}$,
such that $\lim_{n\to{\infty}}\|g_{n}\|_{F_{n}}=0$, the following relations
hold true:

$\displaystyle\liminf_{n\to{\infty}}\|h_{n}+g_{n}\|_{F_{n}}=\liminf_{n\to{\infty}}\|h_{n}\|_{F_{n}},$
(2.2)
$\displaystyle\limsup_{n\to{\infty}}\|h_{n}+g_{n}\|_{F_{n}}=\limsup_{n\to{\infty}}\|h_{n}\|_{F_{n}}.$
(2.3)

######  Proposition 2.3.

(i) If ”norms” $\|\cdot\|_{F_{n}},\,n\in{\mathbb{N}}$, satisfy the symmetry
property (b*) and the ${\beta}$-triangle inequality (c), where
${\beta}\in(0,1]$ is independent of $n$, then
$\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ possesses the C-property.
(ii) If $\|\cdot\|_{F_{n}}$ is a $\beta$-norm or a $\beta$-seminorm,
$n\in{\mathbb{N}},$ where ${\beta}\in(0,1]$ is independent of $n$, then
$\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ possesses the C-property.
(iii) If
$\|\cdot\|_{F_{n}}=\|\cdot\|_{F_{n},\mathrm{triv}},\,n\in{\mathbb{N}},$ then
$\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ possesses the C-property.

###### Proof.

Statement (i) follows immediately from the ${\beta}$-triangle inequalities

$\displaystyle\|h_{n}\|_{F_{n}}^{\beta}-\|g_{n}\|_{F_{n}}^{\beta}\leq\|h_{n}+g_{n}\|_{F_{n}}^{\beta}\leq\|h_{n}\|_{F_{n}}^{\beta}+\|g_{n}\|_{F_{n}}^{\beta},\qquad
n\in{\mathbb{N}},$

while statements (ii) and (iii) are easy consequences of (i). $\Box$

######  Definition 2.4.

Let $F_{n}$ be equipped with two ”norms” $\|\cdot\|_{F_{n}}$ and
$\|\cdot\|_{F_{n}}^{*},\,n\in{\mathbb{N}}$. We say that
$\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ possesses the GC-property
with respect to $\left\\{\|\cdot\|_{F_{n}}^{*}\right\\}_{n=1}^{\infty}$ if for
any $h_{n}\in F_{n},\,g_{n}\in F_{n},\,n\in{\mathbb{N}}$, such that
$\lim_{n\to{\infty}}\|g_{n}\|_{F_{n}}^{*}=0$, relations (2.2) and (2.3) hold
true.

In case of $\|\cdot\|_{F_{n}}=\|\cdot\|_{F_{n}}^{*},\,n\in{\mathbb{N}}$, the
GC-property is reduced to the C-property. In most of our applications we use
sequences of ”norms” with the C-property. However, in the following typical
example (related to approximation theory) we present a sequence of ”norms”
that possesses the GC-property but does not possess the C-property.

######  Example 2.5.

Let $F_{n}\neq\\{0\\}$ and let $\|\cdot\|_{F_{n}}^{*},\,n\in{\mathbb{N}},$
satisfy property (b) of Definition 2.1. Next, let
$\left\\{\|\cdot\|_{F_{n}}^{*}\right\\}_{n=1}^{\infty}$ possess the
C-property. Let us set

$\displaystyle\|f\|_{F_{n}}:=\|{\varphi}_{n}-f\|_{F_{n}}^{*},\qquad f\in
F_{n},$

where ${\varphi}_{n}$ is a fixed element of $F_{n},\,n\in{\mathbb{N}}$. Then
$\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ possesses the GC-property
with respect to $\left\\{\|\cdot\|_{F_{n}}^{*}\right\\}_{n=1}^{\infty}$.
Indeed, for any $h_{n}\in F_{n},\,g_{n}\in F_{n},\,n\in{\mathbb{N}}$, such
that

$\displaystyle\lim_{n\to{\infty}}\|g_{n}\|_{F_{n}}^{*}=\lim_{n\to{\infty}}\|-g_{n}\|_{F_{n}}^{*}=0,$

we have

$\displaystyle\liminf_{n\to{\infty}}\|h_{n}+g_{n}\|_{F_{n}}$ $\displaystyle=$
$\displaystyle\liminf_{n\to{\infty}}(\|{\varphi}_{n}-h_{n})+(-g_{n})\|_{F_{n}}^{*}$
(2.4) $\displaystyle=$
$\displaystyle\liminf_{n\to{\infty}}\|{\varphi}_{n}-h_{n}\|_{F_{n}}^{*}$
$\displaystyle=$ $\displaystyle\liminf_{n\to{\infty}}\|h_{n}\|_{F_{n}}.$

Similar relations hold true if we replace $\liminf$ in (2.4) with $\limsup$.
Hence (2.2) and (2.3) are valid.

On the other hand, since $F_{n}\neq\\{0\\},\,n\in{\mathbb{N}}$, then by
property (b) of Definition 2.1, it is always possible to choose
${\varphi}_{n}\in F_{n},\,n\in{\mathbb{N}}$, such that
$\lim_{n\to{\infty}}\|{\varphi}_{n}\|_{F_{n}}^{*}=a$, where $a\in(0,{\infty})$
is a fixed number. Then for $h_{n}:=(1/3){\varphi}_{n}$ and
$g_{n}:=(1-1/n){\varphi}_{n},\,n\in{\mathbb{N}}$, we have
$\lim_{n\to{\infty}}\|g_{n}\|_{F_{n}}=\lim_{n\to{\infty}}(1/n)\|{\varphi}_{n}\|_{F_{n}}^{*}=0$
and

$\displaystyle\lim_{n\to{\infty}}\|h_{n}+g_{n}\|_{F_{n}}=\lim_{n\to{\infty}}\|{\varphi}_{n}-h_{n}-g_{n}\|_{F_{n}}^{*}=a/3;\quad\lim_{n\to{\infty}}\|h_{n}\|_{F_{n}}=\lim_{n\to{\infty}}\|{\varphi}_{n}-h_{n}\|_{F_{n}}^{*}=2a/3.$

These relations show that in general
$\left\\{\|\cdot\|_{F_{n}}\right\\}_{n=1}^{\infty}$ does not possess the
C-property.

###  2.2. Sharp Constants

For vector spaces $F$ and $H$ equipped with the corresponding ”norms” and for
an operator $L:F\to H$, we define the ”norm” of $L$ by the formula

$\|L\|=\|L\|_{F\to H}:=\sup_{h\in
F,\,\|h\|_{F}>0}\frac{\|L(h)\|_{H}}{\|h\|_{F}}.$ (2.5)

If $B$ is a subset of $F$, we define the ”norm” $\|L\|$ of an operator $L:B\to
H$ on $B$ by the right hand side of (2.5) with $h\in F$ replaced by $h\in B$.

Let $F^{(j)}\neq\\{0\\}$ be a vector space over ${\mathbb{C}}$ or
${\mathbb{R}}$ equipped with a ”norm” $\|\cdot\|_{F^{(j)}},\,j=1,\,2,$ and let
$B$ be a nonempty subset of $F^{(2)}$. Next, let $D:B\to F^{(1)}$ be an
operator. In case of the _imbedding_ operator $D=I$, we assume that
$F^{(1)}\cap F^{(2)}\neq\emptyset$ and $B\subseteq F^{(1)}\cap F^{(2)}$.
Certainly, if $F^{(1)}=F^{(2)}$ and $\|\cdot\|_{F^{(1)}}=\|\cdot\|_{F^{(2)}}$,
then $I$ is the identity operator.

Let us define the following _sharp constant_ :

${\mathcal{C}}={\mathcal{C}}\left(D,B,F^{(1)},F^{(2)}\right):=\sup_{f\in
B,\,\|f\|_{F^{(2)}}>0}\frac{\|D(f)\|_{F^{(1)}}}{\|f\|_{F^{(2)}}},$ (2.6)

which is the ”norm” of $D$ on $B$. Clearly, ${\mathcal{C}}\in[0,{\infty}]$.

In certain cases the following formulae for ${\mathcal{C}}$ are valid:

$\displaystyle{\mathcal{C}}=\sup_{f\in
B,\,\|f\|_{F^{(2)}}=1}\|D(f)\|_{F^{(1)}},$ (2.7)
$\displaystyle{\mathcal{C}}=\sup_{f\in
B,\,\|f\|_{F^{(2)}}>0,\,\|D(f)\|_{F^{(1)}}=1}\frac{1}{\|f\|_{F^{(2)}}}.$ (2.8)

In particular, it is easy to verify the following proposition.

######  Proposition 2.6.

(i) Let $B$ be a subspace of $F^{(2)},\,D$ be a $1$-homogeneous operator (that
is, $D({\alpha}f)={\alpha}D(f),\,{\alpha}>0$), and ”norms”
$\|\cdot\|_{F^{(j)}},\,j=1,\,2,$ satisfy the homogeneity property (b). Then
relations (2.7) and (2.8) hold true.
(ii) If $\|\cdot\|_{F^{(2)}}=\|\cdot\|_{F^{(2)},\mathrm{triv}}$, then (2.7)
holds true.
(iii) If $\|\cdot\|_{F^{(1)}}=\|\cdot\|_{F^{(1)},\mathrm{triv}}$, then (2.8)
holds true.

###  2.3. Sharp Constants in Approximation Theory and Analysis

It turns out that the problem of finding
${\mathcal{C}}={\mathcal{C}}\left(D,B,F^{(1)},F^{(2)}\right)$ includes most
classical problems of approximation theory that are related to sharp
constants. In addition, it includes certain analysis problems as well. Below
we first discuss three major approximation problems A, B, and C as special
cases of the problem of finding ${\mathcal{C}}$.
_A. Inequalities for Approximating Elements._ Let $F^{(j)}$ be a
$\beta_{j}$-normed functional space, $j=1,2;\,B$ be a nontrivial subspace of
$F^{(2)}$; and $D$ be a linear (often differential) operator. Then
${\mathcal{C}}$ is the sharp constant in various inequalities for
approximating elements that we call _Markov-Bernstein type inequalities of
different metrics._ This is one of the most studied topics of approximation
theory. Several special problems are discussed below.
_A1. Markov-Bernstein Type Inequalities for
$\|\cdot\|_{F^{(1)}}=\|\cdot\|_{F^{(2)}}$._
_A1.1. A. A. Markov and V. A. Markov type inequalities_ for the set $B$ of
algebraic polynomials of one or several variables.
_A1.2. Bernstein type inequalities_ for the set $B$ of trigonometric
polynomials, splines, or entire functions of exponential type of one or
several variables.
_A2. Inequalities of Different Metrics for $D=I$ and
$\|\cdot\|_{F^{(1)}}\neq\|\cdot\|_{F^{(2)}}$._
_A2.1. Nikolskii type inequalities_ for the set $B$ of trigonometric or
algebraic polynomials, splines, or entire functions of exponential type of one
or several variables.
_A2.2. Schur type inequalities_ for a nonweighted space $F^{(1)}$, a weighted
space $F^{(2)}$, and the set $B$ of trigonometric or algebraic polynomials or
entire functions of exponential type of one or several variables.
_A2.3. Remez type inequalities_ for a weighted space $F^{(2)}$ when a weight
vanishes outside of a measurable set and for the set $B$ of trigonometric,
algebraic, or Müntz polynomials.

There are numerous publications related to all these inequalities, see, e.g.,
[74, 76, 60, 15, 53, 55, 33, 6, 47, 24, 27, 4, 25, 38, 41, 5, 42, 43] and
references therein.
_B. Approximation of Individual Elements._ Let us define the operator $D:B\to
F^{(1)}$ arbitrarily with the only restriction that $D(f)\neq 0$ for $f\in B$
and let us choose $\|\cdot\|_{F^{(1)}}=\|\cdot\|_{F^{(1)},\mathrm{triv}}.$ We
denote by $F^{(2)}_{*}$ the vector space $F^{(2)}$ equipped with a certain
${\beta}_{2}$-norm $\|\cdot\|^{*}_{F^{(2)}}$ on
$F^{(2)},\,{\beta}_{2}\in(0,1]$. For a fixed element ${\varphi}\in
F^{(2)}_{*}\setminus B$ we choose
$\|f\|_{F^{(2)}}=\|{\varphi}-f\|^{*}_{F^{(2)}},\,f\in F^{(2)}_{*}$. Then

${\mathcal{C}}=\frac{1}{\inf_{f\in B,\,f\neq
0}\|{\varphi}-f\|^{*}_{F^{(2)}}}=\frac{1}{E\left({\varphi},B,F_{*}^{(2)}\right)},$
(2.9)

where $E\left({\varphi},B,F_{*}^{(2)}\right)$ is the error of best
approximation by elements from $B$ in the space $F^{(2)}_{*}$. To have
${\mathcal{C}}<{\infty}$, we assume that
$E\left({\varphi},B,F_{*}^{(2)}\right)>0$.

So the problem of finding ${\mathcal{C}}$ is reduced to finding sharp or
asymptotically sharp value of $E\left({\varphi},B,F_{*}^{(2)}\right)$. These
problems have attracted much attention in classical and contemporary
approximation theory, having started with classical works by Chebyshev, A. A.
Markov, V. A. Markov, Zolotarev, and Bernstein (see, e.g., [74, 1, 23, 60,
15]). Much contributions to this topic, including asymptotic relations (1.1)
through (1.4) and their generalizations, have been made in [7, 8, 11, 13, 74,
1, 29, 30, 31, 32, 70, 37, 20, 19] and references therein.
_C. Approximation on Classes of Elements._ Let $F^{(1)}=F^{(2)},\,D=I$, and
let us choose $\|\cdot\|_{F^{(2)}}=\|\cdot\|_{F^{(2)},\mathrm{triv}}.$ Let
$\|\cdot\|_{F^{(1)}}$ be a $\beta_{1}$-seminorm of the form

$\displaystyle\|f\|_{F^{(1)}}:=E\left(f,G,F^{(1)}_{*}\right)=\inf_{g\in
G}\|f-g\|^{*}_{F^{(1)}},\qquad f\in F^{(1)}_{*},$

where $G$ is a subspace of $F^{(1)}$ and $F^{(1)}_{*}$ is a vector space
$F^{(1)}$ equipped with a certain $\beta_{1}$-norm $\|\cdot\|^{*}_{F^{(1)}}$
on $F^{(1)},\,{\beta}_{1}\in(0,1]$. Then ${\mathcal{C}}=\sup_{f\in
B}E\left(f,G,F^{(1)}_{*}\right)$ is the sharp constant of approximation by
elements from $G$ in the $\beta_{1}$-normed space $F_{*}^{(1)}$ on a class
$B$. Sharp or asymptotically sharp values of this constant of approximation by
trigonometric or algebraic polynomials, splines, and entire functions of
exponential type have been found since the 1930s for various classes, see,
e.g., [2, 62, 9, 12, 74, 1, 51, 26, 52, 31, 32, 35, 39, 40] and references
therein.
We also discuss the following important problem in analysis.
_D. Norms of Operators._ If $F^{(j)}$ is a $\beta_{j}$-normed space,
$j=1,\,2,$ and $B=F^{(2)}$, then ${\mathcal{C}}=\|D\|$ is the ”norm” of the
operator $D:F^{(2)}\to F^{(1)}$. Finding or estimating ${\mathcal{C}}$ is one
of the major problems in functional analysis, Fourier analysis, real and
complex analysis, and other areas. Below we discuss just a few examples of
such a problem.
_D1. Lebesgue constants in Fourier analysis_ , see, e.g., [77, Vol. I, Sect.
2.12] and [59].
_D2. Summability methods_ , see, e.g., [74, Ch. 8] and [75, Ch. 8].
_D3. Hausdorf-Young-F. Riesz inequalities_ , see, e.g., [77, Vol. 2, Sects.
12.2 and 16.3].
_D4. Embedding and extension operators_ , see, e.g., [65, 18, 17].
_D5. Hilbert and maximal Hardy-Littlewood operators_ , see, e.g., [45, Sects
4.3 and 6.6].

##  3\. Problems and Conditions

Here, we discuss major objectives of this paper and introduce conditions that
are needed for our results.

###  3.1. Problems

Let

$\displaystyle{\mathcal{C}}:={\mathcal{C}}\left(D,B,F^{(1)},F^{(2)}\right):=\sup_{f\in
B,\,\|f\|_{F^{(2)}}>0}\frac{\|D(f)\|_{F^{(1)}}}{\|f\|_{F^{(2)}}},$ (3.1)
$\displaystyle{\mathcal{C}}_{n}:={\mathcal{C}}\left(D_{n},B_{n},F^{(1)}_{n},F^{(2)}_{n}\right):=\sup_{f_{n}\in
B_{n},\,\|f_{n}\|_{F^{(2)}_{n}}>0}\frac{\|D_{n}(f_{n})\|_{F^{(1)}_{n}}}{\|f_{n}\|_{F^{(2)}_{n}}}$
(3.2)

be two sharp constants over two families of components
$\left\\{D,B,F^{(1)},F^{(2)}\right\\}$ and
$\left\\{D_{n},B_{n},F^{(1)}_{n},F^{(2)}_{n}\right\\}$ with a running
parameter $n\in{\mathbb{N}}$ attached to the second family.

Our goal is to obtain asymptotic relations of the form:

$\displaystyle{\mathcal{C}}\leq\liminf_{n\to{\infty}}{\mathcal{C}}_{n},$ (3.3)
$\displaystyle{\mathcal{C}}\geq\limsup_{n\to{\infty}}{\mathcal{C}}_{n},$ (3.4)
$\displaystyle{\mathcal{C}}=\lim_{n\to{\infty}}{\mathcal{C}}_{n}.$ (3.5)

In special cases when ${\mathcal{C}}$ and ${\mathcal{C}}_{n}$ are reduced to
approximation of individual elements or approximation on classes of elements,
relations like (3.3)–(3.5) were obtained by Bernstein [9, 10, 11, 12, 13] for
approximation by univariate trigonometric or algebraic polynomials and entire
functions of exponential type. The author extended these results to
multivariate approximation [29, 30] and to a more general setting [31, 32].
Burenkov and Goldman [18] discussed relations like (3.3)–(3.5) between norms
of linear operators on periodic and nonperiodic spaces. Relations like
(3.3)–(3.5) in Markov-Bernstein type inequalities of different metrics for
trigonometric and algebraic polynomials and entire functions of exponential
type were proved by Taikov [72, 73], Gorbachev [47, 48], Levin and Lubinsky
[57, 58], the author and Tikhonov [44], Dai, Gorbachev, and Tikhonov [21, 22],
and the author [38, 41, 42, 43].

Below we introduce some conditions on families
$\left\\{D,B,F^{(1)},F^{(2)}\right\\}$ and
$\left\\{D_{n},B_{n},F^{(1)}_{n},F^{(2)}_{n}\right\\},\,n\in{\mathbb{N}}$,
that are needed for the proof of relations (3.3)–(3.5).

###  3.2. Conditions and their Properties

The first group of conditions is needed for the proof of relation (3.3).
_Condition C1)._ A space $F^{(1)}_{n}$ is equipped with another ”norm”
$\|\cdot\|_{F^{(1)}_{n}}^{*},\,n\in{\mathbb{N}},$ and
$\left\\{\|\cdot\|_{F^{(1)}_{n}}\right\\}_{n=1}^{\infty}$ possesses the GC-
property with respect to
$\left\\{\|\cdot\|_{F^{(1)}_{n}}^{*}\right\\}_{n=1}^{\infty}$ (see Definition
2.4).
_Condition C2)._ There exists a sequence of operators $L_{n}^{(1)}:F^{(1)}\to
F_{n}^{(1)},\,n\in{\mathbb{N}}$, such that the following two conditions hold:
_Condition C3)._ For every $f\in B$ with $\|f\|_{F^{(2)}}>0,$ there exists a
sequence $f_{n}\in B_{n},\,n\in{\mathbb{N}}$, such that

$\limsup_{n\to{\infty}}\|f_{n}\|_{F_{n}^{(2)}}\leq\|f\|_{F^{(2)}}$ (3.6)

and

$\lim_{n\to{\infty}}\left\|L_{n}^{(1)}(D(f))-D_{n}(f_{n})\right\|_{F_{n}^{(1)}}^{*}=0.$
(3.7)

_Condition C4)._ For every $h\in F^{(1)}$,

$\liminf_{n\to{\infty}}\left\|L_{n}^{(1)}(h)\right\|_{F_{n}^{(1)}}\geq\|h\|_{F^{(1)}}.$
(3.8)

In certain cases a slightly different condition is needed.
_Condition C4´)._ Let $B\subseteq F^{(1)}\cap F^{(2)}$ and $D(B)\subseteq B$.
Then for every $h\in B$, inequality (3.8) holds true.

A group of stronger conditions marked with asterisks is given below.
_Condition C1*)._ A space $F^{(j)}_{n}$ is equipped with another ”norm”
$\|\cdot\|_{F^{(j)}_{n}}^{*},\,n\in{\mathbb{N}},$ and
$\left\\{\|\cdot\|_{F^{(j)}_{n}}\right\\}_{n=1}^{\infty}$ possesses the GC-
property with respect to
$\left\\{\|\cdot\|_{F^{(j)}_{n}}^{*}\right\\}_{n=1}^{\infty}$, j=1, 2 (see
Definition 2.4).
_Condition C2*)._ There exist two sequences of operators
$L_{n}^{(j)}:F^{(j)}\to F_{n}^{(j)},\,n\in{\mathbb{N}},\,j=1,2$, such that the
following two conditions hold:
Condition C3*). For every $f\in B$ with $\|f\|_{F^{(2)}}>0,$ there exists a
sequence $f_{n}\in B_{n},\,n\in{\mathbb{N}}$, such that

$\displaystyle\lim_{n\to{\infty}}\left\|L_{n}^{(1)}(D(f))-D_{n}(f_{n})\right\|_{F_{n}^{(1)}}^{*}=0,$
(3.9)
$\displaystyle\lim_{n\to{\infty}}\left\|f_{n}-L_{n}^{(2)}(f)\right\|_{F_{n}^{(2)}}^{*}=0.$
(3.10)

_Condition C4*)._ For every $h\in F^{(2)}$ with $\|h\|_{F^{(2)}}>0$,

$\limsup_{n\to{\infty}}\left\|L_{n}^{(2)}(h)\right\|_{F_{n}^{(2)}}\leq\|h\|_{F^{(2)}}.$
(3.11)

######  Proposition 3.1.

Conditions C1*) through C4*) imply Conditions C1) through C3).

###### Proof.

It suffices to show that C1*), C2*), (3.10) of C3*), and (3.11) of C4*) imply
(3.6). By C2*), C3*), and C4*), for every $f\in B$ there exists a sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ that satisfies (3.10), where
$\left\\{L_{n}^{(2)}\right\\}_{n=1}^{\infty}$ satisfies (3.11). Then using
C1*), (3.10), and (3.11), we obtain

$\displaystyle\limsup_{n\to{\infty}}\|f_{n}\|_{F_{n}^{(2)}}=\limsup_{n\to{\infty}}\left\|L_{n}^{(2)}(f)+\left(f_{n}-L_{n}^{(2)}(f)\right)\right\|_{F_{n}^{(2)}}=\limsup_{n\to{\infty}}\left\|L_{n}^{(2)}(f)\right\|_{F_{n}^{(2)}}\leq\|f\|_{F^{(2)}}.$

Hence (3.6) follows. $\Box$
Next, we introduce the second group of conditions that is needed for the proof
of relation (3.4).
_Condition C5)._ A space $F^{(j)}_{n}$ is equipped with another ”norm”
$\|\cdot\|_{F^{(j)}_{n}}^{*},\,n\in{\mathbb{N}},$ and
$\left\\{\|\cdot\|_{F^{(j)}_{n}}\right\\}_{n=1}^{\infty}$ possesses the GC-
property with respect to
$\left\\{\|\cdot\|_{F^{(j)}_{n}}^{*}\right\\}_{n=1}^{\infty}$, j=1, 2 (see
Definition 2.4).
_Condition C6)._ There exist two sequences of operators
$l_{n}^{(j)}:F^{(j)}\to F_{n}^{(j)},\,n\in{\mathbb{N}},\,j=1,2$, such that the
following two properties hold:
_Condition C6a)._ For every $h\in F^{(1)}$ with $\|h\|_{F^{(1)}}>0$,

$\limsup_{n\to{\infty}}\left\|l_{n}^{(1)}(h)\right\|_{F_{n}^{(1)}}\leq\|h\|_{F^{(1)}}.$
(3.12)

_Condition C6b)._ For every $h\in F^{(2)}$,

$\liminf_{n\to{\infty}}\left\|l_{n}^{(2)}(h)\right\|_{F^{(2)}_{n}}\geq\|h\|_{F^{(2)}}.$
(3.13)

_Condition C7)._ There is a constant $C_{1}\in(0,{\infty})$ such that for
every $h\in F^{(2)}$ with $\|h\|_{F^{(2)}}>0$,

$\limsup_{n\to{\infty}}\left\|l_{n}^{(2)}(h)\right\|_{F_{n}^{(2)}}\leq
C_{1}\|h\|_{F^{(2)}}.$ (3.14)

Next, we need one of the following two conditions associated with relations
like (2.7) and (2.8).
_Condition C8)._ There is a set $B_{n}^{\prime}\subseteq\left\\{f_{n}\in
B_{n}:\|f_{n}\|_{F^{(2)}_{n}}=1\right\\}$ such that the following relations
hold:

$\displaystyle{\mathcal{C}}_{n}=\sup_{f_{n}\in
B_{n}^{\prime}}\|D_{n}(f_{n})\|_{F^{(1)}_{n}}=\sup_{f_{n}\in
B_{n}^{\prime}}\frac{\|D_{n}(f_{n})\|_{F_{n}^{(1)}}}{\|f_{n}\|_{F_{n}^{(2)}}},\qquad
n\in{\mathbb{N}}.$ (3.15)

_Condition C9)._ There is a set $B_{n}^{\prime}\subseteq\left\\{f_{n}\in
B_{n}:\|f_{n}\|_{F^{(2)}_{n}}>0,\,\|D_{n}(f_{n})\|_{F^{(1)}_{n}}=1\right\\}$
such that the following relations hold:

$\displaystyle{\mathcal{C}}_{n}=\sup_{f_{n}\in
B_{n}^{\prime}}\frac{1}{\|f_{n}\|_{F^{(2)}_{n}}}=\sup_{f_{n}\in
B_{n}^{\prime}}\frac{\|D_{n}(f_{n})\|_{F_{n}^{(1)}}}{\|f_{n}\|_{F_{n}^{(2)}}},\qquad
n\in{\mathbb{N}}.$ (3.16)

_Condition C10)._ (Compactness condition) For any sequence
$\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}\in
B_{n}^{\prime}\setminus\\{0\\},\,n\in{\mathbb{N}}$, there exist a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers and an element $f\in B$ such
that

$\displaystyle\lim_{k\to{\infty}}\left\|D_{n_{k}}\left(f_{n_{k}}\right)-l_{n_{k}}^{(1)}(D(f))\right\|_{F_{n_{k}}^{(1)}}^{*}=0,$
(3.17)
$\displaystyle\lim_{k\to{\infty}}\left\|f_{n_{k}}-l_{n_{k}}^{(2)}(f)\right\|_{F_{n_{k}}^{(2)}}^{*}=0.$
(3.18)

Further, we discuss a different group of conditions with a weak compactness
condition that replaces C10).
_Condition C11)._ There exist two families of operators
$l_{n,s}^{(j)}:F_{n}^{(j)}\to
F_{s}^{(j)},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}},\,j=1,2$, such that the
following property holds:
_Condition C11a)._

$\sup_{n\in{\mathbb{N}},\,s\in{\mathbb{N}}}\left\|l_{n,s}^{(2)}\right\|\leq
1.$ (3.19)

We recall that the ”norm” of an operator is defined by (2.5).
_Condition C12)._ (Weak compactness condition) For any sequence
$\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}\in B_{n}^{\prime},\,n\in{\mathbb{N}}$,
there exists $f\in B$ such that for any fixed $s\in{\mathbb{N}}$, there exists
a sequence $\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers
($n_{k}=n_{k}(s),\,k\in{\mathbb{N}}$) such that

$\displaystyle\lim_{k\to{\infty}}\left\|l_{n_{k},s}^{(1)}\left(D_{n_{k}}\left(f_{n_{k}}\right)\right)-l_{s}^{(1)}(D(f))\right\|_{F_{s}^{(1)}}^{*}=0,$
(3.20)
$\displaystyle\lim_{k\to{\infty}}\left\|l_{n_{k},s}^{(2)}(f_{n_{k}})-l_{s}^{(2)}(f)\right\|_{F_{s}^{(2)}}^{*}=0,$
(3.21)

and, in addition,

$\left\|D_{n_{k}}\left(f_{n_{k}}\right)\right\|_{F_{n_{k}}^{(1)}}=\left\|l_{n_{k},s}^{(1)}\left(D_{n_{k}}\left(f_{n_{k}}\right)\right)\right\|_{F_{s}^{(1)}},\quad
k\in{\mathbb{N}}.$ (3.22)

_Condition C13)._ There is a constant $C_{2}\in(0,{\infty})$ such that for
every $h_{n}\in F_{n}^{(2)}$,

$\limsup_{n\to{\infty}}\|h_{n}\|_{F_{n}^{(2)}}\leq
C_{2}\limsup_{s\to{\infty}}\limsup_{n\to{\infty}}\left\|l_{n,s}^{(2)}(h_{n})\right\|_{F_{s}^{(2)}}.$
(3.23)

In Section 5.3 we will need modified conditions C6a), C10), and C12) marked
with asterisks below.
_Condition C6*a)._ There exists a sequence of operators
$l_{n}^{(1)}:F^{(1)}\to F_{n}^{(1)},\,n\in{\mathbb{N}}$, such that for every
$h\in F^{(1)}$ with $\|h\|_{F^{(1)}}>0$,

$\limsup_{n\to{\infty}}\left\|l_{n}^{(1)}(h)\right\|_{F_{n}^{(1)}}\leq\|h\|_{F^{(1)}}.$
(3.24)

_Condition C10*a)._ For any sequence $\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}\in
B_{n}^{\prime}\setminus\\{0\\},\,n\in{\mathbb{N}}$, there exists a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers and an element $f\in B$ such
that

$\lim_{k\to{\infty}}\left\|D_{n_{k}}\left(f_{n_{k}}\right)-l_{n_{k}}^{(1)}(D(f))\right\|_{F_{n_{k}}^{(1)}}^{*}=0.$
(3.25)

_Condition C10*b)._ For any sequence $\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}\in
B_{n}^{\prime}\setminus\\{0\\},\,n\in{\mathbb{N}}$, for the sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers from Condition C10*a), and for
the element $f\in B$ from Condition C10*a), there exists a sequence of
operators $l_{n}^{(2)}:F^{(2)}\to F_{n}^{(2)},\,n\in{\mathbb{N}}$, such that
Conditions C6b) and C7) are satisfied. In addition, the following relation
holds:

$\lim_{k\to{\infty}}\left\|f_{n_{k}}-l_{n_{k}}^{(2)}(f)\right\|_{F_{n_{k}}^{(2)}}^{*}=0.$
(3.26)

_Condition C12*a)._ There exists a family of operators
$l_{n,s}^{(1)}:F_{n}^{(1)}\to
F_{s}^{(1)},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}}$, such that for any sequence
$\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}\in
B_{n}^{\prime}\setminus\\{0\\},\,n\in{\mathbb{N}}$, there exists $f\in B$ such
that for any fixed $s\in{\mathbb{N}}$, there exists a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers
($n_{k}=n_{k}(s),\,k\in{\mathbb{N}}$) such that

$\displaystyle\lim_{k\to{\infty}}\left\|l_{n_{k},s}^{(1)}\left(D_{n_{k}}\left(f_{n_{k}}\right)\right)-l_{s}^{(1)}(D(f))\right\|_{F_{s}^{(1)}}^{*}=0,$
(3.27)
$\displaystyle\left\|D_{n_{k}}\left(f_{n_{k}}\right)\right\|_{F_{n_{k}}^{(1)}}=\left\|l_{n_{k},s}^{(1)}\left(D_{n_{k}}\left(f_{n_{k}}\right)\right)\right\|_{F_{s}^{(1)}},\quad
k\in{\mathbb{N}}.$ (3.28)

_Condition C12*b)._ For any sequence $\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}\in
B_{n}^{\prime}\setminus\\{0\\},\,n\in{\mathbb{N}}$, for the sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers from Condition C12*a) and the
element $f\in B$ from Condition C12*a), there exists a sequence of operators
$l_{n}^{(2)}:F^{(2)}\to F_{n}^{(2)},\,n\in{\mathbb{N}}$, such that Conditions
C6b) and C7) are satisfied and there exists a sequence of operators
$l_{n,s}^{(2)}:F_{n}^{(2)}\to
F_{s}^{(2)},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}}$, such that Conditions C11a)
and C13) are satisfied. In addition, the following relation holds:

$\lim_{k\to{\infty}}\left\|l_{n_{k},s}^{(2)}(f_{n_{k}})-l_{s}^{(2)}(f)\right\|_{F_{s}^{(2)}}^{*}=0,\qquad
s\in{\mathbb{N}}.$ (3.29)

######  Remark 3.2.

Condition C4´) includes inequality (3.8) with a weaker requirement $h\in B$,
so if $B\subseteq F^{(1)}\cap F^{(2)}$ and $D(B)\subseteq B$ (for example, if
$D=I$ is an imbedding operator), then Condition C4´) can efficiently replace
Condition C4), see Theorem 4.2 and Corollaries 4.3, 5.1 (i), 5.5 (i) for
details.

######  Remark 3.3.

Modified compactness conditions C10*b) and C12*b) are characterized by the
dependence of operators $\left\\{l_{n}^{(2)}\right\\}_{n\in{\mathbb{N}}}$ and
$\left\\{l_{n,s}^{(2)}\right\\}_{n\in{\mathbb{N}},s\in{\mathbb{N}}}$ on a
sequence $\\{f_{n}\\}_{n=1}^{\infty}$ ($f_{n}\in
B_{n}^{\prime}\setminus\\{0\\},\,n\in{\mathbb{N}}$), the sequence
$\\{n_{k}\\}_{k=1}^{\infty}$, and $f\in B$, while these operators are defined
in C6) and C11) independently of C10) and C12).

######  Remark 3.4.

The set $B_{n}^{\prime},\,n\in{\mathbb{N}}$, mentioned in the Compactness
Conditions C10), C12), C10*a), C10*b) C12*a), and C12*b) is defined in C8) or
C9). In other words, if any of these Compactness Conditions is satisfied, then
either C8) or C9) must be satisfied as well. Actually, Conditions C8) and C9)
state only the existence of $B_{n}^{\prime},\,n\in{\mathbb{N}}$. So these sets
should be defined for each concrete problem separately (see Chapters 5 and 7).

######  Remark 3.5.

Conditions C4*), C6a), C6*a), and C7) can be replaced by stronger conditions
for the ”norms” of the corresponding operators. For example, if the condition
$\limsup_{n\to{\infty}}\left\|L_{n}^{(2)}\right\|\leq 1$ is satisfied, then
C4*) is satisfied as well. However in general, C4*) does not imply this
condition.

Several conditions defined above are satisfied for certain special ”norms.”

######  Proposition 3.6.

Let $j=1$ or $j=2$ and let ${\beta}_{j}\in(0,1]$ be independent of
$n\in{\mathbb{N}}$. Next, let $F_{n}^{(j)}$ be equipped with two ”norms”
$\left\|\cdot\right\|_{F_{n}^{(j)}}$ and
$\left\|\cdot\right\|_{F_{n}^{(j)}}^{*},\,n\in{\mathbb{N}}$ (see Conditions
C1), C1*), and C5)). In addition, let
$\left\|\cdot\right\|_{F_{n}^{(j)}},\,n\in{\mathbb{N}}$, be one of the
following ”norms:”

  * a)

$\left\|\cdot\right\|_{F_{n}^{(j)}}=\left\|\cdot\right\|_{F_{n}^{(j)}}^{*}$ is
a ${\beta}_{j}$-norm or a ${\beta}_{j}$-seminorm, $n\in{\mathbb{N}}$.

  * b)

$\left\|\cdot\right\|_{F_{n}^{(j)}}=\left\|\cdot\right\|_{F_{n}^{(j)}}^{*}:=\|\cdot\|_{F_{n}^{(j)},\mathrm{triv}},\,n\in{\mathbb{N}}$.

  * c)

$\left\|f_{n}\right\|_{F_{n}^{(j)}}=\|{\varphi}_{n}-f_{n}\|_{F_{n}^{(j)}}^{*},\,f_{n}\in
F_{n}^{(j)}$, where ${\varphi}_{n}$ is a fixed element of $F_{n}^{(j)}$, and
$\|\cdot\|_{F_{n}^{(j)}}^{*}$ is a ${\beta}_{j}$-norm or a
${\beta}_{j}$-seminorm, $n\in{\mathbb{N}}$.

Then the following statements are valid:
(i) If $j=1$, then Condition C1) is satisfied.
(ii) If $j=1,\,2$, then Conditions C1*) and C5) are satisfied.

###### Proof.

In case of ”norms” a) and b), the corresponding sequence
$\left\\{\|\cdot\|_{F_{n}^{(j)}}\right\\}_{n=1}^{\infty}=\left\\{\|\cdot\|_{F_{n}^{(j)}}^{*}\right\\}_{n=1}^{\infty}$
possesses the C-property by statements (ii) and (iii) of Proposition 2.3. In
case of ”norm” c), the sequence
$\left\\{\|\cdot\|^{*}_{F_{n}^{(j)}}\right\\}_{n=1}^{\infty}$ possesses the
C-property by statement (ii) of Proposition 2.3. Then
$\left\\{\|\cdot\|_{F_{n}^{(j)}}\right\\}_{n=1}^{\infty}$ possesses the GC-
property with respect to
$\left\\{\|\cdot\|^{*}_{F_{n}^{(j)}}\right\\}_{n=1}^{\infty}$. Indeed, like in
Example 2.5, for any $h_{n}\in F_{n}^{(j)},\,g_{n}\in
F_{n}^{(j)},\,n\in{\mathbb{N}}$, such that

$\displaystyle\lim_{n\to{\infty}}\|g_{n}\|_{F_{n}^{(j)}}^{*}=\lim_{n\to{\infty}}\|-g_{n}\|_{F_{n}^{(j)}}^{*}=0,$

we have

$\displaystyle\liminf_{n\to{\infty}}\|h_{n}+g_{n}\|_{F_{n}^{(j)}}$
$\displaystyle=$
$\displaystyle\liminf_{n\to{\infty}}(\|{\varphi}_{n}-h_{n})+(-g_{n})\|_{F_{n}^{(j)}}^{*}$
(3.30) $\displaystyle=$
$\displaystyle\liminf_{n\to{\infty}}\|{\varphi}_{n}-h_{n}\|_{F_{n}^{(j)}}^{*}$
$\displaystyle=$ $\displaystyle\liminf_{n\to{\infty}}\|h_{n}\|_{F_{n}^{(j)}}.$

Similar relations hold true if we replace $\liminf$ in (3.30) with $\limsup$.
Hence (2.2) and (2.3) are valid for $F_{n}$ replaced with $F_{n}^{(j)}$.
$\Box$

######  Proposition 3.7.

(i) Let $F_{n}^{(j)},\,n\in{\mathbb{N}}$, be a ${\beta}_{j}$-normed or
${\beta}_{j}$-seminormed space, where ${\beta}_{j}$ is independent of
$n,\,j=1,2.$ In addition, let $B_{n}$ be a subspace of $F_{n}^{(2)}$ and let
$D_{n}$ be a $1$-homogeneous operator (that is,
$D_{n}({\alpha}f)={\alpha}D_{n}(f),\,{\alpha}>0$), $n\in{\mathbb{N}}$. Then
Conditions C8) and C9) are satisfied.
(ii) If
$\left\|\cdot\right\|_{F_{n}^{(2)}}=\|\cdot\|_{F_{n}^{(2)},\mathrm{triv}},\,n\in{\mathbb{N}}$,
then C8) is satisfied.
(iii) If
$\left\|\cdot\right\|_{F_{n}^{(1)}}=\|\cdot\|_{F_{n}^{(1)},\mathrm{triv}},\,n\in{\mathbb{N}}$,
then C9) is satisfied.

###### Proof.

Let us define a set $G_{n},\,n\in{\mathbb{N}}$, by

$\displaystyle G_{n}:=\left\\{\begin{array}[]{ll}\left\\{f_{n}\in
B_{n}:\|f_{n}\|_{F^{(2)}_{n}}=1\right\\}&\mbox{for Condition C8)},\\\
\left\\{f_{n}\in
B_{n}:\|f_{n}\|_{F^{(2)}_{n}}>0,\,\|D_{n}(f_{n})\|_{F^{(1)}_{n}}=1\right\\}&\mbox{for
Condition C9)}.\end{array}\right.$

It follows immediately from Proposition 2.6 that relation (3.15) or relation
(3.16) holds true with $B_{n}^{\prime}$ replaced by
$G_{n},\,n\in{\mathbb{N}}$. Therefore, there exists $B_{n}^{\prime}\subseteq
G_{n},\,n\in{\mathbb{N}}$, such that (3.15) or (3.16) is valid for all
premises of statements (i), (ii), and (iii) of Proposition 3.7 $\Box$

##  4\. Asymptotic Relations between Sharp Constants in Vector Spaces

Here, we prove asymptotic relations (3.3), (3.4), and (3.5). We first discuss
asymptotic relations of the form

${\mathcal{C}}={\mathcal{C}}\left(D,B,F^{(1)},F^{(2)}\right)\leq\liminf_{n\to{\infty}}{\mathcal{C}}\left(D_{n},B_{n},F^{(1)}_{n},F^{(2)}_{n}\right)=\liminf_{n\to{\infty}}{\mathcal{C}}_{n}.$
(4.1)

######  Theorem 4.1.

If Conditions C1) through C4) are satisfied, then (4.1) holds true.

###### Proof.

Let $f\in B$ be a fixed element with $\|f\|_{F^{(2)}}>0$. Then $D(f)\in
F^{(1)}$ and using Condition C2) and inequality (3.8) of Condition C4), we
have

$\|D(f)\|_{F^{(1)}}\leq\liminf_{n\to{\infty}}\left\|L_{n}^{(1)}(D(f))\right\|_{F_{n}^{(1)}}.$
(4.2)

Next by C2) and C3), there exists a sequence $f_{n}\in
B_{n},n\in{\mathbb{N}},$ such that (3.6) and (3.7) hold. Then using C1), we
obtain from (4.2), (3.7), and (3.6)

$\displaystyle\|D(f)\|_{F^{(1)}}\leq\liminf_{n\to{\infty}}\left\|D_{n}(f_{n})+\left(L_{n}^{(1)}(D(f))-D_{n}(f_{n})\right)\right\|_{F_{n}^{(1)}}=\liminf_{n\to{\infty}}\left\|D_{n}(f_{n})\right\|_{F_{n}^{(1)}}$
$\displaystyle\leq\liminf_{n\to{\infty}}\left(\|f_{n}\|_{F^{(2)}_{n}}{\mathcal{C}}_{n}\right)\leq\limsup_{n\to{\infty}}\|f_{n}\|_{F^{(2)}_{n}}\liminf_{n\to{\infty}}{\mathcal{C}}_{n}\leq\|f\|_{F^{(2)}}\liminf_{n\to{\infty}}{\mathcal{C}}_{n}.$

Therefore,
$\|D(f)\|_{F^{(1)}}/\|f\|_{F^{(2)}}\leq\liminf_{n\to{\infty}}{\mathcal{C}}_{n}$
and (4.1) is established. $\Box$
The following theorem can be proved similarly to Theorem 4.1.

######  Theorem 4.2.

If Conditions C1) through C3) and C4´) are satisfied, then (4.1) holds true.

######  Corollary 4.3.

If Conditions C4) (or C4´)) and C1*) through C4*) are satisfied, then (4.1)
holds true.

The corollary follows immediately from Theorems 4.1, 4.2, and Proposition 3.1.

Next, we discuss asymptotic relations of the form

${\mathcal{C}}={\mathcal{C}}\left(D,B,F^{(1)},F^{(2)}\right)\geq\limsup_{n\to{\infty}}{\mathcal{C}}\left(D_{n},B_{n},F^{(1)}_{n},F^{(2)}_{n}\right)=\limsup_{n\to{\infty}}{\mathcal{C}}_{n}.$
(4.3)

In the following Theorems 4.4, 4.5, and 4.7 we take account of Remark 3.4.

######  Theorem 4.4.

Let ${\mathcal{C}}_{n}$ be a finite number for every $n\in{\mathbb{N}}$, and
let either Conditions C7), C8) or Condition C9) be satisfied. In addition, let
C5), C6), and C10) be satisfied. Then (4.3) holds true.

###### Proof.

Assume without loss of generality that ${\mathcal{C}}<{\infty}$. Next, we
assume that Conditions C7) and C8) are satisfied. Since
${\mathcal{C}}_{n}<{\infty}$, we see from C8) that for any fixed
${\varepsilon}>0$ there exists $f_{n}=f_{n,{\varepsilon}}\in B_{n}^{\prime}$
with $\|f_{n}\|_{F^{(2)}_{n}}=1$ such that

${\mathcal{C}}_{n}<\frac{\|D_{n}(f_{n})\|_{F_{n}^{(1)}}}{\|f_{n}\|_{F_{n}^{(2)}}}+{\varepsilon},\qquad
n\in{\mathbb{N}}.$ (4.4)

If Condition C9) is satisfied, then for any fixed ${\varepsilon}>0$ there
exists $f_{n}=f_{n,{\varepsilon}}\in B_{n}^{\prime}$ with
$\|f_{n}\|_{F^{(2)}_{n}}>0$ and $\|D_{n}(f_{n})\|_{F^{(1)}_{n}}=1$ such that
(4.4) holds true as well.

Let $\\{n_{p}\\}_{p=1}^{\infty}$ be a sequence such that

$\limsup_{n\to{\infty}}{\mathcal{C}}_{n}=\lim_{p\to{\infty}}{\mathcal{C}}_{n_{p}}.$
(4.5)

Next, we apply Conditions C6), C6a), C6b) and C10) to a sequence
$\\{f_{n_{p}}\\}_{p=1}^{\infty}$. Then there exist a subsequence
$\\{f_{n_{p_{k}}}\\}_{k=1}^{\infty}$ and an element $f\in B$ such that

$\displaystyle\lim_{k\to{\infty}}\left\|D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)-l_{n_{p_{k}}}^{(1)}(D(f))\right\|_{F_{n_{p_{k}}}^{(1)}}^{*}=0,$
(4.6)
$\displaystyle\lim_{k\to{\infty}}\left\|f_{n_{p_{k}}}-l_{n_{p_{k}}}^{(2)}(f)\right\|_{F_{n_{p_{k}}}^{(2)}}^{*}=0,$
(4.7)

where $l_{n}^{(j)},\,n\in{\mathbb{N}},\,j=1,\,2$, satisfy C6a) and C6b). In
particular, $\|f\|_{F^{(2)}}<{\infty}$ since $B\subseteq F^{(2)}$.

If C7) and C8) are satisfied, then we have from (4.7), C5), and (3.14) of C7)

$\displaystyle
1=\lim_{k\to{\infty}}\left\|f_{n_{p_{k}}}\right\|_{F_{n_{p_{k}}}^{(2)}}$
$\displaystyle=$
$\displaystyle\lim_{k\to{\infty}}\left\|l^{(2)}_{n_{p_{k}}}(f)+\left(f_{n_{p_{k}}}-l^{(2)}_{n_{p_{k}}}(f)\right)\right\|_{F_{n_{p_{k}}}^{(2)}}$
$\displaystyle=$
$\displaystyle\lim_{k\to{\infty}}\left\|l^{(2)}_{n_{p_{k}}}(f)\right\|_{F_{n_{p_{k}}}^{(2)}}\leq
C_{1}\|f\|_{F^{(2)}}.$

Hence there exist $k_{0}\in{\mathbb{N}}$ and constants $C_{3}>0$ and $C_{4}>0$
such that

$\inf_{k\geq
k_{0}}\left\|l^{(2)}_{n_{p_{k}}}(f)\right\|_{F_{n_{p_{k}}}^{(2)}}\geq
C_{3},\qquad\|f\|_{F^{(2)}}\geq C_{4}.$ (4.8)

If C9) is satisfied, then we have from (4.6) and Conditions C5), C6a), and
C6b)

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\lim_{k\to{\infty}}\left\|D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right\|_{F_{n_{p_{k}}}^{(1)}}=\lim_{k\to{\infty}}\left\|l^{(1)}_{n_{p_{k}}}(D(f))+\left(D(f)-l^{(1)}_{n_{p_{k}}}(D(f))\right)\right\|_{F_{n_{p_{k}}}^{(1)}}$
$\displaystyle=$
$\displaystyle\lim_{k\to{\infty}}\left\|l^{(1)}_{n_{p_{k}}}(D(f))\right\|_{F_{n_{p_{k}}}^{(1)}}\leq\|D(f)\|_{F^{(1)}}\leq{\mathcal{C}}\|f\|_{F^{(2)}}\leq{\mathcal{C}}\liminf_{k\to{\infty}}\left\|l^{(2)}_{n_{p_{k}}}(f)\right\|_{F_{n_{p_{k}}}^{(2)}}.$

Hence (4.8) holds true in this case as well with $C_{4}=1/{\mathcal{C}}$.

Next using (4.6) and (4.7) again, we obtain by (4.4), (4.5), and C5),

$\displaystyle\lim_{k\to{\infty}}{\mathcal{C}}_{n_{p_{k}}}\leq\frac{\limsup_{k\to{\infty}}\left\|D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right\|_{F_{n_{p_{k}}}^{(1)}}}{\liminf_{k\to{\infty}}\left\|f_{n_{p_{k}}}\right\|_{F_{n_{p_{k}}}^{(2)}}}+{\varepsilon}=\frac{\limsup_{k\to{\infty}}\left\|l^{(1)}_{n_{p_{k}}}(D(f))\right\|_{F_{n_{p_{k}}}^{(1)}}}{\liminf_{k\to{\infty}}\left\|l^{(2)}_{n_{p_{k}}}(f)\right\|_{F_{n_{p_{k}}}^{(2)}}}+{\varepsilon},$
(4.9)

where
$\liminf_{k\to{\infty}}\left\|l^{(2)}_{n_{p_{k}}}(f)\right\|_{F_{n_{p_{k}}}^{(2)}}>0$
by (4.8). Since $\|f\|_{F^{(2)}}>0$ by (4.8), it follows from (4.5), (4.9),
and Conditions C6a) and C6b) that

$\limsup_{n\to{\infty}}{\mathcal{C}}_{n}\leq\frac{\|D(f)\|_{F^{(1)}}}{\|f\|_{F^{(2)}}}+{\varepsilon}\leq{\mathcal{C}}+{\varepsilon}.$
(4.10)

Finally letting ${\varepsilon}\to 0+$ in (4.10), we arrive at (4.3). $\Box$

######  Theorem 4.5.

Let ${\mathcal{C}}_{n}$ be a finite number for every $n\in{\mathbb{N}}$, and
let either Conditions C7), C8), C13) or Condition C9) be satisfied. In
addition, let C5), C6), C11), and C12) be satisfied. Then (4.3) holds true.

###### Proof.

The proof is similar to that of Theorem 4.4. We can assume without loss of
generality that ${\mathcal{C}}<{\infty}$. Next, we assume that Conditions C7),
C8), and C13) are satisfied. Since ${\mathcal{C}}_{n}<{\infty}$, we see from
C8) that for any fixed ${\varepsilon}>0$ there exists
$f_{n}=f_{n,{\varepsilon}}\in B_{n}^{\prime}$ with $\|f_{n}\|_{F^{(2)}_{n}}=1$
such that

${\mathcal{C}}_{n}<\frac{\|D_{n}(f_{n})\|_{F_{n}^{(1)}}}{\|f_{n}\|_{F_{n}^{(2)}}}+{\varepsilon},\qquad
n\in{\mathbb{N}}.$ (4.11)

If Condition C9) is satisfied, then for any fixed ${\varepsilon}>0$ there
exists $f_{n}=f_{n,{\varepsilon}}\in B_{n}^{\prime}$ with
$\|f_{n}\|_{F^{(2)}_{n}}>0$ and $\|D_{n}(f_{n})\|_{F^{(1)}_{n}}=1$ such that
(4.11) holds true as well.

Let $\\{n_{p}\\}_{p=1}^{\infty}$ be a sequence such that

$\limsup_{n\to{\infty}}{\mathcal{C}}_{n}=\lim_{p\to{\infty}}{\mathcal{C}}_{n_{p}}.$
(4.12)

Next, we apply Conditions C6), C6a), C6b), C11), and C12) to a sequence
$\\{f_{n_{p}}\\}_{p=1}^{\infty}$. Then there exists an element $f\in B$ such
that for any fixed $s\in{\mathbb{N}}$ there exists a subsequence
$\\{f_{n_{p_{k}}}\\}_{k=1}^{\infty}$
($n_{p_{k}}=n_{p_{k}}(s),\,k\in{\mathbb{N}}$) such that

$\displaystyle\lim_{k\to{\infty}}\left\|l_{n_{p_{k}},s}^{(1)}\left(D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right)-l_{s}^{(1)}(D(f))\right\|_{F_{s}^{(1)}}^{*}=0,$
(4.13)
$\displaystyle\lim_{k\to{\infty}}\left\|l_{n_{p_{k}},s}^{(2)}\left(f_{n_{p_{k}}}\right)-l_{s}^{(2)}(f)\right\|_{F_{s}^{(2)}}^{*}=0,$
(4.14)

and in addition,

$\left\|D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right\|_{F_{n_{p_{k}}}^{(1)}}=\left\|l_{n_{p_{k}},s}^{(1)}\left(D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right)\right\|_{F_{s}^{(1)}},\quad
k\in{\mathbb{N}}.$ (4.15)

Here, $l_{n}^{(j)}$ and
$l_{n,s}^{(j)},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}},\,j=1,\,2$, satisfy C6a),
C6b), and C11a). In particular, $\|f\|_{F^{(2)}}<{\infty}$ since $B\subseteq
F^{(2)}$. If C7), C8), and C13) are satisfied, then we have from (3.23) of
C13), Condition C5), (4.14), and (3.14) of C7)

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\lim_{k\to{\infty}}\left\|f_{n_{p_{k}}}\right\|_{F_{n_{p_{k}}}^{(2)}}\leq
C_{2}\limsup_{s\to{\infty}}\limsup_{k\to{\infty}}\left\|l_{n_{p_{k}},s}^{(2)}(f_{n_{p_{k}}})\right\|_{F_{s}^{(2)}}$
$\displaystyle=$ $\displaystyle
C_{2}\limsup_{s\to{\infty}}\limsup_{k\to{\infty}}\left\|l^{(2)}_{s}(f)+\left(l_{n_{p_{k}},s}^{(2)}(f_{n_{p_{k}}})-l^{(2)}_{s}(f)\right)\right\|_{F_{s}^{(2)}}$
$\displaystyle=$ $\displaystyle
C_{2}\limsup_{s\to{\infty}}\left\|l^{(2)}_{s}(f)\right\|_{F_{s}^{(2)}}\leq
C_{1}C_{2}\|f\|_{F^{(2)}}.$

Hence there exist $s_{0}\in{\mathbb{N}}$ and constants $C_{5}>0$ and
$C_{6}=1/(C_{1}C_{2})>0$ such that

$\inf_{s\geq s_{0}}\left\|l^{(2)}_{s}(f)\right\|_{F_{s}^{(2)}}\geq
C_{5},\qquad\|f\|_{F^{(2)}}\geq C_{6}.$ (4.16)

If C9) is satisfied, then by C5) we obtain from (4.15), (4.13), and Conditions
C6a) and C6b)

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\lim_{k\to{\infty}}\left\|D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right\|_{F_{n_{p_{k}}}^{(1)}}=\lim_{k\to{\infty}}\left\|l^{(1)}_{n_{p_{k}},s}\left(D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right)\right\|_{F_{s}^{(1)}}$
$\displaystyle=$
$\displaystyle\left\|l^{(1)}_{n_{p_{k}},s}\left(D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right)+\left(l_{s}^{(1)}(D(f))-l^{(1)}_{n_{p_{k}},s}\left(D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right)\right)\right\|_{F^{(1)}_{s}}$
$\displaystyle=$
$\displaystyle\left\|l_{s}^{(1)}(D(f))\right\|_{F^{(1)}_{s}}\leq\|D(f)\|_{F^{(1)}}\leq{\mathcal{C}}\|f\|_{F^{(2)}}.$

Hence (4.16) holds true in this case as well with $C_{6}=1/{\mathcal{C}}$.

Further, we use (4.11), (4.15), (4.13), (4.14), and Conditions C5) and C11a)
to obtain the following relations for $s\geq s_{0}$:

$\displaystyle\lim_{k\to{\infty}}{\mathcal{C}}_{n_{p_{k}}}$
$\displaystyle\leq$
$\displaystyle\frac{\limsup_{k\to{\infty}}\left\|D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right\|_{F_{n_{p_{k}}}^{(1)}}}{\liminf_{k\to{\infty}}\left\|f_{n_{p_{k}}}\right\|_{F_{n_{p_{k}}}^{(2)}}}+{\varepsilon}$
$\displaystyle\leq$
$\displaystyle\frac{\limsup_{k\to{\infty}}\left\|l_{n_{p_{k}},s}^{(1)}\left(D_{n_{p_{k}}}\left(f_{n_{p_{k}}}\right)\right)\right\|_{F_{s}^{(1)}}}{\liminf_{k\to{\infty}}\left\|l_{n_{p_{k}},s}^{(2)}\left(f_{n_{p_{k}}}\right)\right\|_{F_{s}^{(2)}}}+{\varepsilon}$
$\displaystyle=$
$\displaystyle\frac{\left\|l^{(1)}_{s}(D(f))\right\|_{F_{s}^{(1)}}}{\left\|l^{(2)}_{s}(f)\right\|_{F_{s}^{(2)}}}+{\varepsilon}.$

Therefore for some $f\in B$, satisfying (4.16), we have

$\lim_{k\to{\infty}}{\mathcal{C}}_{n_{p_{k}}}\leq\frac{\limsup_{s\to{\infty}}\left\|l^{(1)}_{s}(D(f))\right\|_{F_{s}^{(1)}}}{\liminf_{s\to{\infty}}\left\|l^{(2)}_{s}(f)\right\|_{F_{s}^{(2)}}}+{\varepsilon},$
(4.17)

where $\liminf_{s\to{\infty}}\left\|l^{(2)}_{s}(f)\right\|_{F_{s}^{(2)}}>0$ by
(4.16). Since $\left\|f\right\|_{F^{(2)}}>0$ by (4.16), it follows from
(4.12), (4.17) and Conditions C6a) and C6b) that

$\limsup_{n\to{\infty}}{\mathcal{C}}_{n}\leq\frac{\|D(f)\|_{F^{(1)}}}{\|f\|_{F^{(2)}}}+{\varepsilon}\leq{\mathcal{C}}+{\varepsilon}.$
(4.18)

Finally letting ${\varepsilon}\to 0+$ in (4.18), we arrive at (4.3). $\Box$

Combining Theorems 4.1, 4.4, 4.5 and Corollary 4.3, we can obtain the relation

${\mathcal{C}}={\mathcal{C}}\left(D,B,F^{(1)},F^{(2)}\right)=\lim_{n\to{\infty}}{\mathcal{C}}\left(D_{n},B_{n},F^{(1)}_{n},F^{(2)}_{n}\right)=\limsup_{n\to{\infty}}{\mathcal{C}}_{n}.$
(4.19)

######  Corollary 4.6.

If all the conditions of Theorems 4.1 or 4.2 or Corollary 4.3 are satisfied
and, in addition, all the conditions of Theorem 4.4 or Theorem 4.5 are
satisfied, then relation (4.19) holds true.

Special versions of Theorems 4.4 and 4.5 are presented below.

######  Theorem 4.7.

Let ${\mathcal{C}}_{n}$ be a finite number for every $n\in{\mathbb{N}}$.
(a) Let Conditions C5), C6*a), C8), C10*a), and C10*b) be satisfied. Then
(4.3) holds true.
(b) Let Conditions C5), C6*a), C8), C12*a), and C12*b) be satisfied. Then
(4.3) holds true.

###### Proof.

The proof of Theorem 4.7 follows that of Theorem 4.4 (or Theorem 4.5) if
Condition C10) (or Condition C12)) is replaced with Conditions C10*a) and
C10*b) (or Conditions C12*a) and C12*b)). The outline of the proof with a new
component of the proof, compared with that of Theorem 4.4 (or Theorem 4.5), is
presented below.

Given ${\varepsilon}>0$ let
$\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}=f_{n,{\varepsilon}}\in B_{n}^{\prime}$
with $\|f_{n}\|_{F^{(2)}_{n}}=1$ be a sequence such that (4.4) (or (4.11))
holds true. In addition, let $\\{n_{p}\\}_{p=1}^{\infty}$ be a sequence such
that (4.5) (or (4.12)) is valid. Next, by Condition C10*a) (or C12*a), there
exist a subsequence $\\{n_{p_{k}}\\}_{k=1}^{\infty}$ and $f\in B$ such that
(4.6) (or (4.13) and (4.15)) hold(s) true. Further, by Condition C10*b) (or
C12*b), there exists a sequence of operators
$\left\\{l_{n}^{(2)}\right\\}_{n=1}^{\infty}$ (or, in addition, a sequence of
operators $\left\\{l_{n,s}^{(2)}\right\\}_{n,s=1}^{\infty}$) such that (4.7)
(or (4.14)) is valid.

The rest of the proof can be copied from the proof of Theorem 4.4 (or Theorem
4.5). $\Box$

######  Remark 4.8.

It is not difficult to extend asymptotic relations of this section to families
$D_{\alpha},\,B_{\alpha},\,F_{\alpha}^{(1)},$ and $F_{\alpha}^{(2)}$ defined
on a set $E$ filtering with respect to a filter $\Phi$. The corresponding
upper and lower limits are considered with respect to the filtering set $E$
(cf. [16, Ch. 4]).

######  Remark 4.9.

We apply the results of this section to Problems A, B, and C of approximation
theory (see Sections 2.3, 5, and 7). However, it is possible to apply them to
Problem D of analysis as well (see Section 2.3). The similar condition-based
approach to norms of linear operators in periodic and nonperiodic spaces was
discussed in [18].

##  5\. Asymptotic Relations in Approximation Theory

We recall that three major Problems A, B, and C of approximation theory
related to sharp constants are described in Section 2.3. Here, we apply
Theorems 4.1, 4.2 4.4, 4.5, and 4.7 and Corollary 4.3 to these problems of
approximation theory. In Problem A, in general, we cannot essentially reduce
the number of conditions in asymptotic results of Section 4. However, we can
do that in Problems B and C.

###  5.1. Inequalities for Approximating Elements (Problem A)

In this case we assume that $F^{(j)}$ and $F^{(j)}_{n}$ are $\beta_{j}$-normed
spaces, where ${\beta}_{j}$ is independent of $n,\,j=1,2,\,n\in{\mathbb{N}}$.
In addition, we set
$\|\cdot\|_{F^{(j)}_{n}}^{*}=\|\cdot\|_{F^{(j)}_{n}},\,j=1,2,\,n\in{\mathbb{N}}$.
Next, let $B$ and $B_{n}$ be nontrivial subspaces of $F^{(2)}$ and
$F^{(2)}_{n}$, respectively, and let $D_{n}$ be a $1$-homogeneous operator
(that is, $D_{n}({\alpha}f)={\alpha}D_{n}(f),\,{\alpha}>0$),
$n\in{\mathbb{N}}$.

Then Conditions C1), C1*), and C5) are satisfied by Proposition 3.6. In
addition, Conditions C8) and C9) are satisfied by Proposition 3.7 (i).
However, it appears that in Problem A Conditions C1) through C3) are more
efficient than Conditions C1*) through C4*) and Condition C9) is more
efficient than Condition C8) (see Corollary 7.1 and Examples 7.2, 7.3, and
7.4).

The following corollary follows from Theorems 4.1, 4.2, 4.4, and 4.5.

######  Corollary 5.1.

(i) If Conditions C2), C3), C4) (or C4´)) are satisfied, then

$\sup_{f\in
B,\,\|f\|_{F^{(2)}}>0}\frac{\|D(f)\|_{F^{(1)}}}{\|f\|_{F^{(2)}}}\leq\liminf_{n\to{\infty}}\sup_{f\in
B_{n},\,\|f\|_{F^{(2)}_{n}}>0}\frac{\|D_{n}(f)\|_{F^{(1)}_{n}}}{\|f\|_{F^{(2)}_{n}}}.$
(5.1)

(ii) Let ${\mathcal{C}}_{n}$ be a finite number for every $n\in{\mathbb{N}}$,
and let Conditions C6) and C10) be satisfied. Then

$\sup_{f\in
B,\,\|f\|_{F^{(2)}}>0}\frac{\|D(f)\|_{F^{(1)}}}{\|f\|_{F^{(2)}}}\geq\limsup_{n\to{\infty}}\sup_{f\in
B_{n},\,\|f\|_{F^{(2)}_{n}}>0}\frac{\|D_{n}(f)\|_{F^{(1)}_{n}}}{\|f\|_{F^{(2)}_{n}}}.$
(5.2)

(iii) Let ${\mathcal{C}}_{n}$ be a finite number for every $n\in{\mathbb{N}}$,
and let Conditions C6), C11), and C12) be satisfied. Then (5.2) holds true.

######  Remark 5.2.

Unlike Problems B and C, inequalities for approximating elements were not
included into the general approach to limit relations between sharp constants
developed in [31, 32].

###  5.2. Approximation of Individual Elements (Problem B)

In this case we set

$F^{(1)}_{n}=F^{(1)};\,D(f):=f_{0},\,f\in B;\,D_{n}(f_{n}):=f_{0},\,f_{n}\in
B_{n};\,n\in{\mathbb{N}},$ (5.3)

where $f_{0}\neq 0$ is a fixed element of $F^{(1)}$. In addition, we set

$\|\cdot\|_{F^{(1)}_{n}}^{*}=\|\cdot\|_{F^{(1)}_{n}}=\|\cdot\|_{F^{(1)}}:=\|\cdot\|_{F^{(1)},\mathrm{triv}},\qquad
n\in{\mathbb{N}},$ (5.4)

where $\|\cdot\|_{F^{(1)},\mathrm{triv}}$ is defined by (2.1). We also assume
that $B$ and $B_{n}$ are proper nonempty subsets of $F^{(2)}$ and
$F^{(2)}_{n}$, respectively, $n\in{\mathbb{N}}$. It follows from (5.3) and
(5.4) that

$\|D(f)\|_{F^{(1)}}=\left\|D_{n}(f_{n})\right\|_{F^{(1)}_{n}}=1,\qquad f\in
B,\quad f_{n}\in B_{n},\quad n\in{\mathbb{N}}.$ (5.5)

Next, for certain ${\beta}_{2}$-norms $\|\cdot\|_{F^{(2)}}^{*}$ and
$\|\cdot\|_{F^{(2)}_{n}}^{*},\,n\in{\mathbb{N}}$, where ${\beta}_{2}\in(0,1]$
is independent of $n$, we define the following ”norms”

$\|h\|_{F^{(2)}}:=\|{\varphi}-h\|_{F^{(2)}}^{*},\quad h\in
F^{(2)};\qquad\|h_{n}\|_{F^{(2)}_{n}}:=\|{\varphi}_{n}-h_{n}\|_{F^{(2)}_{n}}^{*},\quad
h_{n}\in F^{(2)}_{n},$ (5.6)

where ${\varphi}\in F^{(2)}\setminus B$ and ${\varphi}_{n}\in
F^{(2)}_{n}\setminus B_{n},\,n\in{\mathbb{N}}$, are fixed elements. Then by
(2.6), (5.5), and (5.6),

${\mathcal{C}}=1/E\left({\varphi},B,F^{(2)}_{*}\right),\qquad{\mathcal{C}}_{n}=1/E\left({\varphi}_{n},B_{n},F^{(2)}_{*n}\right),$
(5.7)

where $F^{(2)}_{*}$ and $F^{(2)}_{*n}$ are vector spaces $F^{(2)}$ and
$F^{(2)}_{n}$ equipped with the ”norms” $\|\cdot\|_{F^{(2)}}^{*}$ and
$\|\cdot\|_{F^{(2)}_{n}}^{*},\,n\in{\mathbb{N}}$, respectively. In addition,
we recall that

$\displaystyle E\left({\varphi},B,F^{(2)}_{*}\right)=\inf_{f\in
B}\|{\varphi}-f\|^{*}_{F^{(2)}},\quad
E\left({\varphi}_{n},B_{n},F^{(2)}_{*n}\right)=\inf_{f_{n}\in
B_{n}}\|{\varphi}_{n}-f_{n}\|^{*}_{F^{(2)}_{n}},\quad n\in{\mathbb{N}},$

are the errors of best approximation (see (2.9)). We also assume that
$E\left({\varphi},B,F^{(2)}_{*}\right)>0$ and
$E\left({\varphi}_{n},B_{n},F^{(2)}_{*n}\right)>0$, so
${\mathcal{C}}<{\infty}$ and ${\mathcal{C}}_{n}<{\infty},\,n\in{\mathbb{N}}$.
Further, let

$L_{n}^{(1)}=l_{n}^{(1)}=l_{n,s}^{(1)}=I,\,n\in{\mathbb{N}},\,s\in{\mathbb{N}},$
(5.8)

be the corresponding identity operators (because of (5.3) and (5.4)). Since by
(5.5), $B_{n}\setminus\\{0\\}=\left\\{f_{n}\in
B_{n}:\|f_{n}\|_{F^{(2)}_{n}}>0,\,\|D_{n}(f_{n})\|_{F^{(1)}_{n}}=1\right\\}$,
we choose $B_{n}^{\prime}$ from Condition C9) as a subset of
$B_{n}\setminus\\{0\\}$, satisfying the equality

$\displaystyle
E\left({\varphi}_{n},B_{n},F^{(2)}_{*n}\right)=E\left({\varphi}_{n},B_{n}^{\prime},F^{(2)}_{*n}\right),\qquad
n\in{\mathbb{N}}.$

Then Conditions C1), C1*), and C5) are satisfied by Proposition 3.6. In
addition, Condition C9) is satisfied by the choice of
$B_{n}^{\prime},\,n\in{\mathbb{N}}$, and by Proposition 3.7 (iii) if we take
into account (3.16), (5.5), and (5.7). Next, relations (3.7), (3.8), (3.9),
(3.12), (3.17), (3.20), and (3.22) of Conditions C3), C3*), C4), C4´), C6a),
C10), and C12) are trivially satisfied by (5.3), (5.4), (5.5), and (5.8).

In particular, Conditions C1) through C3) can be reduced to inequality (3.6)
from C3), which coincides with (5.9) below. Since (5.9) is one of the
inequalities we wish to prove, Conditions C1) through C3) are obviously
inefficient in Problem B. However, Conditions C1*) through C4*) can be reduced
to more efficient relations (3.10) and (3.11) from C3*) and C4*), respectively
(see Corollary 7.6 (i) and Examples 7.8, 7.9, and 7.10).

The following corollary follows from relations (5.3) through (5.8), Corollary
4.3, and Theorems 4.4 and 4.5.

######  Corollary 5.3.

(i) If there exists a sequence of operators $L_{n}^{(2)}:F^{(2)}\to
F_{n}^{(2)},\,n\in{\mathbb{N}}$, such that relations (3.10) of C3*) and (3.11)
of C4*) are satisfied, then

$\limsup_{n\to{\infty}}E\left({\varphi}_{n},B_{n},F^{(2)}_{*n}\right)\leq
E\left({\varphi},B,F^{(2)}_{*}\right).$ (5.9)

(ii) If there exists a sequence of operators $l_{n}^{(2)}:F^{(2)}\to
F_{n}^{(2)},\,n\in{\mathbb{N}}$, such that C6b) and relation (3.18) of C10)
are satisfied, then

$\liminf_{n\to{\infty}}E\left({\varphi}_{n},B_{n},F^{(2)}_{*n}\right)\geq
E\left({\varphi},B,F^{(2)}_{*}\right).$ (5.10)

(iii) If there exist families of operators $l_{n}^{(2)}:F^{(2)}\to
F_{n}^{(2)},\,n\in{\mathbb{N}}$, and $l_{n,s}^{(2)}:F^{(2)}_{n}\to
F_{s}^{(2)},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}}$, such that C6b), C11a), and
relation (3.21) of C12) are satisfied, then (5.10) holds true.

######  Remark 5.4.

Special cases of Corollary 5.3 were discussed in [31, 32]. In case of linear
operators $L_{n}^{(2)}:F_{*}^{(2)}\to F_{*n}^{(2)}$ with
$\limsup_{n\to{\infty}}\left\|L_{n}^{(2)}\right\|_{F_{*}^{(2)}\to
F_{*n}^{(2)}}\leq 1$ and
${\varphi}_{n}=L_{n}^{(2)}({\varphi}),\,n\in{\mathbb{N}}$, relation (3.11) of
C4*) is satisfied. Indeed, for every $h\in F^{(2)}$,

$\limsup_{n\to{\infty}}\left\|L_{n}^{(2)}(h)\right\|_{F_{n}^{(2)}}=\limsup_{n\to{\infty}}\left\|L_{n}^{(2)}({\varphi})-L_{n}^{(2)}(h)\right\|_{F_{n}^{(2)}}^{*}\leq\|{\varphi}-h\|^{*}_{F^{(2)}}=\|h\|_{F^{(2)}}.$
(5.11)

Next, in case of linear operators $l_{n}^{(2)}:F_{*}^{(2)}\to F_{*n}^{(2)}$
with
$\liminf_{n\to{\infty}}\left\|l_{n}^{(2)}(h)\right\|^{*}_{F_{n}^{(2)}}\geq\|h\|^{*}_{F^{(2)}}$
and ${\varphi}_{n}=l_{n}^{(2)}({\varphi}),\,n\in{\mathbb{N}}$, relation (3.13)
of C6b) is satisfied. The proof of this fact is similar to (5.11).

Further, in case of linear operators $l_{n,s}^{(2)}:F_{*n}^{(2)}\to
F_{*s}^{(2)}$ with
$l_{s}^{(2)}=l_{n,s}^{(2)}\,l_{n}^{(2)},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}};\linebreak\sup_{n\in{\mathbb{N}},s\in{\mathbb{N}}}\left\|l_{n,s}^{(2)}\right\|_{F_{*n}^{(2)}\to
F_{*s}^{(2)}}\leq 1$, and
${\varphi}_{n}=l_{n}^{(2)}({\varphi}),\,n\in{\mathbb{N}}$, relation (3.13) of
C11a) is satisfied. Indeed, for every $h\in F^{(2)}$,

$\displaystyle\left\|l_{n,s}^{(2)}(h)\right\|_{F_{s}^{(2)}}=\left\|l_{s}^{(2)}({\varphi})-l_{n,s}^{(2)}(h)\right\|_{F_{s}^{(2)}}^{*}=\left\|l_{n,s}^{(2)}\left(l_{n}^{(2)}({\varphi})-h\right)\right\|_{F_{s}^{(2)}}^{*}\leq\|l_{n}^{(2)}({\varphi})-h\|^{*}_{F^{(2)}_{n}}=\|h\|_{F^{(2)}_{n}}.$

In all these special cases, Corollary 5.3 was established in [31, Theorems
2.1a) and 2.2a)] for normed spaces $F_{*}^{(2)}$ and
$F_{*n}^{(2)},\,n\in{\mathbb{N}}$. For ${\beta}_{2}$-normed spaces
$F_{*}^{(2)}$ and $F_{*n}^{(2)},\,n\in{\mathbb{N}}$, the corresponding results
were proved in [32, Theorems 1.1a) and 1.2a)].

###  5.3. Approximation on Classes of Elements (Problem C)

We set

$F^{(2)}=F^{(1)},\quad F^{(2)}_{n}=F^{(1)}_{n},\quad
D=D_{n}=I,\quad\,n\in{\mathbb{N}},$ (5.12)

where $I$ is the corresponding imbedding operator, and

$\|\cdot\|_{F^{(2)}}:=\|\cdot\|_{F^{(2)},\mathrm{triv}},\qquad\|\cdot\|_{F^{(2)}_{n}}^{*}=\|\cdot\|_{F^{(2)}_{n}}:=\|\cdot\|_{F^{(2)}_{n},\mathrm{triv}},\quad
n\in{\mathbb{N}}.$ (5.13)

It follows from (5.13) that

$\|f\|_{F^{(2)}}=\left\|f_{n}\right\|_{F^{(2)}_{n}}=1,\qquad f\in
B\setminus\\{0\\},\quad f_{n}\in B_{n}\setminus\\{0\\},\quad
n\in{\mathbb{N}}.$ (5.14)

Next, for ${\beta}_{1}$-norms $\|\cdot\|_{F^{(1)}}^{*}$ and
$\|\cdot\|_{F^{(1)}_{n}}^{*},\,n\in{\mathbb{N}}$, where ${\beta}_{1}\in(0,1]$
is independent of $n$, we define the following ”norms”

$\displaystyle\|h\|_{F^{(1)}}:=E\left(h,G,F^{(1)}_{*}\right)=\inf_{g\in
G}\|h-g\|^{*}_{F^{(1)}},\qquad h\in F^{(1)},$ (5.15)
$\displaystyle\|h_{n}\|_{F^{(1)}_{n}}:=E\left(h_{n},G_{n},F^{(1)}_{*n}\right)=\inf_{g_{n}\in
G_{n}}\|h_{n}-g_{n}\|^{*}_{F^{(1)}_{n}},\qquad h_{n}\in F^{(1)}_{n},\quad
n\in{\mathbb{N}}.$ (5.16)

Here, $F^{(1)}_{*}$ and $F^{(1)}_{*n}$ are vector spaces $F^{(1)}$ and
$F^{(1)}_{n}$ equipped with the ”norms” $\|\cdot\|_{F^{(1)}}^{*}$ and
$\|\cdot\|_{F^{(1)}_{n}}^{*}$, respectively, and, in addition, $G$ and $G_{n}$
are subspaces of $F^{(1)}_{*}$ and $F^{(1)}_{*n}$, respectively,
$n\in{\mathbb{N}}$. Note that $\|\cdot\|_{F^{(1)}}$ and
$\|\cdot\|_{F^{(1)}_{n}},\,n\in{\mathbb{N}}$, defined by (5.15) and (5.16) are
${\beta}_{1}$-seminorms. Then by (2.6), (5.14), (5.15), and (5.16),

${\mathcal{C}}=\sup_{f\in
B\setminus\\{0\\}}E\left(f,G,F^{(1)}_{*}\right),\quad{\mathcal{C}}_{n}=\sup_{f_{n}\in
B_{n}\setminus\\{0\\}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right)=\sup_{f_{n}\in
B_{n}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right)$ (5.17)

are sharp constants of best approximation on classes of elements
$B\setminus\\{0\\}$ and $B_{n},\,n\in{\mathbb{N}}$. In addition, since by
(5.14), $B_{n}\setminus\\{0\\}=\left\\{f_{n}\in
B_{n}:\|f_{n}\|_{F^{(2)}_{n}}=1\right\\}$, we choose $B_{n}^{\prime}$ from
Condition C8) as a subset of $B_{n}\setminus\\{0\\}$, satisfying the equality

$\displaystyle\sup_{f_{n}\in
B_{n}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right)=\sup_{f_{n}\in
B_{n}^{\prime}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right),n\in{\mathbb{N}}.$

Let us also define $l_{n}^{(2)}:F^{(2)}\to F_{n}^{(2)}$ and
$l_{n,s}^{(2)}:F^{(2)}_{n}\to F_{s}^{(2)}$ for each sequence
$\\{f_{n}\\}_{n=1}^{\infty},\,f_{n}\in
B_{n}\setminus\\{0\\},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}}$, and $f\in B$
(mentioned in C10*a) and C12*a)), by the formulae

$l_{n}^{(2)}(h):=\left\\{\begin{array}[]{ll}f_{n},&h=f,\\\ f_{n,0},&h\in
F^{(2)}\setminus\\{f\\},\end{array}\right.\quad
l_{n,s}^{(2)}(h_{n}):=\left\\{\begin{array}[]{ll}f_{s},&h_{n}=f_{n},\\\
f_{s,0},&h_{n}\in F^{(2)}_{n}\setminus\\{f_{n}\\},\end{array}\right.$ (5.18)

where $f_{n,0}\neq 0$ is a fixed element from
$F_{n}^{(2)},\,n\in{\mathbb{N}}$. It follows from (5.13) and (5.18) that for
every $h\in F^{(2)}\setminus\\{f\\}$ and every $h_{n}\in
F^{(2)}_{n}\setminus\\{f_{n}\\}$,

$\left\|l_{n}^{(2)}(h)\right\|_{F_{n}^{(2)}}=\left\|l_{n,s}^{(2)}(h_{n})\right\|_{F_{s}^{(2)}}=1;\qquad
l_{n}^{(2)}(f)=f_{n},\quad l_{n,s}^{(2)}(f_{n})=f_{s},\quad
n\in{\mathbb{N}},\quad s\in{\mathbb{N}}.$ (5.19)

Then Conditions C1), C1*), C5) are satisfied by Proposition 3.6. In addition,
Condition C8) is satisfied by the choice of
$B_{n}^{\prime},\,n\in{\mathbb{N}}$, and by Proposition 3.7 (ii) if we take
into account (3.15), (5.14), and (5.17). Next, relations (3.6), (3.10),
(3.11), (3.13), (3.14) (with $C_{1}=1$), (3.18), (3.19), (3.21), (3.23) (with
$C_{2}=1$), (3.26), and (3.29) of Conditions C3), C3*), C4´), C6b), C7), C10),
C13), C10*b), and C12*b) are trivially satisfied by (5.14) and (5.19).

Since inequality (3.6) of Condition C3) is trivially satisfied, it appears
that in Problem C Conditions C1) through C3) are more efficient than
Conditions C1*) through C4*) (see Corollary 7.13 (i) and Examples 7.15 and
7.16).

The following corollary follows from relations (5.12) through (5.19) and
Theorems 4.2, 4.7 (a), and 4.7 (b).

######  Corollary 5.5.

(i) If relations (3.7) of C3) and (3.8) of C4´) are satisfied, then

$\sup_{f\in
B\setminus\\{0\\}}E\left(f,G,F^{(1)}_{*}\right)\leq\liminf_{n\to{\infty}}\sup_{f_{n}\in
B_{n}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right).$ (5.20)

(ii) Let $\sup_{f_{n}\in B_{n}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right)$ be a
finite number for every $n\in{\mathbb{N}}$. If Conditions C6*a) and C10*a) are
satisfied, then

$\sup_{f\in
B\setminus\\{0\\}}E\left(f,G,F^{(1)}_{*}\right)\geq\limsup_{n\to{\infty}}\sup_{f_{n}\in
B_{n}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right).$ (5.21)

(iii) Let $\sup_{f_{n}\in B_{n}}E\left(f_{n},G_{n},F^{(1)}_{*n}\right)$ be a
finite number for every $n\in{\mathbb{N}}$. If Conditions C6*a) and C12*a) are
satisfied, then (5.21) holds true.

######  Remark 5.6.

Note that relations $B\subseteq F^{(1)}\cap F^{(2)}$ and $D(B)\subseteq B$ of
Condition C4´) are satisfied because $F^{(2)}=F^{(1)}$ and $D=I$ is an
imbedding operator by (5.12).

######  Remark 5.7.

For linear operators $L_{n}^{(1)},\,l_{n}^{(1)}$ and linear operators
$l_{n,s}^{(1)}$ with
$l_{s}^{(1)}=l_{n,s}^{(1)}\,l_{n}^{(1)},\,n\in{\mathbb{N}},\,s\in{\mathbb{N}},$
and
$\sup_{n\in{\mathbb{N}},s\in{\mathbb{N}}}\left\|l_{n,s}^{(1)}\right\|_{F_{*n}^{(1)}\to
F_{*s}^{(1)}}\leq 1$, Corollary 5.5 with slightly different conditions was
obtained in [31, Theorems 2.1 (b) and 2.2 (b)] for normed spaces $F_{*}^{(1)}$
and $F_{*n}^{(1)},\,n\in{\mathbb{N}}$. For ${\beta}_{2}$-normed spaces
$F_{*}^{(1)}$ and $F_{*n}^{(1)},\,n\in{\mathbb{N}}$, the corresponding results
were proved in [32, Theorems 1.1 (b) and 1.2 (b)].

######  Remark 5.8.

Summarizing the applications of the results from Section 4 to Problems A, B,
and C presented in this section, we conclude that Corollary 5.1 is new, while
weaker versions of Corollaries 5.3 and 5.5 were obtained in [31, 32] (see also
Remarks 5.2, 5.4 and 5.7).

##  6\. Function Spaces, Function Classes, and Operators

Applications of Corollaries 5.1, 5.3, and 5.5 to concrete problems of
approximation theory (see Section 7) require more specified spaces. Here, we
introduce some function spaces, function classes, and operators and study
their certain properties.

###  6.1. Notation

Let ${\mathbb{R}}^{m}$ be the Euclidean $m$-dimensional space with elements
$x=(x_{1},\ldots,x_{m}),\,y=(y_{1},\ldots,y_{m}),\,t=(t_{1},\ldots,t_{m})$,
the inner product $t\cdot y:=\sum_{j=1}^{m}t_{j}y_{j}$, and the norm
$|t|:=\sqrt{t\cdot t}$. Next,
${\mathbb{C}}^{m}:={\mathbb{R}}^{m}+i{\mathbb{R}}^{m}$ is the $m$-dimensional
complex space with elements $z=(z_{1},\ldots,z_{m})=x+iy$ and the norm
$|z|:=\sqrt{|x|^{2}+|y|^{2}}$; ${\mathbb{Z}}^{m}$ denotes the set of all
integral lattice points $k=(k_{1},\ldots,k_{m})$ in ${\mathbb{R}}^{m}$; and
${\mathbb{Z}}^{m}_{+}$ is a subset of ${\mathbb{Z}}^{m}$ of all points with
nonnegative coordinates. We also use multi-indices
${\beta}=({\beta}_{1},\ldots,{\beta}_{m})\in{\mathbb{Z}}^{m}_{+}$ and
${\alpha}=({\alpha}_{1},\ldots,{\alpha}_{m})\in{\mathbb{Z}}^{m}_{+}$ with

$\displaystyle|{\beta}|:=\sum_{j=1}^{m}{\beta}_{j},\qquad|{\alpha}|:=\sum_{j=1}^{m}{\alpha}_{j},\qquad
y^{\beta}:=y_{1}^{{\beta}_{1}}\cdot\cdot\cdot y_{m}^{{\beta}_{m}},\qquad
D^{\alpha}:=\frac{\partial^{{\alpha}_{1}}}{\partial
y_{1}^{{\alpha}_{1}}}\cdot\cdot\cdot\frac{\partial^{{\alpha}_{m}}}{\partial
y_{m}^{{\alpha}_{m}}}.$

Let $V$ be a centrally symmetric (with respect to the origin) closed convex
body in ${\mathbb{R}}^{m}$ and $V^{*}:=\\{y\in{\mathbb{R}}^{m}:\forall\,t\in
V,|t\cdot y|\leq 1\\}$ be the polar of $V$. It is well known that $V^{*}$ is a
centrally symmetric (with respect to the origin) closed convex body in
${\mathbb{R}}^{m}$ and $V^{**}=V$ (see, e.g., [67, Sect. 14]). The set $V$
generates the following dual norms on ${\mathbb{R}}^{m}$ and
${\mathbb{C}}^{m}$ by

$\displaystyle\|y\|_{V}^{*}:=\sup_{t\in V}|t\cdot y|,\quad
y\in{\mathbb{R}}^{m};\qquad\|z\|_{V}^{*}:=\sup_{t\in
V}\left|\sum_{j=1}^{m}t_{j}z_{j}\right|,\quad z\in{\mathbb{C}}^{m}.$

Note also that $V^{*}$ is the unit ball in the norm $\|\cdot\|_{V}^{*}$ on
${\mathbb{R}}^{m}$. Examples of sets $V$ and $V^{*}$ and dual norms
$\|\cdot\|_{V}^{*}$ are given below. Given
${\sigma}\in{\mathbb{R}}^{m},\,{\sigma}_{j}>0,\,1\leq j\leq m$, and $M>0$, let
$\Pi^{m}_{\sigma}:=\\{t\in{\mathbb{R}}^{m}:|t_{j}|\leq{\sigma}_{j},1\leq j\leq
m\\},\,Q^{m}_{M}:=\\{t\in{\mathbb{R}}^{m}:|t_{j}|\leq M,1\leq j\leq m\\}$, and
${\mathfrak{B}}^{m}_{M}:=\\{t\in{\mathbb{R}}^{m}:|t|\leq M\\}$ be the
$m$-dimensional parallelepiped, cube, and ball, respectively.

For example, if ${\sigma}\in{\mathbb{R}}^{m},\,{\sigma}_{j}>0,\,1\leq j\leq
m$, and
$V=\left\\{t\in{\mathbb{R}}^{m}:\left(\sum_{j=1}^{m}|t_{j}/{\sigma}_{j}|^{\mu}\right)^{1/\mu}\leq
1\right\\}$, then for
$y\in{\mathbb{R}}^{m},\,\|y\|_{V}^{*}=\left(\sum_{j=1}^{m}|{\sigma}_{j}y_{j}|^{\lambda}\right)^{1/{\lambda}}$
and
$V^{*}=\left\\{y\in{\mathbb{R}}^{m}:\left(\sum_{j=1}^{m}|{\sigma}_{j}y_{j}|^{\lambda}\right)^{1/{\lambda}}\leq
1\right\\}$, where $\mu\in[1,{\infty}],\,{\lambda}\in[1,{\infty}]$, and
$1/\mu+1/{\lambda}=1$. In particular,
$\|y\|_{\Pi^{m}_{\sigma}}^{*}=\sum_{j=1}^{m}{\sigma}_{j}|y_{j}|,\,\|y\|_{Q^{m}_{M}}^{*}=M\sum_{j=1}^{m}|y_{j}|$,
and $\|y\|_{{\mathfrak{B}}^{m}_{M}}^{*}=M|y|$.

We also use the floor function $\lfloor\cdot\rfloor$.

###  6.2. Function spaces

In the capacity of spaces $F^{(1)},\,F^{(2)},\,F^{(1)}_{n},$ and $F^{(2)}_{n}$
from Sections 3, 4, and 5 or their $\ast$-modifications from Section 5, we
mostly use the following function spaces.

Let $\mu$ be a ${\sigma}$-finite positive measure on a ${\sigma}$-algebra
$\Sigma$ over ${\mathbb{R}}^{m}$ and let $S_{\mu}({\mathbb{R}}^{m})$ be a
vector space of all $\mu$-measurable functions
$f:{\mathbb{R}}^{m}\to{\mathbb{C}}^{1}$. Next, let
$F({\mathbb{R}}^{m})=F({\mathbb{R}}^{m},\mu)$ be a ${\beta}$-normed space,
${\beta}\in(0,1]$, of functions from $S_{\mu}({\mathbb{R}}^{m})$. We assume
that the ${\beta}$-norm of $F({\mathbb{R}}^{m})$ possesses properties of
monotonicity and $\boldsymbol{{\omega}}$-continuity.

######  Definition 6.1.

We say that the ${\beta}$-norm $\|\cdot\|_{F({\mathbb{R}}^{m})}$ is _monotone_
if relations $g\in S_{\mu}({\mathbb{R}}^{m}),\,f\in F({\mathbb{R}}^{m})$, and
$|g(x)|\leq|f(x)|,\,x\in{\mathbb{R}}^{m}$, imply $g\in F({\mathbb{R}}^{m})$
and $\|g\|_{F({\mathbb{R}}^{m})}\leq\|f\|_{F({\mathbb{R}}^{m})}$.

Given a $\mu$-measurable set ${\Omega}\subset{\mathbb{R}}^{m}$ and a space
$F({\mathbb{R}}^{m})$, we define a space $F({\Omega})$ of the restrictions
$f\left|{}_{\Omega}\right.$ of functions $f\in S_{\mu}({\mathbb{R}}^{m})$ to
${\Omega}$ with the finite ${\beta}$-norm
$\|f\left|{}_{\Omega}\right.\|_{F({\Omega})}:=\|f\cdot\chi({\Omega})\|_{F({\mathbb{R}}^{m})}$,
where $\chi({\Omega})$ is the characteristic function of ${\Omega}$. If there
is no confusion, we use the notation $f$ instead of
$f\left|{}_{\Omega}\right.$. For a monotone ${\beta}$-norm
$\|\cdot\|_{F({\mathbb{R}}^{m})}$,

$\|f\|_{F({\Omega})}\leq\|f\|_{F({\mathbb{R}}^{m})},\qquad f\in
F({\mathbb{R}}^{m}),$ (6.1)

so if $f\in F({\mathbb{R}}^{m})$, then $f=f\left|{}_{\Omega}\right.\in
F({\Omega})$. In a sense, the next definition allows us to reverse the
inequality sign in (6.1).

######  Definition 6.2.

We say that the monotone ${\beta}$-norm $\|\cdot\|_{F({\mathbb{R}}^{m})}$ is
$\boldsymbol{{\omega}}$-_continuous_ if for any increasing sequence of
$\mu$-measurable sets
$\boldsymbol{{\omega}}:=\\{{\Omega}_{n}\\}_{n=1}^{\infty}$ in
${\mathbb{R}}^{m}$ with $\cup_{n=1}^{\infty}{\Omega}_{n}={\mathbb{R}}^{m}$,
the following relation is valid:

$\lim_{n\to{\infty}}\|f\|_{F({\Omega}_{n})}=\|f\|_{F({\mathbb{R}}^{m})},\qquad
f\in F({\mathbb{R}}^{m}).$ (6.2)

Examples of spaces with monotone and $\boldsymbol{{\omega}}$-continuous norms
include certain Orlicz, Lorentz, and Marcinkiewicz spaces (see, e.g. [50,
Sect. 4.3]). In particular, normed separable spaces $F({\mathbb{R}}^{m})$ or
normed spaces, satisfying the Fatou condition (named C-condition in [50, Sect.
4.3]), have $\boldsymbol{{\omega}}$-continuous norms.

A typical example of a space with a monotone and
$\boldsymbol{{\omega}}$-continuous ${\beta}$-norm is the space
$L_{r,\mu}({\mathbb{R}}^{m}),r\in(0,{\infty}]$, of all $\mu$-measurable
complex-valued functions $f$ on ${\mathbb{R}}^{m}$ with the finite
${\beta}$-norm

$\|f\|_{L_{r,\mu}({\mathbb{R}}^{m})}:=\left\\{\begin{array}[]{ll}\left(\int_{{\mathbb{R}}^{m}}|f(x)|^{r}d\mu(x)\right)^{1/r},&0<r<{\infty},\\\
\operatorname*{ess\,sup}_{x\in{\mathbb{R}}^{m}}|f(x)|,&r={\infty},\end{array}\right.$
(6.3)

where ${\beta}=\min\\{1,r\\}$. For a $\mu$-measurable set
${\Omega}\subset{\mathbb{R}}^{m}$, we also use the ${\beta}$-normed space
$L_{r,\mu}({\Omega})$ ($r\in(0,{\infty}],\,{\beta}=\min\\{1,r\\}$).

In all examples of Chapter 7, $\mu$ is a weighted measure on
${\mathbb{R}}^{m}$, i.e., $d\mu(x)=W(x)dx,\,x\in{\mathbb{R}}^{m}$, where $W$
is a locally Lebesgue-integrable weight on ${\mathbb{R}}^{m}$. In particular,
we use the weighted measure with a univariate power weight $W(x)=|x|^{2\nu+1}$
in Example 7.4. In case of the Lebesgue measure $\mu$ (i.e, $W=1$), we use the
notation $L_{r}({\mathbb{R}}^{m})$ and $L_{r}({\Omega}),\,r\in(0,{\infty}]$,
in Examples 7.2, 7.3, 7.8, 7.9, 7.15, and 7.16.

In Example 7.10 we use a univariate exponential weight
$W(x)=W_{r}(x):=\exp(-rQ(x)),\,r\in(0,{\infty})$, where
$Q:{\mathbb{R}}^{1}\to[0,{\infty})$ is an even differentiable convex function
(see, e.g., [56, 36] for applications of $W_{r}$ in analysis). A typical
example of $W_{r}$ is a Freud weight
$W_{r,{\alpha}}(x):=\exp(-r|x|^{\alpha})),\,{\alpha}>1$. The corresponding
${\beta}$-norm is a version of (6.3) in the following form:

$\|f\|_{L_{r}(W_{r},{\mathbb{R}}^{1})}:=\left(\int_{{\mathbb{R}}^{1}}|f(x)\exp(-Q(x))|^{r}dx\right)^{1/r},\qquad
0<r<{\infty},$ (6.4)

where ${\beta}=\min\\{1,r\\}$. An extension of (6.4) to $r={\infty}$ is given
by

$\|f\|_{L_{{\infty}}(W_{\infty},{\mathbb{R}}^{1})}:=\operatorname*{ess\,sup}_{x\in
R^{1}}|f(x)\exp(-Q(x))|.$ (6.5)

The space $L_{r}(W_{r},{\mathbb{R}}^{1})$
($r\in(0,{\infty}],\,{\beta}=\min\\{1,r\\}$) is the space of all Lebesgue-
measurable complex-valued functions $f$ on ${\mathbb{R}}^{1}$ with the finite
monotone and $\boldsymbol{{\omega}}$-continuous ${\beta}$-norms (6.4) and
(6.5). Note that norm (6.5) is different, compared with norm (6.3) for
$r={\infty}$.

One more typical example of a space with a monotone and
$\boldsymbol{{\omega}}$-continuous norm is the normed space
$C_{0}({\mathbb{R}}^{m})$ of all complex-valued continuous functions $f$ on
${\mathbb{R}}^{m}$ with the ”point” norm
$\|f\|_{C_{0}({\mathbb{R}}^{m})}:=|f(0)|$. For a compact set
${\Omega}\subset{\mathbb{R}}^{m},\,0\in{\Omega},$ we can also define the
normed space $C_{0}({\Omega})$. We use these norms in Examples 7.3 and 7.4.

Given $a>0$, the space of all functions $f$ from
$L_{\infty}({\mathbb{R}}^{m})$ that are $2\pi a$-periodic in each variable is
denoted by $\tilde{L}_{{\infty},2\pi a,m}$ with the norm
$\|f\|_{\tilde{L}_{{\infty},2\pi
a,m}}=\|f\|_{L_{\infty}(R^{m})}=\|f\|_{L_{\infty}(Q^{m}_{\pi a})}$ (see
Example 7.15).

###  6.3. Function classes

In the capacity of sets $B$ and $B_{n}$ from Sections 3, 4, and 5, we use the
following function classes.

Let ${\mathcal{P}}_{n,m}$ be a set of all polynomials
$P(y)=\sum_{|{\beta}|\leq n}c_{\beta}y^{\beta}$ in $m$ variables of degree at
most $n,\,n\in Z^{1}_{+}$, with complex coefficients. Given $a>0$, the set of
all trigonometric polynomials $Q(x)=\sum_{k\in
aV\cap{\mathbb{Z}}^{m}}c_{k}e^{(ik)\cdot x}$ with complex coefficients is
denoted by ${\mathcal{T}}_{aV}$ and the set of all trigonometric polynomials
$T(x)=\sum_{k\in aV\cap{\mathbb{Z}}^{m}}c_{k}e^{(ik)\cdot(x/a)}$ with complex
coefficients is denoted by ${\mathcal{T}}_{a,V}$.

######  Definition 6.3.

We say that an entire function $f:{\mathbb{C}}^{m}\to{\mathbb{C}}^{1}$ has
exponential type $V$ if for any ${\varepsilon}>0$ there exists a constant
$C_{0}({\varepsilon},f)$ such that for all $z\in{\mathbb{C}}$, $|f(z)|\leq
C_{0}({\varepsilon},f)\exp\left((1+{\varepsilon})\|z\|_{V}^{*}\right)$.

The set of all entire function of exponential type $V$ is denoted by $B_{V}$.
The set $B_{V}$ was defined by Stein and Weiss [71, Sect. 3.4]. For
$V=\Pi^{m}_{\sigma},\,V=Q^{m}_{M},$ and $V={\mathfrak{B}}^{m}_{M}$, similar
sets of entire functions were defined by Bernstein [14] and Nikolskii [64],
[65, Sects. 3.1, 3.2.6], see also [24, Definition 5.1]. Properties of
functions from $B_{V}$ have been investigated in numerous publications (see,
e.g., [14, 64, 65, 71, 61, 29, 30, 33] and references therein).

Throughout the remaining sections of the paper, if no confusion may occur, the
same notation is applied to functions $f\in
B_{V},\,f\in{\mathcal{T}}_{aV},\,f\in{\mathcal{T}}_{a,V}$, and
$f\in{\mathcal{P}}_{n,m}$ and their restrictions to ${\mathbb{R}}^{m}$ or to a
subset ${\Omega}$ of ${\mathbb{R}}^{m}$ (e.g., in the form $f\in B_{V}\cap
L_{p}({\mathbb{R}}^{m}))$.

Let $M_{C,N}$ be the set of all Lebesgue measurable functions $f$, satisfying
the inequality

$\displaystyle|f(x)|\leq C(1+|x|)^{N}$

for a. a. $x\in{\mathbb{R}}^{m}$, where $C>0$ and $N\geq 0$ are constants
independent of $x$. In particular, we discuss classes ${\mathcal{T}}_{a,V}\cap
M_{C,N}$ in Examples 7.8 and 7.15.

In addition, let ${\omega}:[0,{\infty})\to[0,{\infty})$ be a modulus of
continuity, i.e., ${\omega}(0)=0$ and ${\omega}$ is a continuous,
nondecreasing, and subadditive function on $[0,{\infty})$. Next, we define the
class $H_{\omega}(\Omega)$ of all functions $f$ on
$\Omega\subseteq{\mathbb{R}}^{m}$, satisfying the inequality

$\displaystyle|f(x)-f(y)|\leq{\omega}(|x-y|),\qquad x\in\Omega,\quad
y\in\Omega.$

We also need a subclass $H_{\omega,0}({\mathbb{R}}^{m}):=\\{f\in
H_{\omega}({\mathbb{R}}^{m}):f(0)=0\\}$ of $H_{\omega}({\mathbb{R}}^{m})$.
Note that

$H_{\omega,0}({\mathbb{R}}^{m})\subseteq M_{{\omega}(2),1}.$ (6.6)

Indeed, if $f\in H_{\omega,0}({\mathbb{R}}^{m})$, then
$|f(x)|\leq{\omega}(|x|),\,x\in{\mathbb{R}}^{m}$. Next,
$\max_{\tau\in[0,2]}{\omega}(\tau)={\omega}(2)$, and if $2^{k}<\tau\leq
2^{k+1},\,k\in{\mathbb{N}}$, then
${\omega}(\tau)\leq{\omega}(2^{k+1})<\tau{\omega}(2)$. Therefore,
$|f(x)|\leq{\omega}(2)(1+|x|),\,x\in{\mathbb{R}}^{m}$, and (6.6) holds true.

In case of
${\omega}(\tau)=\tau^{\lambda},\,\tau\in[0,{\infty}),\,0<{\lambda}\leq 1$, the
class $H^{\lambda}(\Omega):=H_{\omega}(\Omega)$ is called the Hölder class.
For $m=1$ we also define the class $W^{s}H_{\omega}({\mathbb{R}}^{1})$ of all
$s$-differentiable functions on ${\mathbb{R}}^{1}$ with $f^{(s)}\in
H_{\omega}({\mathbb{R}}^{1}),\,s\in{\mathbb{Z}}^{1}_{+}$.

We also recall that the error of best approximation of ${\varphi}\in
F({\Omega})$ by elements from a subspace $B$ of $F({\Omega})$ is defined by

$E({\varphi},B,F({\Omega})):=\inf_{f\in B}\|{\varphi}-f\|_{F({\Omega})}$ (6.7)

(see (2.9)).

Certain properties of functions from defined classes are discussed below. We
first prove multivariate versions of two known univariate results (see, e.g.,
[74, Sects. 4.10.2 and 2.6.21]).

######  Lemma 6.4.

If a function $f\in B_{V}$ has a period of $2\pi a$ in each variable, then
$f\in{\mathcal{T}}_{a,V},\,a>0$.

###### Proof.

The entire function $f$ coincides with its Fourier series
$\sum_{k\in{\mathbb{Z}}^{m}}c_{k}e^{(ik)\cdot(x/a)}$. Since $f\in
L_{\infty}({\mathbb{R}}^{m})$, this function is a tempered distribution on
${\mathbb{R}}^{m}$. Then by the generalized Paley-Wiener theorem [61, p. 13],
the support of the Fourier transform of $f$ is a subset of $V$. So $c_{k}=0$
for $k\in{\mathbb{Z}}^{m}\setminus aV$. $\Box$

######  Lemma 6.5.

If ${\varphi}\in\tilde{L}_{{\infty},2\pi a,m}$, then

$E({\varphi},B_{V},L_{\infty}({\mathbb{R}}^{m}))=E({\varphi},{\mathcal{T}}_{a,V},L_{\infty}({\mathbb{R}}^{m})).$
(6.8)

###### Proof.

Since ${\mathcal{T}}_{a,V}\subseteq B_{V}$, we have

$E({\varphi},B_{V},L_{\infty}({\mathbb{R}}^{m}))\leq
E({\varphi},{\mathcal{T}}_{a,V},L_{\infty}({\mathbb{R}}^{m})).$ (6.9)

Next given ${\varepsilon}\in(0,{\infty})$, let $g_{\varepsilon}\in B_{V}\cap
L_{\infty}({\mathbb{R}}^{m})$ satisfy the relation

$\|{\varphi}-g_{\varepsilon}\|_{L_{\infty}({\mathbb{R}}^{m})}<E({\varphi},B_{V},L_{\infty}({\mathbb{R}}^{m}))+{\varepsilon}.$
(6.10)

Then the function

$\displaystyle G_{N}(x):=(2N+1)^{-m}\sum_{k\in
Q_{N}^{m}}g_{\varepsilon}(x+2\pi ak),\qquad N\in{\mathbb{N}},$

belongs to $B_{V}$ and by (6.10),

$\|{\varphi}-G_{N}\|_{L_{\infty}({\mathbb{R}}^{m})}<E({\varphi},B_{V},L_{\infty}({\mathbb{R}}^{m}))+{\varepsilon}.$
(6.11)

Moreover,
$\sup_{N\in{\mathbb{N}}}\|G_{N}\|_{L_{\infty}({\mathbb{R}}^{m})}\leq\|g_{\varepsilon}\|_{L_{\infty}({\mathbb{R}}^{m})}$.
Applying the compactness theorem (see [41, Lemma 2.3]) to the sequence
$\\{G_{N}\\}_{N=1}^{\infty}$ we see that there exist a sequence of natural
numbers $\\{N_{s}\\}_{s=1}^{\infty}$ and a function $G_{0}\in B_{V}$ such that

$\lim_{s\to{\infty}}G_{N_{s}}(x)=G_{0}(x)$ (6.12)

uniformly on any compact $K\subset{\mathbb{R}}^{m}$. Then (6.11) and (6.12)
imply that

$\|{\varphi}-G_{0}\|_{L_{\infty}({\mathbb{R}}^{m})}\leq
E({\varphi},B_{V},L_{\infty}({\mathbb{R}}^{m}))+{\varepsilon}.$ (6.13)

Next, denoting by $e_{j}$ the $j$th unit vector in ${\mathbb{R}}^{m},\,1\leq
j\leq m$, we have

$\max_{1\leq j\leq m}\left\|G_{N}(\cdot+2\pi
ae_{j})-G_{N}(\cdot)\right\|_{L_{\infty}(R^{m})}\leq C_{7}/N,$ (6.14)

where a constant $C_{7}>0$ is independent of $N$. Setting $N=N_{s}$ and
letting $s\to{\infty}$ in (6.14), we see from (6.12) and (6.14) that $G_{0}$
is $2\pi a$-periodic in each variable. Therefore,
$G_{0}\in{\mathcal{T}}_{a,V}$ by Lemma 6.4, and it follows from (6.13) that

$E({\varphi},{\mathcal{T}}_{a,V},L_{\infty}({\mathbb{R}}^{m}))\leq\|{\varphi}-G_{0}\|_{L_{\infty}({\mathbb{R}}^{m})}\leq
E({\varphi},B_{V},L_{\infty}({\mathbb{R}}^{m}))+{\varepsilon}.$ (6.15)

Letting ${\varepsilon}\to 0+$ in (6.15) and taking account of (6.9), we arrive
at (6.8). $\Box$
We also need the following extension theorem.

######  Lemma 6.6.

Given function $f:Q_{M}^{m}\to{\mathbb{R}}^{1},\,M>0$, there exists a function
$\tilde{f}:{\mathbb{R}}^{m}\to{\mathbb{R}}^{1}$ such that $\tilde{f}=f$ on
$Q_{M}^{m}$ and $\tilde{f}$ is $4M$-periodic in each variable. In addition, if
$f\in H_{\omega}(Q_{M}^{m})$, then $\tilde{f}\in
H_{\omega}({\mathbb{R}}^{m})$.

###### Proof.

Let us set

$h_{p}(x):=(-1)^{k_{p}}(x_{p}-2Mk_{p}),\quad x\in Q_{M}^{m}+2Mk,\quad 1\leq
p\leq m,\quad k=(k_{1},\ldots,k_{m})\in{\mathbb{Z}}^{m}.$ (6.16)

Then $h(x)=(h_{1}(x),\ldots,h_{m}(x)):{\mathbb{R}}^{m}\to Q_{M}^{m}$ is a
vector-valued function. Indeed, this statement is obviously true for any
interior point $x$ of $Q_{M}^{m}+2Mk,\,k\in{\mathbb{Z}}^{m}$. Let $x$ belong
to the boundary of $Q_{M}^{m}+2Mk,\,k\in{\mathbb{Z}}^{m}$. Then without loss
of generality we can assume that there exists $s\in{\mathbb{N}},\,1\leq s\leq
m$, such that

$\displaystyle x_{p}=-M+2Mk_{p}=M+2M(k_{p}-1),\qquad 1\leq p\leq s,$
$\displaystyle x_{p}=x_{p}^{*}+2Mk_{p},\qquad|x_{p}^{*}|<M,\qquad s+1\leq
p\leq m.$

It follows from (6.16) that $h(x_{p})=(-1)^{k_{p}+1}M,\,1\leq p\leq s$, and
$h(x_{p})$ for $s+1\leq p\leq m$ is uniquely defined by (6.16) as well.
Therefore, $h$ is a function.

Next, $h$ is $4M$-periodic in each variable. Indeed, if $x\in
Q_{M}^{m}+2Mk,\,k\in{\mathbb{Z}}^{m}$, then for every
$l=(l_{1},\ldots,l_{m})\in{\mathbb{Z}}^{m}$,

$\displaystyle
h_{p}(x+4Ml)=(-1)^{k_{p}+2l_{p}}(x_{p}+4Ml_{p}-2M(k_{p}+2l_{p}))=h_{p}(x),\qquad
1\leq p\leq m.$

In addition, $h(x)=x$ on $Q_{M}^{m}$. Then
$|h(x)-h(x^{*})|\leq|x-x^{*}|,\,x\in{\mathbb{R}}^{m},\,x^{*}\in{\mathbb{R}}^{m}.$
Indeed, it suffices to show that $y_{p}=h_{p}(x_{p})\in
H^{1}({\mathbb{R}}^{1}),\,1\leq p\leq m$. Since $y_{p}$ is $4M$-periodic on
${\mathbb{R}}^{1}$ and the curve has the slope of $1$ on $[-M,M]$ and the
slope of $-1$ on $[-2M,-M]$ and $[M,2M]$, we see that $y_{p}\in
H^{1}({\mathbb{R}}^{1}),\,1\leq p\leq m$.

Finally, the function $\tilde{f}(x):=f(h(x))$ satisfies all conditions of the
lemma. $\Box$
Note that for $m=1$ this result can be found in [74, Sect. 3.5.71], while for
$m>1$ Shvedov [69, Proof of Theorem 4] presented a geometric construction of
$\tilde{f}$ with no proof provided.

###  6.4. Operators

In the capacity of operators $D$ and $D_{n}$ from Sections 3, 4, and 5, we use
linear operators, in particular, linear differential operators of the form
$d_{N}:=\sum_{|{\alpha}|=N}b_{\alpha}D^{\alpha}$ with constant coefficients
$b_{\alpha}\in{\mathbb{C}}^{1},\,{\alpha}\in{\mathbb{Z}}^{m}_{+},\,|{\alpha}|=N,\,N\in{\mathbb{Z}}^{1}_{+}$.
In case of $N=0$ we assume that $d_{N}$ is the corresponding imbedding (or
identity) operator $I$.

In the capacity of operators
$L_{n}^{(1)},\,L_{n}^{(2)},\,l_{n}^{(1)},\,l_{n}^{(2)},\,l_{n,s}^{(1)}$ and
$l_{n,s}^{(2)}$ from the conditions of Section 3.2 and also from Sections 4
and 5, we use the following linear restriction operator. Let ${\Omega}_{1}$
and ${\Omega}_{2}$ be $\mu$-measurable subsets of ${\mathbb{R}}^{m}$ such that
${\Omega}_{2}\subseteq{\Omega}_{1}$ and let
$R=R({\Omega}_{1},{\Omega}_{2}):F({\Omega}_{1})\to F({\Omega}_{2})$ be the
restriction operator. For $\Omega\subseteq{\mathbb{R}}^{m}$ we also use the
notation $D_{n}\left|{}_{\Omega}\right.:=D_{n}R({\mathbb{R}}^{m},\Omega)$. If
no confusion may occur, we often use the notation $D_{n}$ instead of
$D_{n}\left|{}_{\Omega}\right.$.

The following proposition is a direct consequence of (6.1) and (6.2).

######  Proposition 6.7.

(i) If $F({\mathbb{R}}^{m})$ is a space with a monotone norm, then
$\|R({\Omega}_{1},{\Omega}_{2})\|\leq 1$, i.e., for any $h\in
F({\Omega}_{1})$,

$\displaystyle\|R({\Omega}_{1},{\Omega}_{2})\,(h)\|_{F({\Omega}_{2})}\leq\|h\|_{F({\Omega}_{1})}.$

(ii) Let $\boldsymbol{{\omega}}=\\{{\Omega}_{n}\\}_{n=1}^{\infty}$ be any
increasing sequence of $\mu$-measurable sets in ${\mathbb{R}}^{m}$ with
$\cup_{n=1}^{\infty}{\Omega}_{n}={\mathbb{R}}^{m}$. If $F({\mathbb{R}}^{m})$
is a space with a $\boldsymbol{{\omega}}$-continuous norm, then for any $h\in
F({\mathbb{R}}^{m})$,

$\displaystyle\lim_{n\to{\infty}}\|R({\mathbb{R}}^{m},{\Omega}_{n})\,(h)\|_{F({\Omega}_{n})}=\|h\|_{F({\mathbb{R}}^{m})}.$

In addition, for a set $A$ of functions
$f:{\mathbb{R}}^{m}\to{\mathbb{C}}^{1}$ we define by
$A\left|{}_{\Omega}\right.:=\\{R({\mathbb{R}}^{m},\Omega)(f):f\in A\\}$ the
set of the restrictions to $\Omega\subseteq{\mathbb{R}}^{m}$ of functions from
$A$. In Section 7, if no confusion may occur, we often do not use this
restriction notation for sets
$B_{V},\,{\mathcal{T}}_{aV},\,{\mathcal{T}}_{a,V},\,{\mathcal{P}}_{n,m}$, and
their subsets.

##  7\. Examples of Asymptotic Relations between Sharp Constants in Function
Spaces

In this section we discuss specified conditions and examples of relations
between sharp constants in function spaces separately for each major Problem
A, B, and C of approximation theory described in Sections 2.3 and 5.

Throughout the section $\\{{\Omega}_{n}\\}_{n=1}^{\infty}$ and
$\\{{\Omega}_{n}^{*}\\}_{n=1}^{\infty}$ are increasing sequences of
$\mu$-measurable sets in ${\mathbb{R}}^{m}$ with
$\cup_{n=1}^{\infty}{\Omega}_{n}={\mathbb{R}}^{m}$ and
$\cup_{n=1}^{\infty}{\Omega}_{n}^{*}={\mathbb{R}}^{m}$.

###  7.1. Inequalities for Approximating Elements (Problem A)

Here, we discuss a special case of Corollary 5.1 and three examples.
General result. We assume that all assumptions of Section 5.1 are held in the
following special cases. First, we assume that
$F^{(j)}=F^{(j)}({\mathbb{R}}^{m})=F^{(j)}({\mathbb{R}}^{m},\mu)$ is a
${\beta}_{j}$-normed functional space with the monotone and
$\boldsymbol{{\omega}}$-continuous ${\beta}_{j}$-norm (see Definitions 6.1 and
6.2), ${\beta}_{j}\in(0,1]$, and $F^{(j)}_{n}=F^{(j)}({\Omega}_{n})$ or
$F^{(j)}_{n}=F^{(j)}({\Omega}_{n}^{*}),\,j=1,2,\,n\in{\mathbb{N}}$. Next, we
define the restriction operators
$L_{n}^{(1)}=R({\mathbb{R}}^{m},{\Omega}_{n}^{*}),\,l_{n}^{(1)}=l_{n}^{(2)}=R({\mathbb{R}}^{m},{\Omega}_{n}),\,n\in{\mathbb{N}}$,
and

$\displaystyle
l_{n,s}^{(1)}=l_{n,s}^{(2)}=\left\\{\begin{array}[]{ll}R({\Omega}_{n},{\Omega}_{s}),&n\geq
s,\\\ R({\Omega}_{s},{\Omega}_{n}),&n<s,\end{array}\right.\qquad
n\in{\mathbb{N}},\,s\in{\mathbb{N}}.$

We also assume that $B,\,B_{n}$, and $B_{n}^{*}$ are nontrivial subspaces of
$F^{(2)}({\mathbb{R}}^{m}),\,F^{(2)}({\Omega}_{n})$, and
$F^{(2)}({\Omega}_{n}^{*})$, respectively, $n\in{\mathbb{N}}$. In addition,
let $D:B\to F^{(1)}({\mathbb{R}}^{m}),\,D_{n}:B_{n}\to F^{(1)}({\Omega}_{n})$,
and $D_{n}^{*}:B_{n}^{*}\to F^{(1)}({\Omega}_{n}^{*}),\,n\in{\mathbb{N}}$, be
linear operators. Finally, let $B_{n}^{\prime}\subseteq\left\\{f_{n}\in
B_{n}\setminus\\{0\\}:\|D_{n}(f_{n})\|_{F^{(1)}({\Omega}_{n})}=1\right\\}$ be
a set from Condition C9) such that the following relations hold:

$\displaystyle\sup_{f_{n}\in
B_{n}^{\prime}}\frac{1}{\|f_{n}\|_{F^{(2)}({\Omega}_{n})}}=\sup_{f_{n}\in
B_{n}^{\prime}}\frac{\|D_{n}(f_{n})\|_{F^{(1)}({\Omega}_{n})}}{\|f_{n}\|_{F^{(2)}({\Omega}_{n})}}=\sup_{f\in
B_{n}\setminus\\{0\\}}\frac{\|D_{n}(f)\|_{F^{(1)}({\Omega}_{n})}}{\|f\|_{F^{(2)}({\Omega}_{n})}},\qquad
n\in{\mathbb{N}}.$

The existence of $B_{n}^{\prime},\,n\in{\mathbb{N}}$, follows immediately from
Proposition 3.7 (i).

Below we introduce some conditions on families
$\left\\{D,B,F^{(1)}({\mathbb{R}}^{m}),F^{(2)}({\mathbb{R}}^{m})\right\\},\linebreak\left\\{D_{n},B_{n},F^{(1)}({\Omega}_{n}),F^{(2)}({\Omega}_{n})\right\\}$,
and
$\left\\{D_{n}^{*},B_{n}^{*},F^{(1)}({\Omega}_{n}^{*}),F^{(2)}({\Omega}_{n}^{*})\right\\},\,n\in{\mathbb{N}}$,
that are needed for the validity of relations like (5.1) and/or (5.2) in
function spaces.
_Condition C3.1)._ For every $f\in B\setminus\\{0\\}$ there exists a sequence
$f_{n}^{*}\in B_{n}^{*},\,n\in{\mathbb{N}}$, such that

$\limsup_{n\to{\infty}}\|f_{n}^{*}\|_{F^{(2)}({\Omega}_{n}^{*})}\leq\|f\|_{F^{(2)}({\mathbb{R}}^{m})}$
(7.2)

and

$\lim_{n\to{\infty}}\left\|D(f)-D_{n}^{*}(f_{n}^{*})\right\|_{F^{(1)}({\Omega}_{n}^{*})}=0.$
(7.3)

_Condition C3.1.1)._ Let
${\Omega}_{n}^{*}\subseteq{\Omega}_{n},\,n\in{\mathbb{N}}$. For every $f\in
B\setminus\\{0\\}$ there exists a sequence $f_{n}\in
B_{n},\,n\in{\mathbb{N}}$, such that

$\limsup_{n\to{\infty}}\|f_{n}\|_{F^{(2)}({\Omega}_{n})}\leq\|f\|_{F^{(2)}({\mathbb{R}}^{m})}$
(7.4)

and

$\lim_{n\to{\infty}}\left\|D(f)-D_{n}(f_{n})\right\|_{F^{(1)}({\Omega}_{n}^{*})}=0.$
(7.5)

_Condition C10.1)._ (Compactness condition) For any sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ such that $f_{n}\in
B_{n}^{\prime},\,n\in{\mathbb{N}}$, there exist a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers and an element $f\in B$ such
that

$\displaystyle\lim_{k\to{\infty}}\left\|D_{n_{k}}(f_{n_{k}})-D(f)\right\|_{F^{(1)}\left({\Omega}_{n_{k}}\right)}=0,$
(7.6)
$\displaystyle\lim_{k\to{\infty}}\left\|f_{n_{k}}-f\right\|_{F^{(2)}\left({\Omega}_{n_{k}}\right)}=0.$
(7.7)

_Condition C12.1)._ (Weak compactness condition) For any sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ such that $f_{n}\in
B_{n}^{\prime},\,n\in{\mathbb{N}}$, there exists $f\in B$ such that for any
fixed $s\in{\mathbb{N}}$, there exists a sequence $\\{n_{k}\\}_{k=1}^{\infty}$
of natural numbers ($n_{k}=n_{k}(s),\,k\in{\mathbb{N}}$) such that

$\displaystyle\lim_{k\to{\infty}}\left\|D_{n_{k}}(f_{n_{k}})-D(f)\right\|_{F^{(1)}\left({\Omega}_{s}\right)}=0,$
(7.8)
$\displaystyle\lim_{k\to{\infty}}\left\|f_{n_{k}}-f\right\|_{F^{(2)}\left({\Omega}_{s}\right)}=0,$
(7.9)

and, in addition,

$\left\|D_{n_{k}}(f_{n_{k}})\right\|_{F^{(1)}\left({\Omega}_{n_{k}}\right)}=\left\|D_{n_{k}}(f_{n_{k}})\right\|_{F^{(1)}\left({\Omega}_{s}\right)},\quad
k\in{\mathbb{N}}.$ (7.10)

The following corollary holds true.

######  Corollary 7.1.

(i) If Condition C3.1) is satisfied, then

$\sup_{f\in
B\setminus\\{0\\}}\frac{\|D(f)\|_{F^{(1)}({\mathbb{R}}^{m})}}{\|f\|_{F^{(2)}({\mathbb{R}}^{m})}}\leq\liminf_{n\to{\infty}}\sup_{f\in
B_{n}^{*}\setminus\\{0\\}}\frac{\|D_{n}^{*}(f)\|_{F^{(1)}({\Omega}_{n}^{*})}}{\|f\|_{F^{(2)}({\Omega}_{n}^{*})}}.$
(7.11)

(ii) If Condition C3.1.1) is satisfied, then

$\sup_{f\in
B\setminus\\{0\\}}\frac{\|D(f)\|_{F^{(1)}({\mathbb{R}}^{m})}}{\|f\|_{F^{(2)}({\mathbb{R}}^{m})}}\leq\liminf_{n\to{\infty}}\sup_{f\in
B_{n}\setminus\\{0\\}}\frac{\|D_{n}(f)\|_{F^{(1)}({\Omega}_{n})}}{\|f\|_{F^{(2)}({\Omega}_{n})}}.$
(7.12)

(iii) Let

$\displaystyle{\mathcal{C}}\left(D_{n},B_{n},F^{(1)}({\Omega}_{n}),F^{(2)}({\Omega}_{n})\right)=\sup_{f\in
B_{n}\setminus\\{0\\}}\frac{\|D_{n}(f)\|_{F^{(1)}({\Omega}_{n})}}{\|f\|_{F^{(2)}({\Omega}_{n})}}$

(see (3.2)) be a finite number for every $n\in{\mathbb{N}}$, and let Condition
C10.1) be satisfied. Then

$\sup_{f\in
B\setminus\\{0\\}}\frac{\|D(f)\|_{F^{(1)}({\mathbb{R}}^{m})}}{\|f\|_{F^{(2)}({\mathbb{R}}^{m})}}\geq\limsup_{n\to{\infty}}\sup_{f\in
B_{n}\setminus\\{0\\}}\frac{\|D_{n}(f)\|_{F^{(1)}({\Omega}_{n})}}{\|f\|_{F^{(2)}({\Omega}_{n})}}.$
(7.13)

(iv) Let
${\mathcal{C}}\left(D_{n},B_{n},F^{(1)}({\Omega}_{n}),F^{(2)}({\Omega}_{n})\right)$
be a finite number for every $n\in{\mathbb{N}}$, and let Condition C12.1) be
satisfied. Then (7.13) holds true.
(v) Let ${\Omega}_{n}^{*}\subseteq{\Omega}_{n}$ and let $B_{n}^{*}$ be a set
of the restrictions $(f_{n})^{*}:=R({\mathbb{R}}^{m},\Omega_{n}^{*})(f_{n})$
to ${\Omega}_{n}^{*}$ of all elements $f_{n}$ from $B_{n},\,n\in{\mathbb{N}}$.
In addition, let
$D_{n}^{*}((f_{n})^{*}):=(D_{n}(f_{n}))^{*},\,n\in{\mathbb{N}}$. Next, let
${\mathcal{C}}\left(D_{n},B_{n},F^{(1)}({\Omega}_{n}),F^{(2)}({\Omega}_{n})\right)$
be a finite number for every $n\in{\mathbb{N}}$, and let Condition C3.1) and
either C10.1) or C12.1) be satisfied. Then

$\displaystyle\limsup_{n\to{\infty}}\sup_{f\in
B_{n}\setminus\\{0\\}}\frac{\|D_{n}(f)\|_{F^{(1)}({\Omega}_{n})}}{\|f\|_{F^{(2)}({\Omega}_{n})}}$
$\displaystyle\leq$ $\displaystyle\sup_{f\in
B\setminus\\{0\\}}\frac{\|D(f)\|_{F^{(1)}({\mathbb{R}}^{m})}}{\|f\|_{F^{(2)}({\mathbb{R}}^{m})}}$
(7.14) $\displaystyle\leq$ $\displaystyle\liminf_{n\to{\infty}}\sup_{f\in
B_{n}^{*}\setminus\\{0\\}}\frac{\|D_{n}^{*}(f)\|_{F^{(1)}({\Omega}_{n}^{*})}}{\|f\|_{F^{(2)}({\Omega}_{n}^{*})}}.$

(vi) Let
${\mathcal{C}}\left(D_{n},B_{n},F^{(1)}({\Omega}_{n}),F^{(2)}({\Omega}_{n})\right)$
be a finite number for every $n\in{\mathbb{N}}$, and let Condition C3.1.1) and
either C10.1) or C12.1) be satisfied. Then

$\sup_{f\in
B\setminus\\{0\\}}\frac{\|D(f)\|_{F^{(1)}({\mathbb{R}}^{m})}}{\|f\|_{F^{(2)}({\mathbb{R}}^{m})}}=\lim_{n\to{\infty}}\sup_{f\in
B_{n}\setminus\\{0\\}}\frac{\|D_{n}(f)\|_{F^{(1)}({\Omega}_{n})}}{\|f\|_{F^{(2)}({\Omega}_{n})}}.$
(7.15)

###### Proof.

Note first that in this proof all elements in discussed Conditions C1) through
C13) are replaced without further notice by the corresponding elements that
are used in Conditions C3.1), C3.1.1), C10.1), C12.1) and relations (7.11)
through (7.15). For example, in Condition C3) $B_{n}$ is replaced by
$B_{n}^{*}$.

By the assumptions about the functional spaces and operators introduced above,
Conditions C2), C4), C6), C6a), C6b), C11), and C11a) of Section 3.2 are
satisfied. Indeed, Conditions C2), C6), and C11) are satisfied by the direct
definitions of the operators whose existence is required in these conditions.
In addition, Conditions C4), C6a), C6b), and C11a) are satisfied by
Proposition 6.7. Next, Conditions C3.1), C10.1), and C12.1) of this section
are equivalent to Conditions C3), C10), and C12) of Section 3.2, respectively.
Therefore, statements (i), (iii), and (iv) of Corollary 7.1 follow immediately
from statements (i), (ii), and (iii) of Corollary 5.1, respectively.

To prove statement (ii), we recall that the norm
$\|\cdot\|_{F^{(1)}({\mathbb{R}}^{m})}$ is monotone and
$\boldsymbol{{\omega}}$-continuous. So for any $f\in B\setminus\\{0\\}$, we
obtain by Proposition 6.7 and relation (7.5) of Condition C3.1.1)

$\displaystyle\|D(f)\|_{F^{(1)}({\mathbb{R}}^{m})}=\lim_{n\to{\infty}}\|D(f)\|_{F^{(1)}({\Omega}_{n}^{*})}$
$\displaystyle\leq\liminf_{n\to{\infty}}\left(\|D(f)-D_{n}(f_{n})\|_{F^{(1)}({\Omega}_{n}^{*})}^{{\beta}_{1}}+\|D_{n}(f_{n})\|_{F^{(1)}({\Omega}_{n}^{*})}^{{\beta}_{1}}\right)^{1/{\beta}_{1}}$
$\displaystyle=\liminf_{n\to{\infty}}\left\|D_{n}(f_{n})\right\|_{F^{(1)}({\Omega}_{n}^{*})}\leq\liminf_{n\to{\infty}}\left\|D_{n}(f_{n})\right\|_{F^{(1)}({\Omega}_{n})}$
$\displaystyle\leq\liminf_{n\to{\infty}}\sup_{f\in
B_{n}\setminus\\{0\\}}\frac{\|D_{n}(f)\|_{F^{(1)}({\Omega}_{n})}}{\|f\|_{F^{(2)}({\Omega}_{n})}}\limsup_{n\to{\infty}}\left(\|f_{n}\|_{F^{(2)}({\Omega}_{n})}\right).$
(7.16)

Then (7.12) follows from (7.1) and (7.4). Finally, statement (v) follows from
(i), (iii), and (iv), and statement (vi) follows from (ii), (iii), and (iv).
$\Box$
Examples. There is a number of publications [72, 73, 47, 57, 58, 44, 38, 41,
42, 43] about special cases of Corollary 7.1; here, we call them Bernstein-
Nikolskii type inequalities or inequalities of different metrics (see also
Section 2.3 A). Examples 7.2, 7.3, and 7.4 below discuss inequalities (7.12),
(7.14) and (7.15) proved in these papers in various situations.

######  Example 7.2.

(Multivariate Bernstein-Nikolskii type inequalities). Let
$F^{(1)}({\mathbb{R}}^{m})=L_{q}({\mathbb{R}}^{m}),\,{\beta}_{1}=\min\\{1,q\\},\,F^{(2)}({\mathbb{R}}^{m})=L_{p}({\mathbb{R}}^{m}),\,{\beta}_{2}=\min\\{1,p\\},\,B=B_{V}\cap
L_{p}({\mathbb{R}}^{m}),\,0<p\leq q\leq{\infty},\,{\Omega}_{n}=Q_{\pi
n}^{m},\,{\Omega}_{n}^{*}=Q_{n^{\delta}}^{m},\,{\delta}={\delta}(p,m)\in(0,1),\,B_{n}={\mathcal{T}}_{n,V}\left|{}_{\Omega_{n}}\right.={\mathcal{T}}_{n,V},\,B_{n}^{*}={\mathcal{T}}_{n,V}\left|{}_{\Omega_{n}^{*}}\right.={\mathcal{T}}_{n,V},\,n\in{\mathbb{N}}.$
Next, let $d_{N}=\sum_{|{\alpha}|=N}b_{\alpha}D^{\alpha}$, where
$b_{\alpha}\in{\mathbb{C}}^{1}$
(${\alpha}\in{\mathbb{Z}}^{m}_{+},\,|{\alpha}|=N$), and
$N\in{\mathbb{Z}}^{1}_{+}$, are independent of $n$, and let
$D=d_{N}\left|{}_{{\mathbb{R}}^{m}}\right.=d_{N},\,D_{n}=d_{N}\left|{}_{\Omega_{n}}\right.=d_{N},\,D_{n}^{*}=d_{N}\left|{}_{\Omega_{n}^{*}}\right.=d_{N},\,n\in{\mathbb{N}}$.

Inequality (7.12) for $0<p\leq q\leq{\infty}$ and equality (7.15) for
$0<p\leq{\infty}$ and $q={\infty}$ were proved by the author [41] in the
following form:

$\displaystyle\sup_{f\in(B_{V}\cap
L_{p}({\mathbb{R}}^{m}))\setminus\\{0\\}}\frac{\|d_{N}(f)\|_{L_{q}({\mathbb{R}}^{m})}}{\|f\|_{L_{p}({\mathbb{R}}^{m})}}\leq\liminf_{n\to{\infty}}\sup_{T\in{\mathcal{T}}_{n,V}\setminus\\{0\\}}\frac{\|d_{N}(T)\|_{L_{q}(Q_{\pi
n}^{m})}}{\|T\|_{L_{p}(Q_{\pi n}^{m})}},$ (7.17)
$\displaystyle\sup_{f\in(B_{V}\cap
L_{p}({\mathbb{R}}^{m}))\setminus\\{0\\}}\frac{\|d_{N}(f)\|_{L_{\infty}({\mathbb{R}}^{m})}}{\|f\|_{L_{p}({\mathbb{R}}^{m})}}=\lim_{n\to{\infty}}\sup_{T\in{\mathcal{T}}_{n,V}\setminus\\{0\\}}\frac{\|d_{N}(T)\|_{L_{\infty}(Q_{\pi
n}^{m})}}{\|T\|_{L_{p}(Q_{\pi n}^{m})}}.$ (7.18)

The following asymptotic results (see [41, Theorems 1.2 and 1.3]) are easy
corollaries of (7.17) and (7.18) ($0<p\leq q\leq{\infty}$):

$\displaystyle\sup_{f\in(B_{V}\cap
L_{p}({\mathbb{R}}^{m}))\setminus\\{0\\}}\frac{\|d_{N}(f)\|_{L_{q}({\mathbb{R}}^{m})}}{\|f\|_{L_{p}({\mathbb{R}}^{m})}}\leq\liminf_{n\to{\infty}}n^{-N-m/p+m/q}\sup_{Q\in{\mathcal{T}}_{nV}\setminus\\{0\\}}\frac{\|d_{N}(Q)\|_{L_{q}(Q_{\pi}^{m})}}{\|Q\|_{L_{p}(Q_{\pi}^{m})}},$
(7.19) $\displaystyle\sup_{f\in(B_{V}\cap
L_{p}({\mathbb{R}}^{m}))\setminus\\{0\\}}\frac{\|d_{N}(f)\|_{L_{\infty}({\mathbb{R}}^{m})}}{\|f\|_{L_{p}({\mathbb{R}}^{m})}}=\lim_{n\to{\infty}}n^{-N-m/p}\sup_{Q\in{\mathcal{T}}_{nV}\setminus\\{0\\}}\frac{\|d_{N}(Q)\|_{L_{\infty}(Q_{\pi}^{m})}}{\|Q\|_{L_{p}(Q_{\pi}^{m})}}.$
(7.20)

For $m=1$ and $V=[-{\sigma},{\sigma}]$ relations (7.19) and (7.20) were
established by the author and Tikhonov [44, Theorems 1.4 and 1.5]. Similar
relations for $m=1,\,V=[-{\sigma},{\sigma}]$, and $N=0$ were obtained earlier
by Levin and Lubinsky [57, p. 246], [58, Theorem 1]. A special case of
$m=1,\,V=[-1,1],\,q={\infty},\,p=1$, and $N=0$ was discussed by Taikov [72,
Eq. (3)], [73, Lemma 1] and Gorbachev [47, Theorem 2]. More references can be
found in [44, 41].

Condition C3.1.1) of approximation of entire functions of exponential type $V$
by trigonometric polynomials was verified by using univariate and multivariate
Levitan’s polynomials (see [44, Sect. 2] for $m=1$ and [41, Sect. 3] for
$m>1$).

In addition, we choose

$\displaystyle B_{n}^{\prime}$ $\displaystyle=$
$\displaystyle\Bigg{\\{}f_{n}\in{\mathcal{T}}_{n,V}\setminus\\{0\\}:\Bigg{(}\|d_{N}(f_{n})\|_{L_{\infty}(Q_{\pi
n}^{m})}=|d_{N}(f_{n})(0)|=1\Bigg{)}$ $\displaystyle\wedge$
$\displaystyle\left.\left(\sup_{T\in{\mathcal{T}}_{n,V}\setminus\\{0\\}}\frac{\|d_{N}(T)\|_{L_{\infty}(Q_{\pi
n}^{m})}}{\|T\|_{L_{p}(Q_{\pi n}^{m})}}=\|f_{n}\|_{L_{p}(Q_{\pi
n}^{m})}^{-1}\right)\right\\},\qquad n\in{\mathbb{N}}.$

Since the set ${\mathcal{T}}_{n,V}$ consists of $2\pi n$-periodic
trigonometric polynomials, we have

$\displaystyle\sup_{T\in{\mathcal{T}}_{n,V}\setminus\\{0\\}}\frac{\|d_{N}(T)\|_{L_{\infty}(Q_{\pi
n}^{m})}}{\|T\|_{L_{p}(Q_{\pi
n}^{m})}}=\sup_{T\in{\mathcal{T}}_{n,V}\setminus\\{0\\}}\frac{|d_{N}(T)(0)|}{\|T\|_{L_{p}(Q_{\pi
n}^{m})}},\qquad n\in{\mathbb{N}}.$ (7.21)

Therefore, by (7.21) and by the standard compactness argument (see the proof
of Theorem 1.3 in [41] for details), we conclude that
$B_{n}^{\prime}\neq\emptyset$ and in addition,

$\displaystyle\sup_{T\in{\mathcal{T}}_{n,V}\setminus\\{0\\}}\frac{\|d_{N}(T)\|_{L_{\infty}(Q_{\pi
n}^{m})}}{\|T\|_{L_{p}(Q_{\pi n}^{m})}}=\sup_{f_{n}\in
B_{n}^{\prime}}\frac{\|d_{N}(f_{n})\|_{L_{\infty}(Q_{\pi
n}^{m})}}{\|f_{n}\|_{L_{p}(Q_{\pi n}^{m})}},\qquad n\in{\mathbb{N}}.$

Thus Condition C9) is satisfied for the chosen
$B_{n}^{\prime},\,n\in{\mathbb{N}}$. Next, equality (7.10) of C12.1) holds
true since for any $f_{n}\in B_{n}^{\prime}$ we have
($n\in{\mathbb{N}},\,s\in{\mathbb{N}}$)

$\displaystyle\left\|D_{n}(f_{n})\right\|_{F^{(1)}\left({\Omega}_{n}\right)}=\|d_{N}(f_{n})\|_{L_{\infty}(Q_{\pi
n}^{m})}=\left|d_{N}(f_{n})(0)\right|=\|d_{N}(f_{n})\|_{L_{\infty}(Q_{\pi
s}^{m})}=\left\|D_{n}(f_{n})\right\|_{F^{(1)}\left({\Omega}_{s}\right)}.$

Further, the compactness condition of $\\{B_{n}^{\prime}\\}_{n=1}^{\infty}$,
stated in (7.8) and (7.9) of C12.1) (with a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ that is independent of $s$), is satisfied as
well. Indeed, let $f_{n}\in B_{n}^{\prime},\,n\in{\mathbb{N}}$. Then for
$p\in(0,{\infty})$

$\displaystyle\sup_{n\in{\mathbb{N}}}\|f_{n}\|_{L_{\infty}(Q_{\pi
n}^{m})}\leq\sup_{n\in{\mathbb{N}}}{\mathcal{C}}\left(d_{0},{\mathcal{T}}_{n,V},L_{\infty}(Q_{\pi
n}^{m}),L_{p}(Q_{\pi n}^{m})\right)\|f_{n}\|_{L_{p}(Q_{\pi n}^{m})}$
$\displaystyle=\sup_{n\in{\mathbb{N}}}{\mathcal{C}}\left(d_{0},{\mathcal{T}}_{n,V},L_{\infty}(Q_{\pi
n}^{m}),L_{p}(Q_{\pi
n}^{m})\right)/{\mathcal{C}}\left(d_{N},{\mathcal{T}}_{n,V},L_{\infty}(Q_{\pi
n}^{m}),L_{p}(Q_{\pi n}^{m})\right)=C_{8}<{\infty},$

by the choice of $B_{n}^{\prime}$ and by crude estimates of the constants in
multivariate Bernstein-Nikolskii type inequalities (see [41, Eq. (4.2)]).
Therefore,

$\displaystyle\cup_{n=1}^{\infty}B_{n}^{\prime}$ $\displaystyle\subseteq$
$\displaystyle\\{T\in\cup_{n=1}^{\infty}{\mathcal{T}}_{n,V}:\|T\|_{L_{\infty}(Q_{\pi
n}^{m})}\leq C_{8}\\}$ (7.22) $\displaystyle\subseteq$ $\displaystyle\\{f\in
B_{V}:\|f\|_{L_{\infty}({\mathbb{R}}^{m})}\leq C_{8}\\},$

since ${\mathcal{T}}_{n,V}\subseteq B_{V},\,n\in{\mathbb{N}}$, and the
compactness property of the right-hand side set in (7.22) is proved in [41,
Lemma 2.3] (a special case is discussed in [65, Theorem 3.3.6]). Note also
that by [41, Eq. (4.2)],

$\displaystyle{\mathcal{C}}\left(d_{N},{\mathcal{T}}_{n,V},L_{\infty}(Q_{\pi
n}^{m}),L_{p}(Q_{\pi n}^{m})\right)<{\infty},\qquad n\in{\mathbb{N}}.$

So (7.17) for $0<p\leq q\leq{\infty}$ and (7.18) for $0<p\leq{\infty}$ and
$q={\infty}$ follow from statements (i) and (vi) of Corollary 7.1,
respectively.

######  Example 7.3.

(Multivariate inequalities of different metrics). Let
$F^{(1)}({\mathbb{R}}^{m})=C_{0}({\mathbb{R}}^{m}),\,{\beta}_{1}=1,\,F^{(2)}({\mathbb{R}}^{m})=L_{p}({\mathbb{R}}^{m}),\,{\beta}_{2}=\min\\{1,p\\},\,B=B_{V}\cap
L_{p}({\mathbb{R}}^{m}),\,0<p\leq{\infty},\,{\Omega}_{n}=nV^{*},\,{\Omega}_{n}^{*}=\tau
nV^{*},\,\tau\in(0,1),\,B_{n}={\mathcal{P}}_{n,m}\left|{}_{\Omega_{n}}\right.={\mathcal{P}}_{n,m},\,B_{n}^{*}={\mathcal{P}}_{n,m}\left|{}_{\Omega_{n}^{*}}\right.={\mathcal{P}}_{n,m}$.
Next, let $d_{N}=\sum_{|{\alpha}|=N}b_{\alpha}D^{\alpha}$, where
$b_{\alpha}\in{\mathbb{C}}^{1}$
(${\alpha}\in{\mathbb{Z}}^{m}_{+},\,|{\alpha}|=N$) and
$N\in{\mathbb{Z}}^{1}_{+}$, are independent of $n$, and let
$D=d_{N}\left|{}_{{\mathbb{R}}^{m}}\right.=d_{N},\,D_{n}=d_{N}\left|{}_{\Omega_{n}}\right.=d_{N},\,D_{n}^{*}=d_{N}\left|{}_{\Omega_{n}^{*}}\right.=d_{N},\,n\in{\mathbb{N}}$.

Inequalities (7.14) were proved by the author [43] in the following form:

$\displaystyle\limsup_{n\to{\infty}}\sup_{P\in{\mathcal{P}}_{n,m}\setminus\\{0\\}}\frac{|d_{N}(P)(0)|}{\|P\|_{L_{p}(nV^{*})}}\leq\sup_{f\in(B_{V}\cap
L_{p}({\mathbb{R}}^{m}))\setminus\\{0\\}}\frac{|d_{N}(f)(0)|}{\|f\|_{L_{p}({\mathbb{R}}^{m})}}$
$\displaystyle=\sup_{f\in(B_{V}\cap
L_{p}({\mathbb{R}}^{m}))\setminus\\{0\\}}\frac{\|d_{N}(f)\|_{L_{\infty}({\mathbb{R}}^{m})}}{\|f\|_{L_{p}({\mathbb{R}}^{m})}}\leq\liminf_{n\to{\infty}}\sup_{P\in{\mathcal{P}}_{n,m}\setminus\\{0\\}}\frac{|d_{N}(P)(0)|}{\|P\|_{L_{p}(\tau
nV^{*})}}$ (7.23)

for $\tau\in(0,1)$. By letting $\tau\to 1-$ in (7.3), we arrive at the
following asymptotic result for $0<p\leq{\infty}$ (see [43, Theorem 1.2]):

$\sup_{f\in(B_{V}\cap
L_{p}({\mathbb{R}}^{m}))\setminus\\{0\\}}\frac{\|d_{N}(f)\|_{L_{\infty}({\mathbb{R}}^{m})}}{\|f\|_{L_{p}({\mathbb{R}}^{m})}}=\lim_{n\to{\infty}}n^{-N-m/p}\sup_{Q\in{\mathcal{P}}_{n,m}\setminus\\{0\\}}\frac{|d_{N}(Q)(0)|}{\|Q\|_{L_{p}(V^{*})}}.$
(7.24)

For $m=1,\,V=[-1,1]$, $V^{*}=[-1,1]$, and $d_{N}(f)=d^{N}f/dx^{N}$, equality
(7.24) was established in [38, Theorem 1.1]. In case when $V$ and $V^{*}$ are
the unit ball ${\mathfrak{B}}_{1}^{m}$ and $d_{2N}=\Delta^{N}$ is the
polyharmonic operator, (7.24) was proved in [42, Corollary 3.9] by reducing
the multivariate problem to a univariate one in weighted spaces (see Example
7.4).

Condition C3.1) of approximation of entire functions from $B_{V}\cap
L_{p}({\mathbb{R}}^{m})$ by algebraic polynomials was verified in [29, Lemma
4.4] and [43, Lemma 2.4] by using approximation methods developed by Bernstein
[11] (see also [74, Sect. 5.4.4] and [1, Appendix, Sect. 83]) and by the
author [29, 43].

In addition, we choose

$\displaystyle
B_{n}^{\prime}=\left\\{f_{n}\in{\mathcal{P}}_{n,m}\setminus\\{0\\}:\left(|d_{N}(f_{n})(0)|=1\right)\wedge\left(\sup_{P\in{\mathcal{P}}_{n,m}\setminus\\{0\\}}\frac{|d_{N}(P)(0)|}{\|P\|_{L_{p}(nV^{*})}}=\|f_{n}\|_{L_{p}(nV^{*})}^{-1}\right)\right\\},\,\,n\in{\mathbb{N}}.$

By the standard compactness argument (see the proof of Theorem 1.2 in [43] for
details), we conclude that $B_{n}^{\prime}\neq\emptyset$ and in addition,

$\displaystyle\sup_{P\in{\mathcal{P}}_{n,m}\setminus\\{0\\}}\frac{|d_{N}(P)(0)|}{\|P\|_{L_{p}(nV^{*})}}=\sup_{f_{n}\in
B_{n}^{\prime}}\frac{|d_{N}(f_{n})(0)|}{\|f_{n}\|_{L_{p}(nV^{*})}},\qquad
n\in{\mathbb{N}}.$

Thus Condition C9) is satisfied for the chosen
$B_{n}^{\prime},\,n\in{\mathbb{N}}$. Next, it is easy to verify that equality
(7.10) of C12.1) holds true in this case as well since for any $f_{n}\in
B_{n}^{\prime}$ we have

$\displaystyle\left\|D_{n}(f_{n})\right\|_{F^{(1)}\left({\Omega}_{n}\right)}=\left\|d_{N}(f_{n})\right\|_{C_{0}\left(nV^{*}\right)}=\left\|d_{N}(f_{n})\right\|_{C_{0}\left(sV^{*}\right)}=\left\|D_{n}(f_{n})\right\|_{F^{(1)}\left({\Omega}_{s}\right)},\quad
n\in{\mathbb{N}},\quad s\in{\mathbb{N}}.$

Further, the compactness condition of $\\{B_{n}^{\prime}\\}_{n=1}^{\infty}$,
stated in (7.8) and (7.9) of C12.1) (with a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ that is independent of $s$), is satisfied. The
proof of this fact given in [43] is based on results of Bernstein [11] (for
$m=1$) and the author [43, Lemma 2.2] (for $m>1$). Note also that by a crude
estimate of the constant in a multivariate inequality of different metrics
(see [43, Lemma 2.7 (a)]),
${\mathcal{C}}\left(d_{N},{\mathcal{P}}_{n,m},C_{0}(nV^{*}),L_{p}(nV^{*})\right)<{\infty}$
for $n\in{\mathbb{N}}$.

So (7.3) for $p\in(0,{\infty}]$ follows from Corollary 7.1 (v).

######  Example 7.4.

(Univariate inequalities of different weighted metrics). Let
$d\mu(x)=|x|^{2\nu+1}dx,\,\nu\geq-1/2$, be the univariate weighted measure on
${\mathbb{R}}^{1}$. Next, let
$F^{(1)}({\mathbb{R}}^{1})=C_{0}({\mathbb{R}}^{1}),\,{\beta}_{1}=1,\,F^{(2)}({\mathbb{R}}^{1})=L_{p,\mu}({\mathbb{R}}^{1}),\,{\beta}_{2}=\min\\{1,p\\}$,

$\displaystyle B=B_{1,e}\cap L_{p,\mu}({\mathbb{R}}^{1}):=\\{f\in
B_{[-1,1]}:f(-x)=f(x),\,x\in{\mathbb{R}}^{1}\\}\cap
L_{p,\mu}({\mathbb{R}}^{1}),$

$0<p\leq{\infty},\,{\Omega}_{n}=[-n,n],\,{\Omega}_{n}^{*}=[-\tau n,\tau
n],\,\tau\in(0,1),\,n\in{\mathbb{N}}$.

Next, let
$Be({\varphi})(x):={\varphi}^{\prime\prime}(x)+((2\nu+1)/x){\varphi}^{\prime}(x)$
be the Bessel operator and let
$D=(Be)^{N}\left|{}_{{\mathbb{R}}^{m}}\right.=(Be)^{N},\,D_{n}=(Be)^{N}\left|{}_{\Omega_{n}}\right.=(Be)^{N},\,D_{n}^{*}=(Be)^{N}\left|{}_{\Omega_{n}^{*}}\right.=(Be)^{N},\,N\in{\mathbb{Z}}^{1}_{+},\,n\in{\mathbb{N}}$.
Further, let

$\displaystyle{\mathcal{P}}_{n,1,e}:=\\{P\in{\mathcal{P}}_{n,1}:P(-x)=P(x),\,x\in{\mathbb{R}}^{1}\\}$

be a subset of ${\mathcal{P}}_{n,1}$ of all even univariate polynomials with
complex coefficients, and let
$B_{n}={\mathcal{P}}_{n,1,e}\left|{}_{\Omega_{n}}\right.$ and
$B_{n}^{*}={\mathcal{P}}_{n,1,e}\left|{}_{\Omega_{n}^{*}}\right.,\,n\in{\mathbb{N}}.$

Inequalities (7.14) were proved by the author [42] in the following form:

$\displaystyle\lim_{n\to{\infty}}\sup_{P\in{\mathcal{P}}_{2\lfloor
n/2\rfloor,1,e}\setminus\\{0\\}}\frac{|(Be)^{N}(P)(0)|}{\|P\|_{L_{p,\mu}([-n,n])}}$
$\displaystyle\leq$ $\displaystyle\sup_{f\in(B_{1,e}\cap
L_{p,\mu}({\mathbb{R}}^{1}))\setminus\\{0\\}}\frac{|(Be)^{N}(f)(0)|}{\|f\|_{L_{p,\mu}({\mathbb{R}}^{1})}}$
(7.25) $\displaystyle\leq$
$\displaystyle\lim_{n\to{\infty}}\sup_{P\in{\mathcal{P}}_{2\lfloor
n/2\rfloor,1,e}\setminus\\{0\\}}\frac{|(Be)^{N}(P)(0)|}{\|P\|_{L_{p,\mu}([-\tau
n,\tau n])}}$

for $\tau\in(0,1)$. By letting $\tau\to 1-$ in (7.25), we arrive at the
following asymptotic result for $0<p\leq{\infty}$ (see [42, Theorem 4.1]):

$\displaystyle\sup_{f\in(B_{1,e}\cap
L_{p,\mu}({\mathbb{R}}^{1}))\setminus\\{0\\}}\frac{|(Be)^{N}(f)(0)|}{\|f\|_{L_{p,\mu}({\mathbb{R}}^{1})}}=\lim_{n\to{\infty}}n^{-2N-(2\nu+2)/p}\sup_{P\in{\mathcal{P}}_{2\lfloor
n/2\rfloor,1,e}\setminus\\{0\\}}\frac{|(Be)^{N}(P)(0)|}{\|P\|_{L_{p,\mu}([-1,1])}}.$

Verifications of Conditions C3.1) and C12.1) in weighted spaces are provided
similarly to Example 7.3 (see [42] for details).

So (7.25) for $p\in(0,{\infty}]$ follows from Corollary 7.1 (v).

######  Remark 7.5.

Special cases of relations (7.12), (7.14), and (7.15) that are discussed in
Examples 7.2, 7.3, and 7.4 and published in [44, 38, 41, 42, 43] follow from
statements (ii), (v), and (vi) of Corollary 7.1. In other words, the relations
presented in the aforementioned publications can be proved by verifying the
validity of the corresponding conditions C3.1) (or C3.1.1)) and C12.1).
Despite the fact that these conditions were not formulated explicitly in these
papers, the proofs of special cases of (7.12), (7.14), and (7.15) were
actually based on the verification of these conditions. This observation along
with the general approach to sharp constants of approximation theory developed
in [31, 32] were the major motivations for writing this paper.

###  7.2. Approximation of Individual Elements (Problem B)

Here, we discuss a special case of Corollary 5.3 and three examples.
General result. We assume that all assumptions of Section 5.2 are held in the
following special cases. First, we assume that
$F^{(2)}_{*}=F({\mathbb{R}}^{m})=F({\mathbb{R}}^{m},\mu)$ is a
${\beta}_{2}$-normed functional space with the monotone and
$\boldsymbol{{\omega}}$-continuous ${\beta}_{2}$-norm (see Definitions 6.1 and
6.2), ${\beta}_{2}\in(0,1]$, and $F^{(2)}_{*n}=F({\Omega}_{n})$ or
$F^{(2)}_{*n}=F({\Omega}_{n}^{*}),\,n\in{\mathbb{N}}$. Next, we define the
restriction operators
$L_{n}^{(2)}=R({\mathbb{R}}^{m},{\Omega}_{n}^{*}),\,l_{n}^{(2)}=R({\mathbb{R}}^{m},{\Omega}_{n}),\,n\in{\mathbb{N}}$,
and

$\displaystyle
l_{n,s}^{(2)}=\left\\{\begin{array}[]{ll}R({\Omega}_{n},{\Omega}_{s}),&n\geq
s,\\\ R({\Omega}_{s},{\Omega}_{n}),&n<s,\end{array}\right.\quad
n\in{\mathbb{N}},\quad s\in{\mathbb{N}}.$

We also assume that $B,\,B_{n}$, and $B_{n}^{*}$ are nontrivial subspaces of
$F({\mathbb{R}}^{m}),\ F({\Omega}_{n})$, and $F({\Omega}_{n}^{*})$,
respectively, $n\in{\mathbb{N}}$. Next, for ${\varphi}\in F({\mathbb{R}}^{m})$
we set
${\varphi}_{n}:=L_{n}^{(2)}({\varphi})={\varphi}\left|{}_{\Omega_{n}^{*}}\right.\in
F(\Omega_{n}^{*})$ or
${\varphi}_{n}:=l_{n}^{(2)}({\varphi})={\varphi}\left|{}_{\Omega_{n}}\right.\in
F(\Omega_{n})$. In addition, we assume that ${\varphi}\in
F({\mathbb{R}}^{m})\setminus B$ and ${\varphi}\left|{}_{\Omega_{n}}\right.\in
F(\Omega_{n})\setminus B_{n}$ or
${\varphi}_{n}\left|{}_{\Omega_{n}^{*}}\right.\in F(\Omega_{n}^{*})\setminus
B_{n}^{*}$. Note that to the end of the example we use the notation
${\varphi}$ instead of ${\varphi}\left|{}_{\Omega_{n}}\right.$ or
${\varphi}\left|{}_{\Omega_{n}^{*}}\right.$.

The ”norms” are defined by (5.6) as follows.

$\|h\|_{F^{(2)}}:=\|{\varphi}-h\|_{F({\mathbb{R}}^{m})},\quad h\in
F({\mathbb{R}}^{m});\qquad\|h_{n}\|_{F^{(2)}_{n}}:=\|{\varphi}-h_{n}\|_{F(\Omega)},\quad
h_{n}\in F(\Omega),$ (7.27)

where $\Omega=\Omega_{n}$ or $\Omega=\Omega_{n}^{*}$.

Furthermore, for ${\varphi}\in F({\mathbb{R}}^{m})$ we consider the errors of
best approximation $E({\varphi},B,F({\mathbb{R}}^{m}))$,
$E({\varphi},B_{n},F({\Omega}_{n}))$, and
$E({\varphi},B_{n}^{*},F({\Omega}_{n}^{*})),\,n\in{\mathbb{N}}$ (see (2.9) or
(6.7) for the definition); by the assumptions on ${\varphi}$ and its
restriction to $\Omega_{n}$ or $\Omega_{n}^{*},\,n\in{\mathbb{N}}$, all these
approximation errors are positive.

Finally, let $B_{n}^{\prime}\subseteq B_{n}\setminus\\{0\\}$ be a set from
Condition C9) such that the following relation holds:

$\displaystyle
E({\varphi},B_{n},F({\Omega}_{n}))=E({\varphi},B_{n}^{\prime},F({\Omega}_{n})).$

Below we introduce some conditions on families
$\left\\{B,F({\mathbb{R}}^{m})\right\\},\left\\{B_{n},F({\Omega}_{n})\right\\}$,
and $\left\\{B_{n}^{*},F({\Omega}_{n}^{*})\right\\},\linebreak
n\in{\mathbb{N}}$, that are needed for the validity of relations (5.9) and/or
(5.10) in function spaces.
_Condition C3.2*)._ For every $f\in B\setminus\\{0\\}$ there exists a sequence
$f_{n}^{*}\in B_{n}^{*},\,n\in{\mathbb{N}}$, such that

$\lim_{n\to{\infty}}\left\|f-f_{n}^{*}\right\|_{F({\Omega}_{n}^{*})}=0.$
(7.28)

_Condition C10.2)._ (Compactness condition) For any sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ such that $f_{n}\in
B_{n}^{\prime},\,n\in{\mathbb{N}}$, there exist a sequence
$\\{n_{k}\\}_{k=1}^{\infty}$ of natural numbers and an element $f\in B$ such
that

$\lim_{k\to{\infty}}\left\|f_{n_{k}}-f\right\|_{F\left({\Omega}_{n_{k}}\right)}=0.$
(7.29)

_Condition C12.2)._ (Weak compactness condition) For any sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ such that $f_{n}\in
B_{n}^{\prime},\,n\in{\mathbb{N}}$, there exists $f\in B$ such that for any
fixed $s\in{\mathbb{N}}$, there exists a sequence $\\{n_{k}\\}_{k=1}^{\infty}$
of natural numbers ($n_{k}=n_{k}(s),\,k\in{\mathbb{N}}$) such that

$\lim_{k\to{\infty}}\left\|f_{n_{k}}-f\right\|_{F\left({\Omega}_{s}\right)}=0,$
(7.30)

The following corollary holds true.

######  Corollary 7.6.

(i) If Condition C3.2*) is satisfied, then

$\limsup_{n\to{\infty}}E\left({\varphi},B_{n}^{*},F({\Omega}_{n}^{*})\right)\leq
E\left({\varphi},B,F({\mathbb{R}}^{m})\right).$ (7.31)

(ii) If Condition C10.2) is satisfied, then

$\liminf_{n\to{\infty}}E\left({\varphi},B_{n},F({\Omega}_{n})\right)\geq
E\left({\varphi},B,F({\mathbb{R}}^{m})\right).$ (7.32)

(iii) If Condition C12.2) is satisfied, then (7.32) holds true.
(iv) Let Condition C3.2*) and either C10.2) or C12.2) be satisfied. Then

$\limsup_{n\to{\infty}}E\left({\varphi},B_{n}^{*},F({\Omega}_{n}^{*}\right))\leq
E\left({\varphi},B,F({\mathbb{R}}^{m})\right)\leq\liminf_{n\to{\infty}}E\left({\varphi},B_{n},F({\Omega}_{n}\right)).$
(7.33)

###### Proof.

Note first that in this proof all elements in discussed Conditions C1) through
C13) are replaced without further notice by the corresponding elements that
are used in Conditions C3.2*), C10.2), C12.2) and relations (7.31), (7.32),
(7.33). For example, in Condition C3*) $B_{n}$ is replaced by $B_{n}^{*}$.

By the assumptions about the functional spaces and operators introduced above,
Conditions C4*), C6), C6b), C11), and C11a) of Section 3.2 are satisfied.
Indeed, Conditions C6) and C11) are satisfied by the direct definitions of the
operators whose existence is required in these conditions. In addition,
Conditions C4*), C6b), and C11a) are satisfied by Proposition 6.7. For
example, for every $h\in F^{(2)}$ we obtain by (7.27) and by Proposition 6.7
(ii)

$\displaystyle\liminf_{n\to{\infty}}\left\|l_{n}^{(2)}(h)\right\|_{F^{(2)}_{n}}$
$\displaystyle=$
$\displaystyle\liminf_{n\to{\infty}}\left\|{\varphi}_{n}-l_{n}^{(2)}(h)\right\|_{F(\Omega_{n})}=\liminf_{n\to{\infty}}\left\|l_{n}^{(2)}({\varphi}-h)\right\|_{F(\Omega_{n})}$
$\displaystyle=$
$\displaystyle\|{\varphi}-h\|_{F({\mathbb{R}}^{m})}=\|h\|_{F^{(2)}}.$

Therefore, Condition C6b) is satisfied. Similarly we can prove that Conditions
C4*) and C11a) are satisfied (see also Remark 5.4).

Next, Conditions C3.2*), C10.2), and C12.2) of this section are equivalent to
Conditions C3*), equation (3.18) of C10), and equation (3.21) of C12) from
Section 3.2, respectively. Therefore, statements (i), (ii), and (iii) of
Corollary 7.6 follow immediately from Corollary 5.3, while statement (iv) is a
direct corollary of (i), (ii), and (iii). $\Box$

######  Remark 7.7.

Statements (i) and (ii) of Corollary 7.6 with slightly different conditions
were obtained in [31, Corollary 3.1] for normed spaces and in [32, Sect.
1.4.3] for ${\beta}_{2}$-normed spaces.

Examples. There are numerous publications [7, 8, 10, 11, 13, 74, 1, 66, 29,
30, 31, 32, 34, 36] about special cases of Corollary 7.6 and their
modifications; here, we call them limit theorems of approximation theory (see
also Section 2.3 B). Examples 7.8, 7.9, and 7.10 below discuss relations
(7.33) proved in these papers in various situations.

######  Example 7.8.

(Limit theorems for multivariate periodic approximations). Let
$F({\mathbb{R}}^{m})=L_{p}({\mathbb{R}}^{m}),\linebreak{\beta}_{2}=\min\\{1,p\\}$,
and let ${\varphi}\in L_{p}({\mathbb{R}}^{m})\cap M_{C,N}$ be a fixed
function$,\,0<p\leq{\infty}$. Next, let $B=B_{V}\cap
L_{p}({\mathbb{R}}^{m}),\,0<p\leq{\infty},\,{\Omega}_{n}={\Omega}_{n}^{*}=Q_{{\gamma}(n)}^{m},\,B_{n}=B_{N}^{*}={\mathcal{T}}_{n,V}\cap
M_{C_{9}C,N+2}\left|{}_{Q_{{\gamma}(n)}^{m}}\right.={\mathcal{T}}_{n,V}\cap
M_{C_{9}C,N+2},\,n\in{\mathbb{N}}.$ Here, $C>0$ and $N\geq 0$ are fixed
constants; $C_{9}$ is any constant such that $C_{9}\geq C_{10}>0$, and $C_{9}$
and $C_{10}$ are independent of $n$ and ${\varphi}$. In addition, a number
sequence $\\{{\gamma}(n)\\}_{n=1}^{\infty}$ satisfies the conditions

$0<{\gamma}(n)\leq\pi n,\quad
n\in{\mathbb{N}},\qquad\lim_{n\to{\infty}}{\gamma}(n)={\infty},\qquad\lim_{n\to{\infty}}({\gamma}(n))^{2+m/p}n^{-1/(m+2)}=0.$
(7.34)

Relations (7.33) were proved by the author [31, 32] in the following form:

$\lim_{n\to{\infty}}E\left({\varphi},{\mathcal{T}}_{n,V}\cap
M_{C_{9}C,N+2},L_{p}\left(Q_{{\gamma}(n)}^{m}\right)\right)=E\left({\varphi},B_{V}\cap
L_{p}({\mathbb{R}}^{m}),L_{p}({\mathbb{R}}^{m})\right).$ (7.35)

For $p\in[1,{\infty}]$ equality (7.35) was proved in [31, Theorem 5.1] in a
more general situation; the case of $p\in(0,1)$ was discussed in [32, Sect.
1.4.4.5]. For $m=1,\,V=[-{\sigma},{\sigma}]$, and $p={\infty}$, (7.35) was
established by Bernstein [10, Theorem 5].

Condition C3.2*) of approximation of entire functions of exponential type $V$
by multivariate trigonometric polynomials was verified in [31, Lemma 5.1] (see
also [41, Sect. 3]).

In addition, we choose $B_{n}^{\prime}=B_{n}={\mathcal{T}}_{n,V}\cap
M_{C_{9}C,N+2}$. Since ${\mathcal{T}}_{n,V}\subseteq B_{V}$, the compactness
property of $\cup_{n=1}^{\infty}B_{n}^{\prime}$, stated in C12.2), can be
found in [28, Lemma 2] and [41, Lemma 2.3] (a special case is discussed in
[65, Theorem 3.3.6]).

So (7.35) for $p\in(0,{\infty}]$ follows from Corollary 7.6 (iv).

######  Example 7.9.

(Limit theorems for multivariate polynomial approximations). Let
$F({\mathbb{R}}^{m})=L_{p}({\mathbb{R}}^{m}),\,{\beta}_{2}=\min\\{1,p\\},\,B=B_{V}\cap
L_{p}({\mathbb{R}}^{m}),\,0<p\leq{\infty},\,{\Omega}_{n}=nV^{*},\,{\Omega}_{n}^{*}=\tau
nV^{*},\,\tau\in(0,1),\,B_{n}={\mathcal{P}}_{n,m}\left|{}_{\Omega_{n}}\right.={\mathcal{P}}_{n,m},\,B_{n}^{*}={\mathcal{P}}_{n,m}\left|{}_{\Omega_{n}^{*}}\right.={\mathcal{P}}_{n,m},\,n\in{\mathbb{N}}.$

Relations (7.33) were proved in [29]. The following result is a corollary of
(7.33): if ${\varphi}\in L_{\infty}({\mathbb{R}}^{m})$, then

$\displaystyle E\left({\varphi},B_{V}\cap
L_{\infty}({\mathbb{R}}^{m}),L_{\infty}({\mathbb{R}}^{m})\right)\leq\liminf_{n\to{\infty}}E\left({\varphi},{\mathcal{P}}_{n,m},L_{\infty}\left(nV^{*}\right)\right)$
$\displaystyle\leq\limsup_{n\to{\infty}}E\left({\varphi},{\mathcal{P}}_{n,m},L_{\infty}\left(nV^{*}\right)\right)\leq\lim_{\tau\to
1^{-}}E\left({\varphi},B_{\tau V}\cap
L_{\infty}({\mathbb{R}}^{m}),L_{\infty}({\mathbb{R}}^{m})\right),$ (7.36)

and if ${\varphi}\in L_{p}({\mathbb{R}}^{m})\cap
L_{\infty}({\mathbb{R}}^{m}),\,p\in(0,{\infty})$, then

$\lim_{n\to{\infty}}E\left({\varphi},{\mathcal{P}}_{n,m},L_{p}\left(nV^{*}\right)\right)=E\left({\varphi},B_{V}\cap
L_{p}({\mathbb{R}}^{m}),L_{p}({\mathbb{R}}^{m})\right).$ (7.37)

For $p\in[1,{\infty}]$ relations (7.9) and (7.37) were proved in [29, Theorems
5.1 and 5.2]; the case of $p\in(0,1)$ was discussed in a more general and
precise form in [32, Theorem 1.3]. For $m=1,\,V=[-{\sigma},{\sigma}]$, and
$V^{*}=[-1/{\sigma},1/{\sigma}]$, inequalities (7.9) were established by
Bernstein [11, Theorem 6]. Equality (7.37) for
$m=1,\,V=[-{\sigma},{\sigma}],\,V^{*}=[-1/{\sigma},1/{\sigma}]$, and
$p\in[1,{\infty})$ was proved by Raitsin [66]. More precise relations than
(7.9) were proved in [30].

Condition C3.2*) of approximation of entire functions of exponential type $V$
by multivariate algebraic polynomials was verified in [29, Lemma 4.4] (see
also [43, Lemma 2.4]).

In addition, we choose

$\displaystyle
B_{n}^{\prime}=\left\\{f_{n}\in{\mathcal{P}}_{n,m}:\|f_{n}\|_{L_{p}(nV^{*})}\leq\left(2^{{\beta}_{2}}+1\right)^{1/{\beta}_{2}}\|{\varphi}\|_{L_{p}({\mathbb{R}}^{m})}\right\\},\qquad
n\in{\mathbb{N}}.$

Then

$E({\varphi},{\mathcal{P}}_{n,m},L_{p}(nV^{*}))=E({\varphi},B_{n}^{\prime},L_{p}(nV^{*})),\qquad
n\in{\mathbb{N}}.$ (7.38)

Indeed, given
${\varepsilon}\in\left(0,\|{\varphi}\|_{L_{p}({\mathbb{R}}^{m})}\right)$ and
$n\in{\mathbb{N}}$, let $f_{n,{\varepsilon}}\in{\mathcal{P}}_{n,m}$ satisfy
the inequalities

$\displaystyle\|{\varphi}\|_{L_{p}({\mathbb{R}}^{m})}\geq