One-Step Recurrences for Stationary Random Fields on the Sphere

One-Step Recurrences for Stationary Random Fields
on the Sphere\star\star$\star$This paper is a contribution to the Special
Issue on Orthogonal Polynomials, Special Functions and Applications. The full
collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html

R.K. BEATSON † and W. ZU CASTELL ‡§

R.K. Beatson and W. zu Castell

† School of Mathematics and Statistics, University of Canterbury,
† Private Bag 4800, Christchurch, New Zealand<EMAIL_ADDRESS>http://www.math.canterbury.ac.nz/~r.beatson

‡ Scientific Computing Research Unit, Helmholtz Zentrum München,
‡ Ingolstädter Landstraße 1, 85764 Neuherberg, Germany castell@helmholtz-
muenchen.de http://www.helmholtz-muenchen.de/asc

§ Department of Mathematics, Technische Universität München, Germany

Received January 28, 2016, in final form April 15, 2016; Published online
April 28, 2016

Recurrences for positive definite functions in terms of the space dimension
have been used in several fields of applications. Such recurrences typically
relate to properties of the system of special functions characterizing the
geometry of the underlying space. In the case of the sphere
${\mathbb{S}}^{d-1}\subset{\mathbb{R}}^{d}$ the (strict) positive definiteness
of the zonal function $f(\cos\theta)$ is determined by the signs of the
coefficients in the expansion of $f$ in terms of the Gegenbauer polynomials
$\\{C^{\lambda}_{n}\\}$, with $\lambda=(d-2)/2$. Recent results show that
classical differentiation and integration applied to $f$ have positive
definiteness preserving properties in this context. However, in these results
the space dimension changes in steps of two. This paper develops operators for
zonal functions on the sphere which preserve (strict) positive definiteness
while moving up and down in the ladder of dimensions by steps of one. These
fractional operators are constructed to act appropriately on the Gegenbauer
polynomials $\\{C^{\lambda}_{n}\\}$.

positive definite zonal functions; ultraspherical expansions; fractional
integration; Gegenbauer polynomials

42A82; 33C45; 42C10; 62M30

##  1 Introduction

This paper develops operators for zonal functions on the sphere which preserve
(strict) positive definitenesss while moving up and down in the ladder of
dimensions by steps of one. The operators provide tools for forming families
of (strictly) positive definite zonal functions. Such (strictly) positive
definite zonal functions can be used as covariance models for estimating
regionalized variables and also for interpolation on spheres.

Within a deterministic context, zonal positive definite functions on the
sphere have been used for interpolation or approximation of scattered data
(see [10, 11] and the references therein). The standard ansatz in this setting
is a linear combination of spherical translates of a fixed _$($ zonal$)$ basis
function_. While the present paper could well have been stated within the
context of approximation on the sphere, we rather chose to provide a
probabilistic framework, which is to some extent is equivalent, i.e., the
theory of regionalized variables.

Regionalized variables on spherical domains can nicely be modeled using random
fields on spheres [6, 15]. Such a random field is given through a set of
random variables, $Z(x)$ say, where $x\in{{\mathbb{S}}^{d-1}}$. Assuming the
field to be Gaussian, i.e., for every $n\in{\mathbb{N}}$,
$(Z(x_{1}),\dots,Z(x_{n}))^{T}$ has a multivariate Gaussian distribution for
any choice of $x_{1},\dots,x_{n}\in{{\mathbb{S}}^{d-1}}$, the distribution can
be characterized by its first two moments.

Assuming second order (weak) stationarity, the covariance for an isotropic
model is determined by a function

$\displaystyle\operatorname{Cov}\big{(}Z(x),Z(y)\big{)}=f(\cos\theta),\qquad
x,y\in{{\mathbb{S}}^{d-1}},$

where $\theta=\theta(x,y)=\arccos\big{(}x^{T}y\big{)}$ is the geodesic
distance between the points $x$ and $y$ on the sphere ${{\mathbb{S}}^{d-1}}$.

As a consequence of _Kolmogorov’s extension theorem_ (see [5, Theorem 36.3]),
the set of isotropic Gaussian random fields can be identified with the set of
zonal positive definite functions on the sphere [12, 15]. We note in passing
that Lévy named such processes _Brownian motion_.

######  Definition 1.1.

A continuous function $g\colon[0,\pi]\to{\mathbb{R}}$ is (zonal) positive
definite on the sphere ${{\mathbb{S}}^{d-1}}$ if for all $n\in{\mathbb{N}}$
and all distinct point sets $\\{x_{1},\ldots,x_{n}\\}$ on the sphere, the
inequality

$\displaystyle\sum_{i,j=1}^{n}c_{i}c_{j}g(\theta(x_{i},x_{j}))\geq 0$

holds true for all $c_{1},\dots,c_{n}\in{\mathbb{R}}$. The function is (zonal)
strictly positive definite on ${{\mathbb{S}}^{d-1}}$ if the inequality holds
in the strict sense for all $c_{1},\dots,c_{n}\in{\mathbb{R}}$ not vanishing
simultaneously.

Although the natural distance on the unit sphere is an angle in $[0,\pi]$, it
is convenient for the purpose of this paper to consider functions in
$x=\cos\theta\in[-1,1]$, instead. Thus, by $\Lambda_{d-1}$ we will denote the
cone of all functions $f\in C[-1,1]$ such that $f(\cos\,\cdot\,)$ is positive
definite on ${{\mathbb{S}}^{d-1}}$. $\Lambda^{+}_{d-1}$ will denote the
subcone of all strictly positive definite functions in $\Lambda_{d-1}$.

Gaussian random fields have been widely applied to statistically analyze
spatial phenomena [8, 16, 23]. In particular, kriging allows prediction of
spatial variables from given samples at arbitrary locations. The key
ingredient for such an approach lies in determining a suitable model for the
covariance function of the spherical random field. Commonly, such a model can
be inferred from given data through fitting a parametric family of models
(i.e., estimation of the covariance).

Models for covariance functions have further been used for simulation of
stationary random fields. Matheron [17] suggested a method based on proper
averaging of stationary random fields on a lower dimensional space. In the
Euclidean setting this so-called _turning bands method_ works as follows:

Given a stationary random field $Z_{1}$ on the real line with covariance
function $C_{1}$ and a randomly chosen direction $\xi\in{{\mathbb{S}}^{d-1}}$,
$Z_{\xi}=Z_{1}(x^{T}\xi)$ defines a stationary random field on
${{\mathbb{R}}^{d}}$. Averaging over all directions $\xi$ leads to a
stationary field on ${{\mathbb{R}}^{d}}$ the covariance function of which,
$C_{d}$ say, relates to $C_{1}$ via the so-called _turning bands operator_

$\displaystyle
C_{d}(t)=\operatorname{const}\int_{0}^{\infty}\big{(}1-\tau^{2}\big{)}_{+}^{\frac{d-3}{2}}C_{1}(t\tau)d\tau,\qquad
t\in{\mathbb{R}}_{+}.$

The turning bands operator represents one example out of a suite of operators,
mapping radial positive definite functions on ${{\mathbb{R}}^{d}}$ onto such
functions on a higher or lower dimensional space. Wendland [25], Wu [26] and
Gneiting [13] used such operators to derive compactly supported functions of a
given smoothness. Recurrences for radial positive definite functions in
general have been studied by several authors [22, 27]. Due to Schoenberg’s
characterization of radial positive definite functions and the fact that scale
mixtures of such functions preserve positive definiteness, recurrence
operators can be derived from corresponding relations between special
functions. In the case of radial functions on ${{\mathbb{R}}^{d}}$, the
appropriate fundamental relation is _Sonine’s first integral_ for Bessel
functions of the first kind (see [27]).

In a recent paper [4] the authors applied similar operators to derive
parametrized families of suitable locally supported covariance models for
stationary random fields on the sphere ${{\mathbb{S}}^{d-1}}$. These operators
are based on properties of Gegenbauer polynomials, appearing in Schoenberg’s
characterization [24] of zonal positive definite functions on the sphere.

######  Theorem 1.2.

Let $\lambda=(d-2)/2$, and consider a continuous function $f$ on $[-1,1]$. The
function $f(\cos\,\cdot\,)$ is positive definite on ${{\mathbb{S}}^{d-1}}$,
i.e., $f\in\Lambda_{d-1}$, if and only if $f$ has an ultraspherical expansion

$\displaystyle f(x)\sim\sum_{n=0}^{\infty}a_{n}C^{\lambda}_{n}(x),\qquad
x\in[-1,1],$ (1.1)

in which all the coefficients $a_{n}$ are nonnegative, and the series
converges at the point $x=1$. If this is the case, the series converges
absolutely and uniformly on the whole interval.

Chen, Menegatto and Sun [7] showed that a necessary and sufficient condition
for $f\circ\cos$ to be strictly positive definite on ${{\mathbb{S}}^{d-1}}$,
$d\geq 3$, is that, in addition to the conditions of Theorem 1.2, infinitely
many of the Gegenbauer coefficients $a_{n}$ with odd index, and infinitely
many of those with even index, are positive. In the case $d=2$ the criteria is
necessary but not sufficient for $f\circ\cos$ to be strictly positive
definite. A characterization in this case has been given by Menegatto,
Oliveira & Peron [18], although the criterion is a little more involved (see
also [3] for further details on these issues).

In the same spirit as for the turning bands method, a zonal function defined
on a lower dimensional sphere ${\mathbb{S}}^{d-\kappa}$ can be lifted up to
${{\mathbb{S}}^{d-1}}$ through averaging over the set of all copies of
${\mathbb{S}}^{d-\kappa}$ contained in ${{\mathbb{S}}^{d-1}}$. In [4] it is
shown that the analogues for the sphere of Matheron’s _montée_ and _descente_
operators (see [16]) for ${\mathbb{R}}^{d}$ are the operators

$\displaystyle(\operatorname{{{\mathcal{I}}}}f)(x)=\int_{-1}^{x}f(u)du,\qquad
x\in[-1,1],$

and

$\displaystyle(\operatorname{{{\mathcal{D}}}}f)(x)=f^{\prime}(x),\qquad
x\in[-1,1].$

Paralleling the behaviour of Matheron’s operators in the Euclidean case the
operators move in the ladder of dimensions by steps of two. Specifically, the
$\operatorname{{{\mathcal{I}}}}$ and $\operatorname{{{\mathcal{D}}}}$
operators map zonal positive definite functions $f(\cos\cdot)$ on
${\mathbb{S}}^{d}$ onto ones on ${\mathbb{S}}^{d-2}$ and ${\mathbb{S}}^{d+2}$,
respectively (see [4] for details). Therefore, the natural question arises,
whether it would also be possible to proceed through steps by one within the
ladder of dimensions. While in the Euclidean case this could be achieved using
fractional differentiation and integration (see [27]), the situation is more
intriguing in the spherical setting. The reason lies in the fact that the
characterizing special functions for the sphere are polynomials, which are not
preserved through fractional integration. Thus, one has to work with
combinations of fractional operators in order to guarantee that the operators
are mapping into the space of polynomials.

In the present paper, we provide a suite of four operators which can be used
to define a  _clavier_ (cf. [16]) for the sphere. The main results are given
in Theorems 2.3, 2.4 and 2.8, below. We start with introducing the appropriate
fractional operators in the following section and studying their action on
ultraspherical expansions. The action of the operators on Gegenbauer
polynomials shown in the last section is derived using properties of
hypergeometric ${}_{2}F_{1}$-functions.

##  2 Definition of the half-step operators

In the expansion (1.1) the dimension $d$ appears in the parameter
$\lambda=(d-2)/2$ of the Gegenbauer polynomials. This relation between
$\lambda$ and $d$ will be fixed throughout the paper.

From $\operatorname{{{\mathcal{D}}}}C_{n}^{\lambda}=2\lambda
C_{n-1}^{\lambda+1}$ (cf. [9, 10.9(23)]) we see that classical differentiation
and its inverse, integration, alter the parameter $\lambda$ by an integer.
This is why the operators $\operatorname{{{\mathcal{I}}}}$ and
$\operatorname{{{\mathcal{D}}}}$ traverse the ladder of dimensions in steps of
two (see [4]). At the same time, $\operatorname{{{\mathcal{I}}}}$ and
$\operatorname{{{\mathcal{D}}}}$ change the degree of polynomials by one.
Therefore, in order to obtain a one-step operator in the dimension, we have to
consider fractional integration and differentiation, a fact which perfectly
parallels the Euclidean setting (see [20, 27]).

We are now ready to define the half-step operators and discuss their action on
positive definite functions on ${{\mathbb{S}}^{d-1}}$.

######  Definition 2.1.

For $f\in L^{1}[-1,1]$ and $\lambda\geq 0$, define

$\displaystyle
I^{\lambda}_{+}f(x)=I^{\lambda,\frac{1}{2}}_{+}f(x)=(1+x)^{-\lambda+\frac{1}{2}}\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\lambda}f(\tau)d\tau,$
(2.1) $\displaystyle
I^{\lambda}_{-}f(x)=I^{\lambda,\frac{1}{2}}_{-}f(x)=(1-x)^{-\lambda+\frac{1}{2}}\int_{x}^{1}(\tau-x)^{-\frac{1}{2}}(1-\tau)^{\lambda}f(\tau)d\tau.$
(2.2)

Using these, we further define

$\displaystyle{\mathcal{I}}^{\lambda}_{+}=I^{\lambda}_{+}+I^{\lambda}_{-}\qquad\text{and}\qquad{\mathcal{I}}^{\lambda}_{-}=I^{\lambda}_{+}-I^{\lambda}_{-}.$
(2.3)

Apart from the additional factor $(1\pm x)^{-\lambda+\frac{1}{2}}$ in front of
the integral and the weight $(1\pm\tau)^{\lambda}$, the operators
$I^{\lambda}_{\pm}$ are classical _Riemann–Liouville fractional integrals_ of
order $\frac{1}{2}$ (cf. [21, Definition 2.1]) on the interval $[-1,1]$. To
define inverse operators, we use the corresponding _Riemann–Liouville
fractional derivates_ (cf. [21, Definition 2.2]).

######  Definition 2.2.

Let $f$ be absolutely continuous on $[-1,1]$ and $\lambda\geq 0$. Then

$\displaystyle
D^{\lambda}_{+}f(x)=D^{\lambda,\frac{1}{2}}_{+}f(x)=(1+x)\frac{d}{dx}\left\\{(1+x)^{-\lambda}\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\lambda-\frac{1}{2}}f(\tau)d\tau\right\\},$
$\displaystyle
D^{\lambda}_{-}f(x)=D^{\lambda,\frac{1}{2}}_{-}f(x)=(1-x)\frac{d}{dx}\left\\{(1-x)^{-\lambda}\int_{x}^{1}(\tau-x)^{-\frac{1}{2}}(1-\tau)^{\lambda-\frac{1}{2}}f(\tau)d\tau\right\\}.$

Using these, we further define

$\displaystyle{\mathcal{D}}^{\lambda}_{+}=D^{\lambda}_{+}+D^{\lambda}_{-}\qquad\text{and}\qquad{\mathcal{D}}^{\lambda}_{-}=D^{\lambda}_{+}-D^{\lambda}_{-}.$

The main results of this paper are the following two theorems giving precise
statements of the dimension hopping and positive definiteness preserving
properties of the operators ${\mathcal{I}}^{\lambda}_{\pm}$ and
${\mathcal{D}}^{\lambda}_{\pm}$. These are one-step analogues of Theorems 2.2
and 2.3 in [4]. Since in the light of Theorem 1.2 the statements can be
considered as statements concerning ultraspherical expansions without
referring back to a sphere, we are considering $m=d-1$ to be a positive
integer.

######  Theorem 2.3.

Let $m$ be a positive integer and $\lambda=(m-1)/2$.

  * $(a)$

    * $(i)$

Let $f\in\Lambda_{m+1}$, $m\geq 1$. Then
${\mathcal{I}}^{\lambda}_{\pm}f\in\Lambda_{m}$.

    * $(ii)$

Let $f\in\Lambda_{m+1}^{+}$, $m\geq 2$. Then
${\mathcal{I}}^{\lambda}_{\pm}f\in\Lambda_{m}^{+}$.

  * $(b)$

Let $m\geq 1$, $f\in\Lambda_{m+1}^{+}$ be nonnegative, and $f$ have Gegenbauer
expansion,

$\displaystyle f\sim\sum_{n=0}^{\infty}a_{n}C^{\lambda+\frac{1}{2}}_{n},$

with all coefficients, $\\{a_{n}\\}_{n=0}^{\infty}$, positive. Then
${\mathcal{I}}^{\lambda}_{+}f$ is also nonnegative,
${\mathcal{I}}^{\lambda}_{+}f\in\Lambda_{m}^{+}$, and all the coefficients
$b_{n}$ in the expansion

$\displaystyle{\mathcal{I}}^{\lambda}_{+}f\sim\sum_{n=0}^{\infty}b_{n}C^{\lambda}_{n},$

are positive.

###### Proof.

The proofs for the statements are almost identical with those of the
corresponding parts of Proposition 2.2 in [4], provided that proper analogues
for certain statements on Gegenbauer polynomials are given. We therefore
restrict ourselves to pointing out where adaptations of the proof given in [4]
are needed.

One of these details concerns the boundedness of the operators
${\mathcal{I}}^{\lambda}_{\pm}$ as operators from $C[-1,1]$ to $C[-1,1]$. This
follows from the definitions of $I^{\lambda}_{\pm}$ and
${\mathcal{I}}^{\lambda}_{\pm}$ in equations (2.1), (2.2) and (2.3), combined
with the beta integrals

$\displaystyle\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\nu}d\tau=(1+x)^{\nu+\frac{1}{2}}B\left(\tfrac{1}{2},\nu+1\right)=(1+x)^{\nu+\frac{1}{2}}\frac{\Gamma(\frac{1}{2})\Gamma(\nu+1)}{\Gamma(\nu+\frac{3}{2})},$
(2.4)

and

$\displaystyle\int_{x}^{1}(\tau-x)^{-\frac{1}{2}}(1-\tau)^{\nu}d\tau=(1-x)^{\nu+\frac{1}{2}}B\left(\tfrac{1}{2},\nu+1\right)=(1-x)^{\nu+\frac{1}{2}}\frac{\Gamma(\frac{1}{2})\Gamma(\nu+1)}{\Gamma(\nu+\frac{3}{2})}.$

Similarly, positivity of the operator ${\mathcal{I}}^{\lambda}_{+}$ follows
from the definitions (2.1), (2.2) and (2.3). The main ingredient thus
remaining to be shown is the action of ${\mathcal{I}}^{\lambda}_{\pm}$ on the
Gegenbauer polynomial $C^{\lambda+\frac{1}{2}}_{n}$. This part is given in
Theorem 3.3, below. Note that in contrast to the operators studied in [4],
there is no need to deal with an extra constant in statements (i) and (ii).
This follows from Theorem 3.3, showing that the operators
${\mathcal{I}}^{\lambda}_{\pm}$ do not introduce an additional constant. ∎

######  Theorem 2.4.

Suppose that $f\in\Lambda_{m}$, $m\geq 1$, and let $\lambda=(m-1)/2$. Then, if
both functions ${\mathcal{D}}^{\lambda}_{\pm}f\in C[-1,1]$, then
${\mathcal{D}}^{\lambda}_{\pm}f\in\Lambda_{m+1}$. If, in addition,
$f\in\Lambda_{m}^{+}$, then
${\mathcal{D}}^{\lambda}_{\pm}\in\Lambda_{m+1}^{+}$.

######  Remark 2.5.

Since the operators defined above can be seen as standard operators of
fractional integration/differentiation, classical results from fractional
calculus can be applied. For example, if
$(1+\tau)^{-1/2}f(\tau)\in\operatorname{Lip}\alpha$ for some
$\alpha>\frac{1}{2}$, in particular, if $f\in\operatorname{Lip}\alpha$ and
supp$f\subset(-1,1]$, then by Theorem 19 in [14] ${\mathcal{D}}_{\pm}^{0}f$
exists and is continuous. An analogous statement holds for general $\lambda$.

The proof of Theorem 2.4 depends heavily on a multiplier relationships between
the Gegenbauer coefficients of $f$ and those of
${\mathcal{D}}^{\lambda}_{\pm}f$. The details of these relationship, and the
proof of Theorem 2.4, will be deferred to the next subsection.

Let us finish the section with considering an example.

######  Example 2.6.

Consider the operator $I_{+}^{\lambda}$. In view of its definition (2.1) this
operator maps functions locally supported near one to functions locally
supported near one. Also, since
$I_{+}^{\lambda}=({\mathcal{I}}_{+}^{\lambda}+{\mathcal{I}}_{-}^{\lambda})/2$
this operator preserves (strict) positive definiteness by Theorem 2.3.

Note that by a change of variables

$\displaystyle\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\lambda}f(\tau)d\tau=(x+1)^{\lambda+\frac{1}{2}}\int_{0}^{1}(1-s)^{-\frac{1}{2}}s^{\lambda}f\big{(}(x+1)s-1\big{)}ds.$

Therefore, if $f$ were such that $f\big{(}(x+1)s-1\big{)}=(1-ys)^{-a}$, the
integral becomes

$\displaystyle\int_{0}^{1}(1-s)^{-\frac{1}{2}}s^{\lambda}(1-ys)^{-a}ds=\frac{\Gamma(\lambda+1)\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\lambda+\frac{3}{2}\right)}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}a,\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|y\right]},$ (2.7)

being a special case of _Euler’s integral_ for hypergeometric functions (cf.
[19, (15.6.1)]).

Now consider the _Cauchy family_

$\displaystyle\varphi_{\alpha,\beta}(r)=(1+r^{\alpha})^{-\frac{\beta}{\alpha}},\qquad
0<\alpha\leq 2,\qquad\beta>0,$

which is strictly positive definite on ${\mathbb{R}}^{d}$ for all $d\geq 1$
(cf. [13]). Choosing $\alpha=2$ and restricting the function
$\varphi_{2,\beta}$ to the sphere, we obtain (setting $\tau=\cos\theta$)

$\displaystyle\varphi_{\beta}(\tau)=\varphi_{2,\beta}\big{(}\sqrt{2-2\cos\theta}\big{)}=(3-2\tau)^{-\frac{\beta}{2}},\qquad\beta>0.$

Thus,

$\displaystyle\varphi_{\beta}\big{(}(x+1)s-1\big{)}=5^{-\frac{\beta}{2}}\left(1-\tfrac{2}{5}(x+1)s\right)^{-\frac{\beta}{2}}.$

Therefore, from (2.7) we have that

$\displaystyle
I_{+}^{\lambda}\varphi_{\beta}(x)=\frac{\sqrt{\pi}\Gamma(\lambda+1)}{5^{\frac{\beta}{2}}\Gamma\left(\lambda+\frac{3}{2}\right)}(x+1){{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}\frac{\beta}{2},\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|\tfrac{2}{5}(x+1)\right]}.$

Since (see [19, (15.4.6)])

$\displaystyle(1-z)^{-a}={{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}a,b\\\
b\\\ \end{array}\right|z\right]},$ (2.10)

we can choose $\beta=2\lambda+3$, yielding

$\displaystyle
I_{+}^{\lambda}\varphi_{2\lambda+3}(x)=\sqrt{\frac{\pi}{5}}\frac{\Gamma(\lambda+1)}{\Gamma\left(\lambda+\frac{3}{2}\right)}(x+1)(3-2x)^{-(\lambda+1)}.$
(2.11)

Therefore, the function given in (2.11) is strictly positive definite on
${\mathbb{S}}^{2\lambda+1}$ by the remark at the start of the example.

We can follow the same line of argument applying $I_{+}^{\mu}$ to
$I_{+}^{\lambda}\varphi_{2\lambda+3}$. Again, the result is a hypergeometric
function. Interestingly, for $\mu=\lambda-\frac{3}{2}$ the series is of the
form (2.10) resulting in

$\displaystyle
I_{+}^{\lambda-\frac{3}{2}}\big{(}I_{+}^{\lambda}\varphi_{2\lambda+3}\big{)}(x)=\frac{\pi}{5}\frac{\Gamma\left(\lambda+\frac{1}{2}\right)}{\Gamma\left(\lambda+\frac{3}{2}\right)}\,(x+1)^{2}(3-2x)^{-\left(\lambda+\frac{1}{2}\right)}.$

In general, for a function

$\displaystyle g_{m,\gamma}(x)=(x+1)^{m}(3-2x)^{-\gamma},\qquad
m\in{\mathbf{N}}_{0},\qquad\gamma>0,$

we obtain that

$\displaystyle
I_{+}^{\gamma-m-\frac{3}{2}}g_{m,\gamma}(x)=\sqrt{\frac{\pi}{5}}\frac{\Gamma\left(\gamma-\frac{1}{2}\right)}{\Gamma(\gamma)}(x+1)^{m+1}(3-2x)^{-\left(\gamma-\frac{1}{2}\right)}.$

###  2.1 Ultraspherical expansions of $\boldsymbol{f}$ and
$\boldsymbol{{\mathcal{D}}^{\lambda}_{\pm}f}$

The main results of this section will be Theorems 2.7 and 2.8 giving
multiplier relationships between the Gegenbauer coefficients of the (formal)
series of $f$ and those of the (formal) series of
${\mathcal{D}}^{\lambda}_{+}f$ and ${\mathcal{D}}^{\lambda}_{-}f$. These
relationships will later be used to show that the operators
${\mathcal{D}}^{\lambda}_{+}$ and ${\mathcal{D}}^{\lambda}_{-}$ have the
positive definiteness preserving properties given in Theorem 2.4.

The first statement shows that the operators ${\mathcal{D}}^{0}_{+}$,
${\mathcal{D}}^{0}_{-}$ can be applied term by term to a Chebyshev series to
obtain the formal Legendre series of ${\mathcal{D}}^{0}_{\pm}f$.

######  Theorem 2.7.

Let $f\in C[-1,1]$ with $($formal$)$ Chebyshev series

$\displaystyle f\sim\sum_{n=0}^{\infty}a_{n}T_{n}.$

If both functions ${\mathcal{D}}^{0}_{\pm}f\in C[-1,1]$, then the $($formal$)$
Legendre series

$\displaystyle{\mathcal{D}}^{0}_{+}f\sim\sum_{n=0}^{\infty}b_{n}P_{n}$

has coefficients

$\displaystyle b_{n}=(n+1)\pi a_{n+1},\qquad n\in{\mathbb{N}}_{0},$

and the $($formal$)$ Legendre series

$\displaystyle{\mathcal{D}}^{0}_{-}f\sim\sum_{n=0}^{\infty}c_{n}P_{n}$

has coefficients

$\displaystyle c_{n}=n\pi a_{n},\qquad n\in{\mathbb{N}}_{0}.$

Similar relations between coefficients in ultraspherical expansions hold for
higher order Gegenbauer polynomials.

######  Theorem 2.8.

Let $\lambda>0$, and let $f\in C[-1,1]$ have a $($formal$)$ Gegenbauer series

$\displaystyle f\sim\sum_{n=0}^{\infty}a_{n}C^{\lambda}_{n}.$

If both functions ${\mathcal{D}}^{\lambda}_{\pm}f\in C[-1,1]$, then the
$($formal$)$ Gegenbauer series

$\displaystyle{\mathcal{D}}^{\lambda}_{+}f\sim\sum_{n=0}^{\infty}b_{n}C_{n}^{\lambda+\frac{1}{2}}$

has coefficients

$\displaystyle
b_{n}=\frac{\Gamma(\lambda+\frac{1}{2})\sqrt{\pi}}{\Gamma(\lambda)}\,\frac{2(n+2\lambda+1)}{n+\lambda+1}a_{n+1},\qquad
n\in{\mathbb{N}}_{0},$ (2.12)

and the $($formal$)$ Gegenbauer series

$\displaystyle{\mathcal{D}}^{\lambda}_{-}f\sim\sum_{n=0}^{\infty}c_{n}C^{\lambda+\frac{1}{2}}_{n},$

has coefficients

$\displaystyle
c_{n}=\frac{\Gamma(\lambda+\frac{1}{2})\sqrt{\pi}}{\Gamma(\lambda)}\frac{2n}{n+\lambda}a_{n},\qquad
n\in{\mathbb{N}}_{0}.$ (2.13)

######  Remark 2.9.

Theorem 2.7 is the limiting case of Theorem 2.8 under the limit

$\displaystyle C_{n}^{0}(x)=\lim_{\lambda\to
0^{+}}\frac{1}{\lambda}C_{n}^{\lambda}(x),\qquad n>0,\qquad\text{and}\qquad
C_{0}^{0}(x)=T_{0}(x)=1.$ (2.14)

Furthermore, we have the special cases

$\displaystyle C_{n}^{0}(x)=\frac{2}{n}T_{n}(x),\qquad n>0,$

while

$\displaystyle C_{n}^{\frac{1}{2}}(x)=P_{n}(x)\qquad\text{and}\qquad
C_{n}^{1}(x)=U_{n}(x),\qquad n\geq 0.$

Before proving the theorems we state the following technical lemma.

######  Lemma 2.10.

For $\lambda>0$, $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$,

$\displaystyle\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}=(n+1)C_{n}^{\lambda}(x)+2\sum_{k=0}^{n-1}(k+\lambda)C_{k}^{\lambda}(x),$
(2.15)
$\displaystyle\frac{d}{dx}\big{\\{}(1-x)C_{n}^{\lambda}(x)\big{\\}}=-(n+1)C_{n}^{\lambda}(x)+2\sum_{k=0}^{n-1}(-1)^{k+n+1}(k+\lambda)C_{k}^{\lambda}(x).$
(2.16)

###### Proof.

Formula (2.16) can be obtained from equation (2.15) by using the reflection
formula for Gegenbauer polynomials

$\displaystyle C^{\lambda}_{n}(-x)=(-1)^{n}C^{\lambda}_{n}(x).$ (2.17)

and the change of variables $y=-x$.

For the proof of formula (2.15) we will use two recurrences involving
derivatives of Gegenbauer polynomials which can be found, for example in [9,
10.9(35)]. For notational convenience we use the (non-standard) notation
$D_{n}^{\lambda}(x)=\frac{d}{dx}C_{n}^{\lambda}(x)$ within the proof of the
lemma. Then,

$\displaystyle
nC^{\lambda}_{n}(x)=xD^{\lambda}_{n}(x)-D^{\lambda}_{n-1}(x),\qquad\mbox{
and}$ (2.18)
$\displaystyle\big{(}1-x^{2}\big{)}D^{\lambda}_{n}(x)=\big{(}1-x^{2}\big{)}2\lambda
C_{n-1}^{\lambda+1}(x)=(n+2\lambda-1)C^{\lambda}_{n-1}(x)-nxC^{\lambda}_{n}(x).$
(2.19)

Turn now to an induction proof of formula (2.15). The statement is clearly
true for $C^{\lambda}_{0}(x)=1$, adopting the convention that the sum then is
empty.

Now assume that $n\in{\mathbb{N}}$ and that the first statement is true for
$n-1$. Consider

$\displaystyle\frac{d}{dx}\big{\\{}(1+x)C^{\lambda}_{n}(x)\big{\\}}=C_{n}^{\lambda}(x)+(1+x)D_{n}^{\lambda}(x).$
(2.20)

Using (2.19) and then (2.18) we obtain that

$\displaystyle(1+x)D_{n}^{\lambda}(x)=\big{(}1-x^{2}\big{)}D_{n}^{\lambda}(x)+(1+x)xD_{n}^{\lambda}(x)$
$\displaystyle\hphantom{(1+x)D_{n}^{\lambda}(x)}{}=(n+2\lambda-1)C_{n-1}^{\lambda}(x)-nxC_{n}^{\lambda}(x)+(1+x)\big{(}nC_{n}^{\lambda}(x)+D_{n-1}^{\lambda}(x)\big{)}$
$\displaystyle\hphantom{(1+x)D_{n}^{\lambda}(x)}{}=(n+2\lambda-1)C_{n-1}^{\lambda}(x)+nC_{n}^{\lambda}(x)+(1+x)D_{n-1}^{\lambda}(x).$

Therefore, applying (2.20),

$\displaystyle\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}=(n+2\lambda-1)C_{n-1}^{\lambda}(x)+(n+1)C_{n}^{\lambda}(x)+(1+x)D_{n-1}^{\lambda}(x)$
$\displaystyle\hphantom{\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}}{}=(n+2\lambda-1)C_{n-1}^{\lambda}(x)+(n+1)C_{n}^{\lambda}(x)$
$\displaystyle\hphantom{\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}=}{}+\frac{d}{dx}\big{\\{}(1+x)C_{n-1}^{\lambda}(x)\big{\\}}-C_{n-1}^{\lambda}(x)$
$\displaystyle\hphantom{\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}}{}=(n+2\lambda-2)C_{n-1}^{\lambda}(x)+(n+1)C_{n}^{\lambda}(x)+\frac{d}{dx}\big{\\{}(1+x)C_{n-1}^{\lambda}(x)\big{\\}}.$

Using the induction hypothesis gives

$\displaystyle\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}=(n+1)C_{n}^{\lambda}(x)+(n+2\lambda-2)C_{n-1}^{\lambda}(x)$
$\displaystyle\hphantom{\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}=}{}+nC_{n-1}^{\lambda}(x)+2\sum_{k=0}^{n-2}(k+\lambda)C_{k}^{\lambda}(x)$
$\displaystyle\hphantom{\frac{d}{dx}\big{\\{}(1+x)C_{n}^{\lambda}(x)\big{\\}}}{}=(n+1)C_{n}^{\lambda}(x)+2\big{[}(n-1)+\lambda\big{]}C_{n-1}^{\lambda}(x)+2\sum_{k=0}^{n-2}(k+\lambda)C_{k}^{\lambda}(x),$

which completes the proof. ∎

###### Proof of Theorem 2.7.

Note that the continuity of the two functions ${\mathcal{D}}^{0}_{\pm}f$
implies that of the functions $D^{0}_{\pm}f$. Then, proceeding by integration
by parts, the coefficient $b_{n}$ of $P_{n}$ in the formal Legendre expansion
of $D_{+}^{0}f$ is $(2n+1)/2$ times

$\displaystyle H_{+}=\int_{-1}^{1}P_{n}(x)\left(D_{+}^{0}f\right)(x)dx$
$\displaystyle\hphantom{H_{+}}{}=\int_{-1}^{1}P_{n}(x)(1+x)\frac{d}{dx}\left(\int_{-1}^{x}(x-t)^{-1/2}(1+t)^{-1/2}f(t)dt\right)dx$
$\displaystyle\hphantom{H_{+}}{}=\left[P_{n}(x)(1+x)\int_{-1}^{x}(x-t)^{-1/2}(1+t)^{-1/2}f(t)dt\right]_{-1}^{1}$
$\displaystyle\hphantom{H_{+}=}{}-\int_{-1}^{1}\int_{-1}^{x}(x-t)^{-1/2}(1+t)^{-1/2}f(t)dt\frac{d}{dx}\big{\\{}P_{n}(x)(1+x)\big{\\}}dx.$

In view of the formula [19, (5.12.1)],

$\displaystyle\int_{-1}^{x}(x-t)^{-1/2}(1+t)^{-1/2}dt=\pi,$

for all $-1<x\leq 1$, the limit as $x$ tends to $-1$ of the quantity within
the square brackets vanishes. Hence,

$\displaystyle H_{+}=2P_{n}(1)\int_{-1}^{1}\big{(}1-t^{2}\big{)}^{-1/2}f(t)dt$
$\displaystyle\hphantom{H_{+}=}{}-\int_{-1}^{1}\int_{t}^{1}(x-t)^{-1/2}\frac{d}{dx}\\{P_{n}(x)(1+x)\\}dx(1+t)^{-1/2}f(t)dt$
$\displaystyle\hphantom{H_{+}}{}=2\pi
a_{0}-\int_{-1}^{1}\int_{t}^{1}(x-t)^{-1/2}\left[(n+1)P_{n}(x)+\sum_{k=0}^{n-1}(2k+1)P_{k}(x)\right]dx(1+t)^{-1/2}f(t)dt,$

where the last step follows from an application of formula (2.15).

Noting the relationship (cf. [19, (18.17.46)])

$\displaystyle\int_{t}^{1}(x-t)^{-1/2}P_{k}(x)dx=\frac{1}{(k+\frac{1}{2})}\frac{1}{\sqrt{1-t}}\big{[}T_{k}(t)-T_{k+1}(t)\big{]},$
(2.21)

after some straightforward calculation, the double integral above turns into
the form

$\displaystyle
H_{+}=\frac{2n}{2n+1}\int_{-1}^{1}\big{(}1-t^{2}\big{)}^{-\frac{1}{2}}T_{n}(t)f(t)dt+\frac{2(n+1)}{2n+1}\int_{-1}^{1}\big{(}1-t^{2}\big{)}^{-\frac{1}{2}}T_{n+1}(t)f(t)dt.$

Therefore,

$\displaystyle H_{+}=\frac{n}{2n+1}\pi a_{n}+\frac{n+1}{2n+1}\pi a_{n+1}.$

Analogously, we define

$\displaystyle H_{-}=\int_{-1}^{1}P_{n}(x)\left(D_{-}^{0}f\right)(x)dx.$

A similar integration by parts argument, but now using the formula (2.16), and
the relationship (cf. [19, (18.17.45)])

$\displaystyle\int_{-1}^{t}(t-x)^{-1/2}P_{k}(x)dx=\frac{1}{(k+\frac{1}{2})}\frac{1}{\sqrt{1+t}}\big{[}T_{k}(t)+T_{k+1}(t)\big{]},$
(2.22)

shows that

$\displaystyle H_{-}=-\frac{n}{2n+1}\pi a_{n}+\frac{n+1}{2n+1}\pi a_{n+1}.$

Since $b_{n}=\frac{2n+1}{2}(H_{+}+H_{-})$ we finally obtain

$\displaystyle b_{n}=(n+1)\pi a_{n+1},\qquad n\in{\mathbb{N}}_{0}.$

This completes the proof of the part of the theorem concerning
${\mathcal{D}}^{0}_{+}f$. The proof of the part of the theorem concerning
${\mathcal{D}}^{0}_{-}f$ is similar and will be omitted. ∎

The proof of Theorem 2.8 relies on a kind of fractional integration by parts.
Before going into details, we will state some technical lemmas.

######  Lemma 2.11.

For $\lambda\geq 1/2$, $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$,

$\displaystyle\frac{d}{dx}\big{\\{}(1+x)^{\lambda+1}(1-x)^{\lambda}C^{\lambda+\frac{1}{2}}_{n}(x)\big{\\}}=(1+x)\big{(}1-x^{2}\big{)}^{\lambda-1}Q_{n+1}(x),$
(2.23)

where

$\displaystyle
Q_{n+1}(x)=(1-x)C_{n}^{\lambda+\frac{1}{2}}(x)-(n+1)\frac{2\lambda+n}{2\lambda-1}C_{n+1}^{\lambda-\frac{1}{2}}(x).$

###### Proof.

Note the formula (see [1, (22.13.2)] or [19, (18.17.1)] for the general Jacobi
case)

$\displaystyle
n\left(1+\frac{n}{2\lambda}\right)\int_{0}^{x}\big{(}1-t^{2}\big{)}^{\lambda-\frac{1}{2}}C^{\lambda}_{n}(t)dt=C^{\lambda+1}_{n-1}(0)-\big{(}1-x^{2}\big{)}^{\lambda+\frac{1}{2}}C^{\lambda+1}_{n-1}(x),$

which implies

$\displaystyle\frac{d}{dx}\big{\\{}\big{(}1-x^{2}\big{)}^{\lambda}C^{\lambda+\frac{1}{2}}_{n}(x)\big{\\}}=-(n+1)\left(1+\frac{n+1}{2(\lambda-\frac{1}{2})}\right)\big{(}1-x^{2}\big{)}^{\lambda-1}C^{\lambda-\frac{1}{2}}_{n+1}(x),\qquad\lambda>1/2.$

Employing the relationship above, computing the derivative on the left hand
side of (2.23) yields

$\displaystyle\frac{d}{dx}\big{\\{}\big{(}1-x^{2}\big{)}^{\lambda}C^{\lambda+\frac{1}{2}}_{n}(x)(1+x)\big{\\}}=(1-x^{2})^{\lambda}C^{\lambda+\frac{1}{2}}_{n}(x)+(1+x)\frac{d}{dx}\big{\\{}\big{(}1-x^{2}\big{)}^{\lambda}C^{\lambda+\frac{1}{2}}_{n}(x)\big{\\}}$
$\displaystyle=\big{(}1-x^{2}\big{)}^{\lambda}C^{\lambda+\frac{1}{2}}_{n}(x)-(1+x)(n+1)\left(1+\frac{n+1}{2\lambda-1}\right)\big{(}1-x^{2}\big{)}^{\lambda-1}C^{\lambda-\frac{1}{2}}_{n+1}(x)$
$\displaystyle=(1+x)\big{(}1-x^{2}\big{)}^{\lambda-1}\left\\{(1-x)C^{\lambda+\frac{1}{2}}_{n}(x)-(n+1)\left(1+\frac{n+1}{2\lambda-1}\right)C^{\lambda-\frac{1}{2}}_{n+1}(x)\right\\},\qquad\lambda>1/2.$

Setting

$\displaystyle
Q_{n+1}(x)=(1-x)C^{\lambda+\frac{1}{2}}_{n}(x)-(n+1)\left(1+\frac{n+1}{2\lambda-1}\right)C^{\lambda-\frac{1}{2}}_{n+1}(x),$

completes the proof when $\lambda>1/2$. The limit relation (2.14) implies the
result for $\lambda=1/2$. ∎

######  Lemma 2.12.

Let $\lambda>1$, $n\in{\mathbb{N}}$ and $x\in[-1,1]$. Then

$\displaystyle(n+2\lambda-1)C_{n+1}^{\lambda-1}(x)-(n+2)C_{n+2}^{\lambda-1}(x)=(2\lambda-2)(1-x)\big{[}C_{n+1}^{\lambda}(x)+C_{n}^{\lambda}(x)\big{]}.$
(2.24)

###### Proof.

Since $\frac{d}{dx}C_{n}^{\lambda}(x)=2\lambda C_{n-1}^{\lambda+1}(x)$, we
have that

$\displaystyle(2\lambda-2)\big{[}C_{n+1}^{\lambda}(x)+C_{n}^{\lambda}(x)\big{]}=\frac{d}{dx}C_{n+2}^{\lambda-1}(x)+\frac{d}{dx}C_{n+1}^{\lambda-1}(x).$

Using (cf. [9, 10.9(25), (15)])

$\displaystyle\frac{d}{dx}C_{n+1}^{\lambda}(x)=x\frac{d}{dx}C_{n}^{\lambda}(x)+(2\lambda+n)C_{n}^{\lambda}(x)$

and

$\displaystyle\big{(}1-x^{2}\big{)}\frac{d}{dx}C_{n}^{\lambda}(x)=(n+2\lambda)xC_{n}^{\lambda}(x)-(n+1)C_{n+1}^{\lambda}(x)$

we can proceed, obtaining

$\displaystyle(1-x)(2\lambda-2)\big{[}C_{n+1}^{\lambda}(x)+C_{n}^{\lambda}(x)\big{]}=(1-x)(2\lambda+n-1)C_{n+1}^{\lambda-1}(x)$
$\displaystyle\hphantom{(1-x)(2\lambda-2)\big{[}C_{n+1}^{\lambda}(x)+C_{n}^{\lambda}(x)\big{]}=}{}+(n+2\lambda-1)xC_{n+1}^{\lambda-1}(x)-(n+2)C_{n+1}^{\lambda-1}(x),$

from which the statement follows. ∎

The following proposition is the limit case of (2.24) taking the limit
$\lambda\to 1^{+}$ after multiplying either side with $1/(\lambda-1)$.

######  Proposition 2.13.

Let $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$. Then

$\displaystyle T_{n+1}(x)-T_{n+2}(x)=(1-x)\big{[}U_{n+1}(x)+U_{n}(x)\big{]}.$

###### Proof.

From [9, 10.11(3)] we have that

$\displaystyle T_{n+1}(x)=U_{n+1}(x)-xU_{n}(x).$

Furthermore, [9, 10.11(37)] yields

$\displaystyle xU_{n+1}(x)=U_{n+2}(x)-T_{n+2}(x),\qquad
2T_{n+2}(x)=U_{n+2}(x)-U_{n}(x).$

The claim follows from using these relations to rewrite the right hand side in
terms of Chebyshev polynomials of the first kind. ∎

The following lemma states the analogues of equations (2.21) and (2.22) for
general $\lambda>0$. Note that the first integral given in the lemma is a
special case of _Bateman’s integral_ (see [19, (18.17.9)]). Using this, the
statement follows from [19, (18.9.4)]. Rather than using Bateman’s integral
for Jacobi polynomials, we provide a proof staying within the family of
Gegenbauer polynomials, only.

######  Lemma 2.14.

Let $\lambda>0$, $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$. Then

$\displaystyle\frac{1}{\Gamma(\lambda)}\int_{x}^{1}(1-t)^{\lambda}(t-x)^{-\frac{1}{2}}C_{n}^{\lambda+\frac{1}{2}}(t)dt$
$\displaystyle\qquad{}=\frac{\sqrt{\pi}}{2\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}(1-x)^{\lambda-\frac{1}{2}}\big{[}(n+2\lambda)C_{n}^{\lambda}(x)-(n+1)C_{n+1}^{\lambda}(x)\big{]},$

and

$\displaystyle\frac{1}{\Gamma(\lambda)}\int_{-1}^{x}(1+t)^{\lambda}(x-t)^{-\frac{1}{2}}C_{n}^{\lambda+\frac{1}{2}}(t)dt$
$\displaystyle\qquad{}=\frac{\sqrt{\pi}}{2\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}(1+x)^{\lambda-\frac{1}{2}}\big{[}(n+2\lambda)C_{n}^{\lambda}(x)+(n+1)C_{n+1}^{\lambda}(x)\big{]}.$

###### Proof.

From the definition we have that

$\displaystyle\int_{x}^{1}(1-t)^{\lambda}(t-x)^{-\frac{1}{2}}C_{n}^{\lambda+\frac{1}{2}}(t)dt=(1-x)^{\lambda-\frac{1}{2}}I_{-}^{\lambda}C_{n}^{\lambda+\frac{1}{2}}(x),$

and
$2I_{-}^{\lambda}={\mathcal{I}}_{+}^{\lambda}-{\mathcal{I}}_{-}^{\lambda}$. We
can therefore use Theorem 3.3 below to compute

$\displaystyle
2I_{-}^{\lambda}C_{n}^{\lambda+\frac{1}{2}}(x)=\frac{\sqrt{\pi}\Gamma(\lambda)}{\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}\big{[}(n+2\lambda)C_{n}^{\lambda}(x)-(n+1)C_{n+1}^{\lambda}(x)\big{]},$

from which the first statement follows. The second integral follows from the
first by a change of variables $\tau=-t$ and using the reflection formula
(2.17). ∎

######  Remark 2.15.

Taking the limit (2.14) readily leads to (2.21) and (2.22).

######  Corollary 2.16.

Let $\lambda\geq 1$, $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$. Then

$\displaystyle\frac{1}{\Gamma(\lambda)}\int_{x}^{1}(1-t)^{\lambda-1}(t-x)^{-\frac{1}{2}}C_{n+1}^{\lambda-\frac{1}{2}}(t)dt$
$\displaystyle\qquad{}=\frac{\sqrt{\pi}}{\Gamma\left(\lambda-\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}(1-x)^{\lambda-\frac{1}{2}}\big{[}C_{n+1}^{\lambda}(x)+C_{n}^{\lambda}(x)\big{]},$

and

$\displaystyle\frac{1}{\Gamma(\lambda)}\int_{-1}^{x}(1+t)^{\lambda-1}(x-t)^{-\frac{1}{2}}C_{n+1}^{\lambda-\frac{1}{2}}(t)dt$
$\displaystyle\qquad{}=\frac{\sqrt{\pi}}{\Gamma\left(\lambda-\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}(1+x)^{\lambda-\frac{1}{2}}\big{[}C_{n+1}^{\lambda}(x)-C_{n}^{\lambda}(x)\big{]}.$

###### Proof.

The result follows from setting $\lambda-1$ and $n+1$ instead of $\lambda$ and
$n$, respectively in Lemma 2.14 and applying Lemma 2.12. ∎

######  Remark 2.17.

Formulæ (2.21) and (2.22) are the special cases for $\lambda=1$.

###### Proof of Theorem 2.8.

The Fourier–Gegenbauer coefficients of the function
${\mathcal{D}}_{+}^{\lambda}f$ are given by

$\displaystyle
b_{n}=\frac{1}{h_{n}^{\lambda+\frac{1}{2}}}\int_{-1}^{1}{\mathcal{D}}_{+}^{\lambda}f(t)C_{n}^{\lambda+\frac{1}{2}}(t)\big{(}1-t^{2}\big{)}^{\lambda}dt,$

where

$\displaystyle
h_{n}^{\lambda+\frac{1}{2}}=\frac{\pi\Gamma(2\lambda+n+1)}{2^{2\lambda}n!\left(\lambda+n+\frac{1}{2}\right)\Gamma^{2}\left(\lambda+\frac{1}{2}\right)}.$

Therefore,

$\displaystyle
h_{n}^{\lambda+\frac{1}{2}}b_{n}=\int_{-1}^{1}D_{+}^{\lambda}f(t)C_{n}^{\lambda+\frac{1}{2}}(t)\big{(}1-t^{2}\big{)}^{\lambda}dt+\int_{-1}^{1}D_{-}^{\lambda}f(t)C_{n}^{\lambda+\frac{1}{2}}(t)\big{(}1-t^{2}\big{)}^{\lambda}dt.$

Let us denote the first integral by $H_{+}$ and the second by $H_{-}$. From
integration by parts it follows that

$\displaystyle
H_{+}=\int_{-1}^{1}(1+x)\frac{d}{dx}\left\\{(1+x)^{-\lambda}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}(1+t)^{\lambda-\frac{1}{2}}f(t)dt\right\\}C_{n}^{\lambda+\frac{1}{2}}(x)\big{(}1-x^{2}\big{)}^{\lambda}dx$
$\displaystyle\hphantom{H_{+}}{}=\left[(1+x)^{-\lambda}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}(1+t)^{\lambda-\frac{1}{2}}f(t)dt(1+x)\big{(}1-x^{2}\big{)}^{\lambda}C_{n}^{\lambda+\frac{1}{2}}(x)\right]_{-1}^{1}$
$\displaystyle\hphantom{H_{+}=}{}-\int_{-1}^{1}(1+x)^{-\lambda}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}(1+t)^{\lambda-\frac{1}{2}}f(t)dt\frac{d}{dx}\big{\\{}(1+x)\big{(}1-x^{2}\big{)}^{\lambda}C_{n}^{\lambda+\frac{1}{2}}(x)\big{\\}}dx.$

The constant term vanishes. Applying Lemma 2.11 we can decompose $H_{+}$ into
the sum of the two integrals

$\displaystyle
I_{1}=-\int_{-1}^{1}(1+x)^{-\lambda}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}(1+t)^{\lambda-\frac{1}{2}}f(t)dt\big{(}1-x^{2}\big{)}^{\lambda}C_{n}^{\lambda+\frac{1}{2}}(x)dx$
$\displaystyle\hphantom{I_{1}}{}=-\int_{-1}^{1}\int_{t}^{1}(1-x)^{\lambda}(x-t)^{-\frac{1}{2}}C_{n}^{\lambda+\frac{1}{2}}(x)dx(1+t)^{\lambda-\frac{1}{2}}f(t)dt$

and

$\displaystyle
I_{2}=(n+1)\frac{2\lambda+n}{2\lambda-1}\int_{-1}^{1}(1+x)^{-\lambda+1}\int_{-1}^{x}(x-t)^{-\frac{1}{2}}(1+t)^{\lambda-\frac{1}{2}}f(t)dt\big{(}1-x^{2}\big{)}^{\lambda-1}C_{n+1}^{\lambda-\frac{1}{2}}(x)dx$
$\displaystyle\hphantom{I_{2}}{}=(n+1)\frac{2\lambda+n}{2\lambda-1}\int_{-1}^{1}\int_{t}^{1}(1-x)^{\lambda-1}(x-t)^{-\frac{1}{2}}C_{n+1}^{\lambda-\frac{1}{2}}(x)dx(1+t)^{\lambda-\frac{1}{2}}f(t)dt.$

In an analogous way, we decompose $H_{-}$ into a sum of the integrals

$\displaystyle
I_{3}=\int_{-1}^{1}\int_{-1}^{t}(1+x)^{\lambda}(t-x)^{-\frac{1}{2}}C_{n}^{\lambda+\frac{1}{2}}(x)dx(1-t)^{\lambda-\frac{1}{2}}f(t)dt$

and

$\displaystyle
I_{4}=(n+1)\frac{2\lambda+n}{2\lambda-1}\int_{-1}^{1}\int_{-1}^{t}(1+x)^{\lambda-1}(t-x)^{-\frac{1}{2}}C_{n+1}^{\lambda-\frac{1}{2}}(x)dx(1-t)^{\lambda-\frac{1}{2}}f(t)dt.$

The inner integrals in $I_{1}$ and $I_{3}$ can be computed using Lemma 2.14,
whereas the corresponding formulæ for $I_{2}$ and $I_{4}$ are stated in
Corollary 2.16. Doing so, and using the definition of the coefficients

$\displaystyle
a_{n}=\frac{1}{h_{n}^{\lambda}}\int_{-1}^{1}f(t)C_{n}^{\lambda}(t)\big{(}1-t^{2}\big{)}^{\lambda-\frac{1}{2}}dt,\qquad\text{where}\qquad
h_{n}^{\lambda}=\frac{\pi\Gamma(2\lambda+n)}{2^{2\lambda-1}n!(\lambda+n)\Gamma^{2}(\lambda)},$

we obtain that

$\displaystyle
I_{1}=\frac{\pi^{\frac{3}{2}}}{2^{2\lambda}n!\Gamma(\lambda)\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}$
$\displaystyle\hphantom{I_{1}=}{}\times\left[\frac{\Gamma(2\lambda+n+1)}{\lambda+n+1}a_{n+1}-(2\lambda+n)\frac{\Gamma(2\lambda+n)}{\lambda+n}a_{n}\right],$
$\displaystyle
I_{2}=\frac{\pi^{\frac{3}{2}}}{2^{2\lambda}n!\Gamma(\lambda)\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}$
$\displaystyle\hphantom{I_{2}=}{}\times\left[(2\lambda+n)\frac{\Gamma(2\lambda+n+1)}{\lambda+n+1}a_{n+1}+(2\lambda+n)\frac{\Gamma(2\lambda+n)}{\lambda+n}a_{n}\right],$
$\displaystyle
I_{3}=\frac{\pi^{\frac{3}{2}}}{2^{2\lambda}n!\Gamma(\lambda)\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}$
$\displaystyle\hphantom{I_{3}=}{}\times\left[(n+2\lambda)\frac{\Gamma(2\lambda+n)}{\lambda+n}a_{n}+\frac{\Gamma(2\lambda+n+1)}{\lambda+n+1}a_{n+1}\right],$
$\displaystyle
I_{4}=\frac{\pi^{\frac{3}{2}}}{2^{2\lambda}n!\Gamma(\lambda)\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)}$
$\displaystyle\hphantom{I_{4}=}{}\times\left[(2\lambda+n)\frac{\Gamma(2\lambda+n+1)}{\lambda+n+1}a_{n+1}-(2\lambda+n)\frac{\Gamma(2\lambda+n)}{\lambda+n}a_{n}\right].$

Therefore,

$\displaystyle
H_{+}+H_{-}=\frac{\pi^{\frac{3}{2}}\Gamma(2\lambda+n+2)}{2^{2\lambda-1}n!\Gamma(\lambda)\Gamma\left(\lambda+\frac{1}{2}\right)\left(n+\lambda+\frac{1}{2}\right)(\lambda+n+1)}a_{n+1},$

from which it follows that

$\displaystyle
b_{n}=\frac{2\sqrt{\pi}\Gamma\left(\lambda+\frac{1}{2}\right)}{\Gamma(\lambda)}\frac{2\lambda+n+1}{\lambda+n+1}a_{n+1}.$

This completes the proof of the part of the theorem concerning
${\mathcal{D}}_{+}^{\lambda}f$. Again, the proof of the part concerning
${\mathcal{D}}^{\lambda}_{-}f$ is similar and will therefore be omitted. ∎

###### Proof of Theorem 2.4.

From the continuity assumption we have that the Gegenbauer series of the
functions ${\mathcal{D}}^{\lambda}_{\pm}f$ are Abel summable. Since $f$ is
positive definite by hypothesis, Theorem 2.8 shows that the Gegenbauer
coefficients of ${\mathcal{D}}^{\lambda}_{\pm}f$ are non-negative. Hence the
Gegenbauer series restricted to $x=1$ are series of non-negative terms. For
such series of non-negative terms Abel summability implies summability. Since
$C_{n}^{\lambda+\frac{1}{2}}$ attains its maximum at the point 1 we can apply
the Weierstraß M-test to show that the Gegenbauer series of
${\mathcal{D}}^{\lambda}_{\pm}f$ converge uniformly on $[-1,1]$. Furthermore,
the multipliers given in equations (2.12) and (2.13) preserve the sign of the
coefficients. Therefore, ${\mathcal{D}}_{\pm}^{\lambda}f\in\Lambda_{m+1}$ by
Theorem 1.2. The statement about strict positive-definiteness then follows
from the same relation and the discussion on strict positive definiteness in
the paragraph following Theorem 1.2. ∎

##  3 The action of $\boldsymbol{{\mathcal{I}}^{\lambda}_{\pm}}$ and
$\boldsymbol{{\mathcal{D}}^{\lambda}_{\pm}}$ on Gegenbauer polynomials

The claim that the operators ${\mathcal{I}}^{\lambda}_{\pm}$ and
${\mathcal{D}}^{\lambda}_{\pm}$ map Gegenbauer polynomials onto Gegenbauer
polynomials with a changed parameter is based upon contiguous relations for
hypergeometric functions. We will thus first state a proposition showing that
the images of $C_{n}^{\lambda+\frac{1}{2}}$ under the operators
$I_{\pm}^{\lambda}$ are hypergeometric polynomials.

######  Proposition 3.1.

Let $\lambda\geq 0$, $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$. Then

$\displaystyle
I^{\lambda}_{+}C_{n}^{\lambda+\frac{1}{2}}(x)=c_{n,\lambda}\frac{\lambda+\frac{1}{2}}{\lambda+n+{\frac{1}{2}}}(1+x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{1}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]},$ (3.3)
$\displaystyle
I^{\lambda}_{-}C_{n}^{\lambda+\frac{1}{2}}(x)=c_{n,\lambda}(1-x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]},$ (3.6)

where

$\displaystyle
c_{n,\lambda}=\frac{\sqrt{\pi}(2\lambda+1)_{n}}{n!}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}.$

Before proving the proposition let us state a technical lemma.

######  Lemma 3.2.

Let $\lambda>-\frac{1}{2}$, $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$. Then

$\displaystyle\left(I^{\lambda}_{\pm}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+1\\\ \end{array}\right|\frac{1\pm\mathchoice{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\displaystyle\cdot$}}}}\hss}}}{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\textstyle\cdot$}}}}\hss}}}{{\hbox
to4.16661pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptstyle\cdot$}}}}\hss}}}{{\hbox
to3.61105pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptscriptstyle\cdot$}}}}\hss}}}}{2}\right]}\right)(x)$
$\displaystyle\qquad{}=\sqrt{\pi}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}(1\pm
x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|\frac{1\pm x}{2}\right]}.$

###### Proof.

We will prove the statement in the ‘$+$’ case (the proof of ‘$-$’ case being
analogous):

$\displaystyle\left(I^{\lambda}_{+}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+1\\\ \end{array}\right|\frac{1+\mathchoice{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\displaystyle\cdot$}}}}\hss}}}{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\textstyle\cdot$}}}}\hss}}}{{\hbox
to4.16661pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptstyle\cdot$}}}}\hss}}}{{\hbox
to3.61105pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptscriptstyle\cdot$}}}}\hss}}}}{2}\right]}\right)(x)$
$\displaystyle\qquad{}=(1+x)^{-\lambda+\frac{1}{2}}\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\lambda}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+1\\\ \end{array}\right|\frac{1+\tau}{2}\right]}d\tau$
$\displaystyle\qquad{}=\sum_{k=0}^{n}\frac{(-n)_{k}(n+2\lambda+1)_{k}}{(\lambda+1)_{k}k!2^{k}}(1+x)^{-\lambda+\frac{1}{2}}\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\lambda+k}d\tau.$

Using the beta integral (2.4) the expression above reduces to

$\displaystyle\sum_{k=0}^{n}\frac{(-n)_{k}(n+2\lambda+1)_{k}}{(\lambda+1)_{k}k!2^{k}}\frac{\Gamma(\frac{1}{2})\Gamma(\lambda+k+1)}{\Gamma(\lambda+k+\frac{3}{2})}(1+x)^{k+1}$
$\displaystyle\qquad{}=\sqrt{\pi}(1+x)\sum_{k=0}^{n}\frac{(-n)_{k}(n+2\lambda+1)_{k}\Gamma(\lambda+1)}{k!2^{k}}\frac{(1+x)^{k}}{(\lambda+\frac{3}{2})_{k}\Gamma(\lambda+\frac{3}{2})}$
$\displaystyle\qquad{}=\sqrt{\pi}\;\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}(1+x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|\frac{1+x}{2}\right]}.$ ∎

The following proof of Proposition 3.1 uses an identity due to Pfaff (cf. [2,
2.3.14])

$\displaystyle{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-m,b\\\ c\\\
\end{array}\right|z\right]}=\frac{(c-b)_{m}}{(c)_{m}}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-m,b\\\
b-c-m+1\\\ \end{array}\right|1-z\right]},\qquad m\in{\mathbb{N}}_{0},$ (3.11)

which will occur frequently later on.

###### Proof of Proposition 3.1.

From the representation of the Gegenbauer polynomials as hypergeometric
functions [19, (18.5.9)], and the reflection formula (2.17) we have

$\displaystyle
I^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)=(-1)^{n}\frac{\Gamma(n+2\lambda+1)}{n!\Gamma(2\lambda+1)}\left(I^{\lambda}_{+}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+1\\\ \end{array}\right|\frac{1+\mathchoice{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\displaystyle\cdot$}}}}\hss}}}{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\textstyle\cdot$}}}}\hss}}}{{\hbox
to4.16661pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptstyle\cdot$}}}}\hss}}}{{\hbox
to3.61105pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptscriptstyle\cdot$}}}}\hss}}}}{2}\right]}\right)(x).$

Applying Lemma 3.2

$\displaystyle
I^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)=(-1)^{n}\frac{\Gamma(n+2\lambda+1)}{n!\Gamma(2\lambda+1)}\sqrt{\pi}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}(1+x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|\frac{1+x}{2}\right]}$ (3.14)
$\displaystyle\hphantom{I^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)}{}=\frac{\sqrt{\pi}(-1)^{n}(2\lambda+1)_{n}}{n!}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}(1+x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|\frac{1+x}{2}\right]}.$ (3.17)

Now (3.11) implies

$\displaystyle{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{3}{2}\\\
\end{array}\right|\frac{1+x}{2}\right]}=\frac{(-\lambda-n+\frac{1}{2})_{n}}{(\lambda+\frac{3}{2})_{n}}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{1}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]}.$ (3.22)

Substituting (3.22) into (3.17) noting that $(a-n)_{n}=(-1)^{n}(1-a)_{n}$
shows

$\displaystyle
I^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)=\frac{\sqrt{\pi}(2\lambda+1)_{n}}{n!}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}\frac{(\lambda+\frac{1}{2})_{n}}{(\lambda+\frac{3}{2})_{n}}(1+x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{1}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]}$
$\displaystyle\hphantom{I^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)}{}=\frac{\sqrt{\pi}(2\lambda+1)_{n}}{n!}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}\frac{\lambda+\frac{1}{2}}{\lambda+n+{\frac{1}{2}}}(1+x)\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{1}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]}.$

The proof of equation (3.6) is almost identical to that part of the proof of
equation (3.3) up to equation (3.17). It will therefore be omitted. ∎

######  Theorem 3.3.

Let $\lambda>0$ and $n\in{\mathbb{N}}_{0}$. Then

$\displaystyle{\mathcal{I}}^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)=\frac{\sqrt{\pi}\Gamma(\lambda)}{\Gamma({\lambda+\frac{1}{2}})}\frac{n+2\lambda}{n+\lambda+{\frac{1}{2}}}C^{\lambda}_{n}(x),\qquad
x\in[-1,1],$ (3.23)
$\displaystyle{\mathcal{I}}^{\lambda}_{-}C^{\lambda+\frac{1}{2}}_{n}(x)=\frac{\sqrt{\pi}\Gamma(\lambda)}{\Gamma({\lambda+\frac{1}{2}})}\frac{n+1}{n+\lambda+{\frac{1}{2}}}C^{\lambda}_{n+1}(x),\qquad
x\in[-1,1].$ (3.24)

######  Remark 3.4.

Taking the limit of the last two relations as $\lambda\rightarrow 0^{+}$ (see
(2.14)) gives

$\displaystyle{\mathcal{I}}^{0}_{+}P_{n}(x)=\frac{2}{n+\frac{1}{2}}T_{n}(x),\qquad\text{and}\qquad{\mathcal{I}}^{0}_{-}P_{n}(x)=\frac{2}{n+\frac{1}{2}}T_{n+1}(x),\qquad
n\in{\mathbb{N}}_{0}.$

###### Proof of Theorem 3.3.

From (3.3) and (3.6) we obtain that

$\displaystyle{\mathcal{I}}^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)=\big{\\{}I^{\lambda}_{+}+I^{\lambda}_{-}\big{\\}}C^{\lambda+\frac{1}{2}}_{n}(x)$
$\displaystyle\hphantom{{\mathcal{I}}^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)}{}=\frac{2\sqrt{\pi}(2\lambda+1)_{n}}{(\lambda+n+\frac{1}{2})n!}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}\left\\{\left(\lambda+\frac{1}{2}\right)\left(\frac{1+x}{2}\right){{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{1}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]}\\!\right.$
$\displaystyle\left.\hphantom{{\mathcal{I}}^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)=}{}+\left(\lambda+n+\frac{1}{2}\right)\left(\frac{1-x}{2}\right){{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda+1\\\
\lambda+\frac{3}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]}\right\\}.$

An application of the contiguous relation [1, (15.2.25)] then shows

$\displaystyle{\mathcal{I}}^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)=\frac{2\sqrt{\pi}(2\lambda+1)_{n}}{(\lambda+n+\frac{1}{2})n!}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{3}{2})}\left(\lambda+\frac{1}{2}\right){{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda\\\
\lambda+\frac{1}{2}\\\ \end{array}\right|\frac{1-x}{2}\right]}$
$\displaystyle\hphantom{{\mathcal{I}}^{\lambda}_{+}C^{\lambda+\frac{1}{2}}_{n}(x)}{}=\frac{2\sqrt{\pi}(2\lambda+1)_{n}}{(\lambda+n+\frac{1}{2})n!}\frac{\Gamma(\lambda+1)}{\Gamma(\lambda+\frac{1}{2})}\frac{n!}{(2\lambda)_{n}}C^{\lambda}_{n}(x)=\frac{\sqrt{\pi}(2\lambda+n)\Gamma(\lambda)}{(\lambda+n+\frac{1}{2})\Gamma(\lambda+\frac{1}{2})}C^{\lambda}_{n}(x),$

which is equation (3.23).

The proof of equation (3.24) is analogous except that it uses the contiguous
relation [19, (15.5.16)]. Hence, it will be omitted. ∎

For the operators ${\mathcal{D}}^{\lambda}_{\pm}$ we obtain the following
result.

######  Theorem 3.5.

Let $\lambda>0$ and $n\in{\mathbb{N}}_{0}$. Then

$\displaystyle{\mathcal{D}}^{\lambda}_{+}C^{\lambda}_{n}(x)=\frac{\sqrt{\pi}\Gamma(\lambda+\frac{1}{2})}{\Gamma({\lambda})}\frac{2(n+2\lambda)}{n+\lambda}C^{\lambda+\frac{1}{2}}_{n-1}(x),\qquad
x\in[-1,1],$ (3.25)
$\displaystyle{\mathcal{D}}^{\lambda}_{-}C^{\lambda}_{n}(x)=\frac{\sqrt{\pi}\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda)}\frac{2n}{n+\lambda}C^{\lambda+\frac{1}{2}}_{n}(x),\qquad
x\in[-1,1].$ (3.26)

######  Remark 3.6.

Taking the limit of the last two relations as $\lambda\rightarrow 0^{+}$ (see
(2.14)) gives

$\displaystyle{\mathcal{D}}^{0}_{+}T_{n}(x)=n\pi
P_{n-1}(x)\qquad\text{and}\qquad{\mathcal{D}}^{0}_{-}T_{n}(x)=n\pi
P_{n}(x),\qquad n\in{\mathbb{N}}_{0}.$

Again, we first prove a preparatory proposition.

######  Proposition 3.7.

Let $\lambda>0$, $n\in{\mathbb{N}}_{0}$ and $x\in[-1,1]$. Then

$\displaystyle
D^{\lambda}_{+}C_{n}^{\lambda}(x)=c_{n,\lambda}\frac{n(n+2\lambda)}{n+\lambda}\frac{1+x}{2}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+1\\\ \end{array}\right|\frac{1-x}{2}\right]},$ (3.29) $\displaystyle
D^{\lambda}_{-}C_{n}^{\lambda}(x)=c_{n,\lambda}\frac{n(n+2\lambda)}{\lambda+1}\frac{1-x}{2}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+2\\\ \end{array}\right|\frac{1-x}{2}\right]},$ (3.32)

where

$\displaystyle
c_{n,\lambda}=\frac{\sqrt{\pi}(2\lambda)_{n}}{n!}\frac{\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda+1)}.$

###### Proof.

Consider the first equation

$\displaystyle\frac{1}{1+x}D^{\lambda}_{+}C^{\lambda}_{n}(x)=\frac{d}{dx}\left\\{(1+x)^{-\lambda}\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\lambda-\frac{1}{2}}C^{\lambda}_{n}(\tau)d\tau\right\\}$
$\displaystyle=\frac{(-1)^{n}(2\lambda)_{n}}{n!}\frac{d}{dx}\left\\{(1+x)^{-\lambda}\int_{-1}^{x}(x-\tau)^{-\frac{1}{2}}(1+\tau)^{\lambda-\frac{1}{2}}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda\\\
\lambda+\frac{1}{2}\\\
\end{array}\right|\frac{1+\tau}{2}\right]}d\tau\right\\}$
$\displaystyle=\frac{(-1)^{n}(2\lambda)_{n}}{n!}\frac{d}{dx}\left\\{\frac{1}{1+x}I^{\lambda-{\frac{1}{2}}}_{+}\left({{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda\\\
\lambda+\frac{1}{2}\\\ \end{array}\right|\frac{1+\mathchoice{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\displaystyle\cdot$}}}}\hss}}}{{\hbox
to4.99995pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\textstyle\cdot$}}}}\hss}}}{{\hbox
to4.16661pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptstyle\cdot$}}}}\hss}}}{{\hbox
to3.61105pt{\hss\leavevmode\hbox{\raisebox{0.0pt}{\scalebox{2.0}{\raisebox{0.0pt}{$\scriptscriptstyle\cdot$}}}}\hss}}}}{2}\right]}\right)(x)\right\\}.$

Applying Lemma 3.2

$\displaystyle\frac{1}{1+x}D^{\lambda}_{+}C^{\lambda}_{n}(x)=\frac{(-1)^{n}(2\lambda)_{n}}{n!}\sqrt{\pi}\frac{\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda+1)}\frac{d}{dx}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n,n+2\lambda\\\
\lambda+1\\\ \end{array}\right|\frac{1+x}{2}\right]}.$

The formula (cf. [19, (15.2.1)])

$\displaystyle\frac{d}{dx}{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}a,b\\\
c\\\
\end{array}\right|x\right]}=\frac{(a)_{1}(b)_{1}}{(c)_{1}}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}a+1,b+1\\\
c+1\\\ \end{array}\right|x\right]}$

then implies

$\displaystyle
D^{\lambda}_{+}C^{\lambda}_{n}(x)=\frac{(-1)^{n}(2\lambda)_{n}}{n!}\sqrt{\pi}\frac{\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda+1)}\frac{1+x}{2}\frac{(-n)(n+2\lambda)}{(\lambda+1)}$
$\displaystyle\hphantom{D^{\lambda}_{+}C^{\lambda}_{n}(x)=}{}\times{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+2\\\ \end{array}\right|\frac{1+x}{2}\right]}$ (3.35)
$\displaystyle\hphantom{D^{\lambda}_{+}C^{\lambda}_{n}(x)}{}=\frac{\sqrt{\pi}(2\lambda)_{n}\Gamma(\lambda+\frac{1}{2})}{n!\Gamma(\lambda+1)}(-1)^{n+1}\frac{1+x}{2}\frac{(n)(n+2\lambda)}{(\lambda+1)}$
$\displaystyle\hphantom{D^{\lambda}_{+}C^{\lambda}_{n}(x)=}{}\times{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+2\\\ \end{array}\right|\frac{1+x}{2}\right]}.$ (3.38)

Now from equation (3.11)

$\displaystyle{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+2\\\
\end{array}\right|\frac{1+x}{2}\right]}=\frac{(-1)^{n-1}(1+\lambda)}{n+\lambda}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+1\\\ \end{array}\right|\frac{1-x}{2}\right]}.$

Substituting into equation (3.38) gives equation (3.29).

The proof of equation (3.32) is almost identical to that of equation (3.29).
It will therefore be omitted. ∎

###### Proof of Theorem 3.5.

From (3.29) and (3.32)

$\displaystyle{\mathcal{D}}^{\lambda}_{+}C_{n}^{\lambda}(x)=\frac{\sqrt{\pi}(2\lambda)_{n}}{n!}\frac{\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda+1)}\frac{n(n+2\lambda)}{(n+\lambda)(\lambda+1)}$
$\displaystyle\hphantom{{\mathcal{D}}^{\lambda}_{+}C_{n}^{\lambda}(x)=}{}\times\left\\{(\lambda+1)\left(\frac{1+x}{2}\right){{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+1\\\ \end{array}\right|\frac{1-x}{2}\right]}\right.$
$\displaystyle\left.\hphantom{{\mathcal{D}}^{\lambda}_{+}C_{n}^{\lambda}(x)=}{}+(n+\lambda)\left(\frac{1-x}{2}\right){{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda+1\\\
\lambda+2\\\ \end{array}\right|\frac{1-x}{2}\right]}\right\\}.$

An application of the contiguous relation [1, (15.2.25)] then shows

$\displaystyle{\mathcal{D}}^{\lambda}_{+}C_{n}^{\lambda}(x)=\frac{\sqrt{\pi}(2\lambda)_{n}}{n!}\frac{\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda+1)}\frac{n(n+2\lambda)}{(n+\lambda)}\,{{{}_{2}F_{1}}\left[\left.\begin{array}[]{@{}c@{}}-n+1,n+2\lambda\\\
\lambda+1\\\ \end{array}\right|\frac{1-x}{2}\right]},$

and rewriting the hypergeometric function on the right as a Gegenbauer
polynomial gives (3.25).

The proof of equation (3.26) is analogous except that it uses contiguous
relation [19, (15.5.16)]. Hence, it will be omitted. ∎

### Acknowledgements

The authors thank the editor and the referees for their helpful suggestions.
WzC acknowledges support from a University of Canterbury Visiting Erskine
Fellowship. RKB is grateful for the hospitality provided by the Helmholtz
Zentrum München.

## References

  * [1] Abramowitz M., Stegun I.A. (Editors), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, New York, 1972\.
  * [2] Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  * [3] Barbosa V.S., Menegatto V.A., Strictly positive definite kernels on two-point compact homogeneous spaces, Math. Inequal. Appl. 19 (2016), 743–756, arXiv:1505.00591.
  * [4] Beatson R.K., zu Castell W., Dimension hopping and families of strictly positive definite radial functions on spheres, arXiv:1510.08658.
  * [5] Billingsley P., Probability and measure, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, New York – Chichester – Brisbane, 1979.
  * [6] Bingham N.H., Positive definite functions on spheres, Proc. Cambridge Philos. Soc. 73 (1973), 145–156.
  * [7] Chen D., Menegatto V.A., Sun X., A necessary and sufficient condition for strictly positive definite functions on spheres, Proc. Amer. Math. Soc. 131 (2003), 2733–2740.
  * [8] Chilès J.P., Delfiner P., Geostatistics. Modeling spatial uncertainty, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1999.
  * [9] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G., Higher transcendental functions, Vol. II, Bateman Manuscript Project, McGraw-Hill Book Co., New York, 1953.
  * [10] Fasshauer G.E., Schumaker L.L., Scattered data fitting on the sphere, in Mathematical Methods for Curves and Surfaces, II (Lillehammer, 1997), Innov. Appl. Math., Vanderbilt University Press, Nashville, TN, 1998, 117–166.
  * [11] Freeden W., Gervens T., Schreiner M., Constructive approximation on the sphere. With applications to geomathematics, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998.
  * [12] Gangolli R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters, Ann. Inst. H. Poincaré Sect. B (N.S.) 3 (1967), 121–226.
  * [13] Gneiting T., Compactly supported correlation functions, J. Multivariate Anal. 83 (2002), 493–508.
  * [14] Hardy G.H., Littlewood J.E., Some properties of fractional integrals. I, Math. Z. 27 (1928), 565–606.
  * [15] Lévy P., Le mouvement Brownien fonction d’un point de la sphère de Riemann, Rend. Circ. Mat. Palermo 8 (1959), 297–310.
  * [16] Matheron G., Les variables régionalisées et leur estimation: une application de la théorie des fonctions aléatoires aux sciences de la nature, Masson, Paris, 1965.
  * [17] Matheron G., The intrinsic random functions and their applications, Adv. in Appl. Probab. 5 (1973), 439–468.
  * [18] Menegatto V.A., Oliveira C.P., Peron A.P., Strictly positive definite kernels on subsets of the complex plane, Comput. Math. Appl. 51 (2006), 1233–1250.
  * [19] NIST digital library of mathematical functions, available at http://dlmf.nist.gov/.
  * [20] Porcu E., Gregori P., Mateu J., La descente et la montée étendues [extended rises and descents]: the spatially $d$-anisotropic and the spatio-temporal case, Stoch. Environ. Res. Risk Assess. 21 (2007), 683–693.
  * [21] Samko S.G., Kilbas A.A., Marichev O.I., Fractional integrals and derivatives. Theory and applications, Gordon and Breach Science Publishers, Yverdon, 1993.
  * [22] Schaback R., Wu Z., Operators on radial functions, J. Comput. Appl. Math. 73 (1996), 257–270.
  * [23] Schlather M., Construction of covariance functions and unconditional simulation of random fields, in Advances and Challenges in Space-Time Modelling of Natural Events, Lecture Notes in Statistics, Vol. 207, Editors E. Porcu, J. Montero, M. Schlather, Springer, Berlin, 2012, 25–54.
  * [24] Schoenberg I.J., Positive definite functions on spheres, Duke Math. J. 9 (1942), 96–108.
  * [25] Wendland H., Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Adv. Comput. Math. 4 (1995), 389–396.
  * [26] Wu Z.M., Compactly supported positive definite radial functions, Adv. Comput. Math. 4 (1995), 283–292.
  * [27] zu Castell W., Recurrence relations for radial positive definite functions, J. Math. Anal. Appl. 271 (2002), 108–123.


note-name = , use-sort-key = false, placement = mixed

# Cross-scale energy transfer from fluid-scale Alfvén waves to kinetic-scale
ion acoustic waves in the Earth’s magnetopause boundary layer

Xin An<EMAIL_ADDRESS>Department of Earth, Planetary, and Space Sciences,
University of California, Los Angeles, CA, 90095, USA  Anton Artemyev
Department of Earth, Planetary, and Space Sciences, University of California,
Los Angeles, CA, 90095, USA  Vassilis Angelopoulos  Department of Earth,
Planetary, and Space Sciences, University of California, Los Angeles, CA,
90095, USA  Terry Z. Liu  Department of Earth, Planetary, and Space Sciences,
University of California, Los Angeles, CA, 90095, USA  Ivan Vasko  Department
of Physics, University of Texas at Dallas, Richardson, TX, 75080, USA  David
Malaspina  Astrophysical and Planetary Sciences Department, University of
Colorado, Boulder, CO, 80305, USA  Laboratory for Atmospheric and Space
Physics, University of Colorado, Boulder, CO, 80303, USA

###### Abstract

In space plasmas, large-amplitude Alfvén waves can drive compressive
perturbations, accelerate ion beams, and lead to plasma heating and the
excitation of ion acoustic waves at kinetic scales. This energy channelling
from fluid to kinetic scales represents a complementary path to the classical
turbulent cascade. Here, we present observational and computational evidence
to validate this hypothesis by simultaneously resolving the fluid-scale Alfvén
waves, kinetic-scale ion acoustic waves, and their imprints on ion velocity
distributions in the Earth’s magnetopause boundary layer. We show that two
coexisting compressive modes, driven by the magnetic pressure gradients of
Alfvén waves, not only accelerate the ion tail population to the Alfvén
velocity, but also heat the ion core population near the ion acoustic velocity
and generate Debye-scale ion acoustic waves. Thus, Alfvén-acoustic energy
channeling emerges as a viable mechanism for plasma heating near plasma
boundaries where large-amplitude Alfvén waves are present.

††preprint:

Understanding the multiscale energy transfer from magnetohydrodynamic
electromagnetic perturbations and convective flows to kinetic field structures
and plasma heating is of crucial importance in space and many other plasmas.
The classical picture suggests that turbulent energy cascades from large to
successively smaller scales through nonlinear interactions between counter-
propagating Alfvén waves Iroshnikov (1964); Kraichnan (1965); Goldreich and
Sridhar (1997); Boldyrev (2006); Howes  _et al._ (2012). A complementary path
of energy transfer involves the electrostatic coupling between Alfvénic and
ion acoustic fluctuations. In this scenario, magnetic pressure gradients of
Alfvén waves induce density perturbations and electric fields parallel to the
background magnetic field (e.g., Hollweg, 1971), which accelerate ion beams
through nonlinear Landau resonance Medvedev  _et al._ (1998); Araneda  _et
al._ (2008); Matteini  _et al._ (2010). These ion beams then excite kinetic-
scale ion acoustic waves (IAWs), which in turn relax ion beams, leading to
parallel ion heating and terminating the energy transfer Valentini  _et al._
(2008); Valentini and Veltri (2009); Valentini  _et al._ (2011, 2014). The
implications of Alfvén-acoustic channeling on multiscale energy transfer span
across space, astrophysical, and fusion plasmas. In the solar wind, IAWs and
associated ion beams are observed around magnetic discontinuities or
switchbacks Graham  _et al._ (2021); Mozer  _et al._ (2020); Malaspina  _et
al._ (2023, 2024), which likely evolve from outward-propagating Alfvén waves
Squire  _et al._ (2020); Mallet  _et al._ (2021); Tenerani  _et al._ (2021),
resulting in significant ion heating Mozer  _et al._ (2022); Kellogg  _et al._
(2024); Woodham  _et al._ (2021). In Tokamak plasmas, Alfvén waves driven by
suprathermal ions transfer some of their energy to IAWs, which are heavily
Landau-damped by thermal ions, thereby heating the thermal ions Gorelenkov
_et al._ (2009); Curran  _et al._ (2012); Bierwage  _et al._ (2015); Chen and
Zonca (2016).

Despite the importance of Alfvén-acoustic energy channeling, direct
observational evidence of this process has been difficult to obtain, mainly
because of the instrumentation required to resolve ion velocity distributions
at a high time cadence, as well as the vast scale separation between Alfvén
waves and IAWs. In this Letter, we test the hypothesis of electrostatic-
driven, cross-scale energy transfer from Alfvén waves to IAWs using the
Earth’s magnetopause boundary layer as a natural laboratory, where the solar
wind interaction with magnetosphere generates a plethora of large-amplitude
Alfvén waves. The four-satellite Magnetospheric Multiscale (MMS) mission Burch
_et al._ (2016) provides simultaneous measurements for fluid-scale Alfvén
waves (spatial scales resolved by inter-spacecraft interferometry), kinetic-
scale IAWs (spatial scales resolved by inter-antenna interferometry with a
single spacecraft), and ion velocity distributions (enabled by 3D measurements
of ion velocity space with a high time resolution). The interpretation of
these data is supported by event-oriented kinetic simulations.

On 8 September 2015, the MMS constellation traversed from the duskside
magnetosphere to the magnetosheath between 09:10 and 11:40 UT. A surface wave,
generated by the Kelvin-Helmholtz instability, is seen through repetitive
crossings of current sheets, which separate the relatively cold, dense
magnetosheath from the hot, tenuous magnetopause boundary layer Eriksson  _et
al._ (2016). Thanks to the long interval ($\sim 80$ minutes) of continuous
burst-mode data collected during boundary layer crossing, a rich variety of
complex plasma dynamics was observed, including Alfvénic turbulence Stawarz
_et al._ (2016), ion beams and plasma heating Sorriso-Valvo  _et al._ (2019),
and large-amplitude ($\sim 100$ mV/m) electrostatic waves Wilder  _et al._
(2016).

Figure 1 shows an example of magnetopause crossings between 10:35:35 and
10:36:05 UT (see Supplemental Materials See Supplemental Materials for the
repetitive crossings of the magnetopause boundary layer over the $2.5$ hours
interval. for the whole event of magnetopause crossing). Within the
magnetopause boundary layer, enhanced Alfvén waves with a normalized amplitude
$|\delta B/B_{0}|\sim 0.2$ are identified by the correlated perturbations
between magnetic and velocity fields. Here $B_{0}$ is the background magnetic
field averaged in the magnetopause boundary layer. The magnetic field data
from the four MMS spacecraft show clear time shifts among them, due to wave
propagation and plasma flows. Using four-spacecraft interferometry (see
explanations in Supplemental Materials See the four-spacecraft interferometry
analysis of Alfvén waves in Supplemental Materials and Ref. Paschmann and
Schwartz (2000)), we determine the parallel wave propagation velocity to be
$539\,\mathrm{km/s}\approx v_{\mathrm{A}}$ in the plasma rest frame and the
corresponding wavelength to be $2059\,\mathrm{km}\approx 30\,d_{i}$ for the
dominant frequency $0.05\,\mathrm{Hz}=0.05f_{ci}$ in the spacecraft frame
($0.26\,\mathrm{Hz}=0.26f_{ci}$ in the plasma rest frame). Here, $d_{i}=68$
km, $f_{ci}=1$ Hz and $v_{\mathrm{A}}=506$ km/s are the ion inertial length,
ion gyrofrequency and Alfvén velocity, respectively, averaged over the
interval 10:35:40–10:36:00 UT. These large-amplitude Alfvén waves steepen into
current sheets with strong gradients in the total magnetic field (e.g., around
10:35:46 UT), which are colocated with density bumps and low-frequency
parallel electric fields [Figures 1(b) and 1(c)]. Such density and electric
field perturbations are likely driven by gradients in the wave magnetic-field
pressure [Figure 1(c)], and can be viewed as the ion acoustic mode in the
long-wavelength limit Hollweg (1971).

Figure 1: An example of magnetopause boundary layer crossing by MMS1 on 8
September 2015. The boundary layer is observed between 10:35:33 and 10:36:08
UT. (a) Three components of magnetic field in Geocentric Solar Ecliptic (GSE)
coordinate system measured by the Fluxgate Magnetometer at $128$
Samples/second Russell  _et al._ (2016). The total magnetic field strength is
shown in black. (b) Ion and electron densities measured by the Fast Plasma
Investigation (FPI) instrument at time cadences of $150$ ms and $30$ ms,
respectively Pollock  _et al._ (2016). (c) Parallel spatial gradients of
magnetic field pressure and smoothed parallel electric fields. (d) Reduced ion
parallel velocity distributions obtained from integrating phase space
densities measured in the 3D velocity space (energy, pitch angle, gyrophase)
by the FPI instrument. The phase space density is coded in color. The three
solid lines from top to bottom indicate Alfvén and ion acoustic velocities
relative to the ion core, and the maximum phase space density at each time,
respectively. (e) Wavelet analysis of electric field measured by the Electric
Double Probes at a sampling rate $8192$ Samples/second Lindqvist  _et al._
(2016); Ergun  _et al._ (2016). The spectral density is coded in color. The
black line tracks the cone of influence, below which the stretched wavelets
extend beyond the edges of the observation interval.

Ion beams (local maximums in phase space density in $v_{\parallel}$ aside from
the ion core) are seen in the reduced velocity distributions at variable
parallel velocities up to $v_{\mathrm{A}}=506$ km/s relative the ion core
[Figure 1(d)]. These ion beams are believed to be driven by resonant
interactions between ions and Alfvénic fluctuations Sorriso-Valvo  _et al._
(2019). Intense broadband electrostatic waves appear during each boundary
layer crossing throughout the $\sim 2$-hour event [Figure 1(e)], with
frequencies ranging from $10$ Hz to the ion plasma frequency $f_{pi}=700$ Hz.
Identified as the ion acoustic mode, these waves are likely excited by the ion
beams Wilder  _et al._ (2016). By analyzing time delays between the voltage
signals in each pair of opposing voltage-sensitive probes in three orthogonal
directions (with tip-to-tip effective distances of $120$ m in the spin plan
and $30$ m along the spin axis), we perform interferometry for short-
wavelength IAWs at various locations in the boundary layer (see analysis in
Supplemental Materials see Supplemental Materials for the detailed inter-
antenna interferometry analysis and Refs. Vasko  _et al._ (2018, 2020)). The
IAWs propagate within $15^{\circ}$ relative to $B_{0}$ or $-B_{0}$, with phase
speeds $0.1$–$1.5\,c_{s}$ in the plasma rest frame. They have wavelengths of
$1$–$30\,\lambda_{D}$, where $c_{s}$ is the ion acoustic velocity and
$\lambda_{D}$ the local Debye length. The antiparallel-propagating IAWs are
consistent with ion beams in the antiparallel direction, although these beams
are weaker than those in the parallel direction.

It is important to determine the ordering of ion acoustic and Alfvén
velocities, $c_{s}$ and $v_{\mathrm{A}}$, relative to the thermal velocity of
core ions $v_{\mathrm{Ti}}$ (excluding beam ions). The electron-to-ion
temperature ratio is $T_{e}/T_{i}\approx 0.25$, and the ion beta is
$\beta_{i}=n_{0}T_{i}/(B_{0}^{2}/8\pi)\approx 0.15$, which yields
$c_{s}/v_{\mathrm{Ti}}=\sqrt{(5/3)+(T_{e}/T_{i})}\approx 1.4$ and
$v_{\mathrm{A}}/v_{\mathrm{Ti}}=\sqrt{2/\beta_{i}}\approx 3.7$. Here, $n_{0}$
is the background density averaged in the magnetopause boundary layer. Thus,
the ion beams at $v_{\mathrm{A}}=3.7v_{\mathrm{Ti}}=506$ km/s (i.e., the tail
of the velocity distribution) cannot be in Landau-resonance with the IAWs
propagating along the field at $c_{s}=1.4v_{\mathrm{Ti}}=191$ km/s (i.e., the
core of the velocity distribution). Nevertheless, there must be some ion beams
present at $c_{s}$ to drive the IAWs; otherwise, those waves would be heavily
damped by thermal core ions through Landau damping. This expectation is
strongly supported by the presence of ion beams at $\sim 200$ km/s relative to
the ion core in Figure 1(d).

To facilitate the interpretation of in situ observation data, we perform
event-oriented kinetic simulations using the Hybrid-VPIC code Le  _et al._
(2023). This code treats ions as kinetic particles and electrons as a massless
fluid. Our simulation spans two dimensions $[0\leq x\leq 120\,d_{i},0\leq
y\leq 15\,d_{i}]$ in configuration space and three dimensions
$(v_{x},v_{y},v_{z})$ in velocity space. The cell size is $\Delta x=\Delta
y=0.059d_{i}$. The time step is $\Delta t=0.004\omega_{ci}^{-1}$ for particle
push, which is divided to $10$ substeps for the field solver to satisfy the
Courant condition $(\Delta t/10)\cdot\omega_{ci}<(\Delta x/d_{i})^{2}/\pi$ for
the fastest eigenmode, the whistler mode. Periodic boundary conditions are
applied to both fields and particles. All numerical values for fields and
particles are based on MMS observations in Figure 1. A uniform background
magnetic field $B_{0}$ is applied along the $+x$ direction. Initially, the
system is perturbed by a spectrum of parallel-propagating Alfvén waves
prescribed by $\delta B_{x}=0$, $\delta B_{y}=-\delta
B_{\perp}\sum_{m=3}^{5}\sin(2\pi mx/L_{x}+\phi_{i})$, and $\delta B_{z}=\delta
B_{\perp}\sum_{m=3}^{5}\cos(2\pi mx/L_{x}+\phi_{i})$, where $\delta
B_{\perp}/B_{0}=0.15$ is the wave amplitude, $L_{x}$ is the system size in the
$x$ direction, and $m$ is the mode number. The initial wave phases are
uniformly distributed between $0$ and $2\pi$ as $\phi_{1}=0$,
$\phi_{2}=2\pi/3$, and $\phi_{3}=4\pi/3$. Ions are initialized as a drifting
Maxwellian with thermal velocity $v_{\mathrm{Ti}}=0.27\,v_{\mathrm{A}}$ and
perturbed transverse fluid velocities commensurate with Alfvén waves, $\delta
v_{x}=0$, $\delta v_{y}=\delta v_{\perp}\sum_{m=3}^{5}\sin(2\pi
mx/L_{x}+\phi_{i})$, and $\delta v_{z}=-\delta
v_{\perp}\sum_{m=3}^{5}\cos(2\pi mx/L_{x}+\phi_{i})$, where $\delta
v_{\perp}/v_{\mathrm{A}}\approx\delta B_{\perp}/B_{0}=0.15$. The uniform ion
density $n_{0}$ at $t=0$ is sampled by $400$ particles in each cell. The
electron-to-ion temperature ratio is $0.25$. Because the electron thermal
velocity is much greater than the Alfvén and ion acoustic velocities, the
equation of state for electrons is isothermal, i.e.,
$T_{e}=\mathrm{constant}$. The results are presented in normalized units: time
to $\omega_{ci}^{-1}$, lengths to $d_{i}$, velocities to $v_{\mathrm{A}}$,
densities to $n_{0}$, electric fields to $v_{\mathrm{A}}B_{0}/c$ (where $c$ is
the speed of light), and magnetic fields to $B_{0}$.

The large-amplitude, broadband Alfvén waves naturally exhibit magnitude
modulations, leading to phase-steepened wavefronts (see Figure 2(a) and Refs.
Cohen and Kulsrud (1974); González  _et al._ (2021)). At these steepened
wavefronts, rotational discontinuities are observed in the rapid phase changes
of $B_{y}$ and $B_{z}$ over a distance of $\sim 5d_{i}$ [Figure 3(a)]. The
gradients of magnetic pressure drive compressive perturbations, as described
by Hollweg (1971)

$\frac{\partial^{2}\delta\rho}{\partial
t^{2}}-c_{s}^{2}\frac{\partial^{2}\delta\rho}{\partial
x^{2}}=\frac{1}{8\pi}\frac{\partial(\delta B_{\perp})^{2}}{\partial x},$ (1)

where $\delta\rho$ is the mass density perturbation, and $\delta B_{\perp}$ is
the perpendicular magnetic field of the Alfvén waves. Consequently, a density
bump of $\delta\rho/\rho_{0}\lesssim 0.5$ forms at the steepened wavefront,
accompanied by a parallel ion flow with a streaming velocity $\delta
v/v_{\mathrm{A}}=\delta\rho/\rho_{0}\lesssim 0.5$ [at $x\approx 15\,d_{i}$ in
Figures 3(a)]. These streaming ions are unstable to the current-driven
instability Buneman (1959); Davidson  _et al._ (1970), and are expected to
generate IAWs with a maximum growth rate at the wavelength $\lambda\sim
c_{s}/\omega_{pi}=\lambda_{D}\sqrt{(5T_{i}/3T_{e})+1}=2.7\lambda_{D}$. Indeed,
short-wavelength IAWs are observed near the steepened wavefront, appearing as
short packets with spiky electric fields [Figure 3(a)]. The expected
wavelength is consistent with that measured by MMS ($1$–$30\,\lambda_{D}$)
IAWs are allowed to propagate in the hybrid-kinetic scheme. However, the
Debye-scale IAWs cannot be resolved in this scheme, because there is no other
intrinsic, physical spatial scale in the system smaller than the ion inertial
length..

Figure 2: Propagation characteristics of Alfvén waves and IAWs in the
spatiotemporal domain in the simulation. The horizontal axis represents the
$x$ direction parallel to $B_{0}$. The vertical axis represents time. The
amplitudes of oscillations are shown at the top of each panel. (a) The
perpendicular magnetic field $\delta B_{\perp}$. (b) The full parallel
electric field $\delta E_{x}$. (c) The parallel electric field $\langle\delta
E_{x}\rangle$ averaged over the $y$ direction. The short-wavelength IAWs are
canceled out by averaging in the $y$ direction, whereas the long-wavelength
electric fields survive. Figure 3: The steepened Alfvén waves and the
associated ion beams and kinetic-scale IAWs in the simulation. The shown time
snapshots are at (a) $t=800\,\omega_{ci}^{-1}$ and (b)
$t=1300\,\omega_{ci}^{-1}$. In each snapshot, the four panels from top to
bottom are (1) the three components of magnetic field and the total strength,
(2) the plasma density, (3) the ion phase portrait color-coded by the phase
space density, and (4) the parallel electric field, respectively. The
horizontal axis for all panels are the $x$ direction parallel to $B_{0}$.

Equation (1) reveals two compressive modes driven by large-amplitude Alfvén
waves. The first mode is a particular solution to the equation, which is
localized and attached to the steepened wavefront propagating at
$v_{\mathrm{A}}$. The second mode is a solution to the homogeneous part of the
equation, which is periodically emitted by the steepend wavefront once a
density perturbation is established, propagating at $c_{s}$. These two modes
are observed in the long-wavelength electric fields [Figure 2(c)].

The two compressive modes are also manifested in the ion phase portrait.
First, the localized electrostatic field at the steepened wavefront
accelerates the ion tail population to $v_{\mathrm{A}}$ through Landau
resonance (see Figure 3(a) and Refs. Araneda  _et al._ (2008); Valentini  _et
al._ (2008); Matteini  _et al._ (2010)). Second, the periodic electrostatic
field propagating at $c_{s}=1.4\,v_{\mathrm{Ti}}=0.38\,v_{\mathrm{A}}$ traps
the ion thermal population also through nonlinear Landau resonance and forms
phase space vortexes (or beams), leading to substantial heating of thermal
ions [Figure 3(b)]. In addition, the two coexisting ion beams within the
resonant islands (centered at $c_{s}$ and $v_{\mathrm{A}}$) excite short-
wavelength IAWs, propagating at variable phase velocities comparable to Alfvén
and ion acoustic velocities [Figure 2(b)], which agree with the phase
velocities measured by MMS based on the interferometry analysis.
Interestingly, the amplitudes of the kinetic-scale electrostatic fields are
much larger than their fluid-scale counterparts (see Figures 1(c), 1(e), 2(b),
2(c), and Ref. Wilder  _et al._ (2016)).

Figure 4 compares the electromagnetic power spectra from MMS observations and
simulations. Both observed and simulated spectra transition from an
electromagnetic regime with $c|\delta E|/v_{\mathrm{A}}|\delta B|\sim 1$ in
the low-frequency ($\omega/\omega_{ci}<1$) or long-wavelength ($kd_{i}<1$)
limit, to an electrostatic one with $c|\delta E|/v_{\mathrm{A}}|\delta B|\gg
1$ in the high-frequency ($\omega\lesssim\omega_{pi}$) or short-wavelength
($k\lambda_{D}\lesssim 1$) limit.

In the electromagnetic regime, the dominant Alfvén wave at $0.07\omega_{ci}$
from the MMS observation generates second and third harmonics at
$\omega/\omega_{ci}=0.14,0.28$ due to phase steepening, similar to the
simulation. The simulation shows that long-wavelength electrostatic field
energy can be comparable to transverse electric field energy [Figures 2(a)
2(c), and 4(a)–(b)], with the ordering $\langle\delta
E_{x}\rangle^{2}\sim\delta E_{\perp}^{2}\sim
0.01(v_{\mathrm{A}}B_{0}/c)^{2}\sim(v_{\mathrm{A}}\delta
B_{\perp}/c)^{2}\ll(\delta B_{\perp})^{2}$. However, these relatively small-
amplitude, long-wavelength electrostatic fields mediate energy transfer from
fluid-scale Alfvén waves to kinetic-scale IAWs and core ion heating.

In the electrostatic regime, the Debye-scale IAWs exhibit a relatively flat
electric field spectrum compared to the magnetic field spectrum. In the
hybrid-kinetic simulation, electrostatic energy accumulates near the grid
scale due to the absence of an intrinsic Debye scale to terminate the energy
transfer. Nevertheless, the transition from electromagnetic fluctuations at
the fluid scale to electrostatic fluctuations at the kinetic scale is clearly
demonstrated in both the observations and simulations.

Figure 4: Comparison of electromagnetic power spectra between the MMS
observation and the simulation. (a) The electric and magnetic power spectra as
functions of wave frequency and (c) the ratio between the two spectra from the
MMS observation in Figure 1. Note that the shown wave frequency is in the
spacecraft frame, which is Doppler-shifted from that in the plasma rest frame.
There is no simple conversion between the two frequencies, because the amount
of Doppler shift varies with wave frequency or wavenumber. To facilitate the
mapping between wave frequencies in the spacecraft frame and wavenumbers in
the plasma frame, we mark the wavenumbers of Alfvén waves and IAWs at their
corresponding frequencies in the spacecraft frame. (b) The electric and
magnetic power spectra as functions of wavenumber and (d) the ratio between
the two spectra from the simulation at the end of the simulation
$t=1600\,\omega_{ci}^{-1}$.

In summary, we provide direct observational and computational evidence
supporting electrostatic-driven energy transfer from fluid-scale Alfvén waves
to kinetic-scale IAWs. In a realistic parameter regime of $v_{\mathrm{Ti}}\sim
c_{s}<v_{\mathrm{A}}$, two ion beams centered at $c_{s}$ and $v_{\mathrm{A}}$
are simultaneously accelerated through nonlinear Landau resonance by two
coexisting compressive electrostatic modes, both modes driven by phase-
steepened Alfvén waves. This process results in the generation of kinetic-
scale IAWs, substantial heating of thermal ions, and acceleration of
suprathermal ions up to $v_{\mathrm{A}}$. More broadly, the Alfvén-acoustic
channelling of ion energy may contribute to ion heating near various plasma
boundaries in the interplanetary space, such as magnetic discontinuities
Woodham  _et al._ (2021); Malaspina  _et al._ (2023), and low-velocity regions
ahead of high-speed streams Gurnett  _et al._ (1979). Moreover, given that the
wavelengths of kinetic-scale IAWs are comparable to the electron thermal
gyroradius Kamaletdinov  _et al._ (2022, 2024), the Alfvén-acoustic
channelling may also contribute to the momentum exchange between ions and
electrons and electron heating Mozer  _et al._ (2022), thereby providing a
collisionless dissipation mechanism to allow fast magnetic reconnection in
current sheets Coroniti and Eviatar (1977); Sagdeev (1979); Smith and Priest
(1972); Coppi and Friedland (1971).

###### Acknowledgements.

This work was supported by NASA grant NO. 80NSSC22K1634 and NSF grant NO.
2108582. We acknowledge MMS data (including FGM, EDP, and FPI) obtained from
https://lasp.colorado.edu/mms. Data access and processing was done using
SPEDAS V4.1 Angelopoulos  _et al._ (2019). We would like to acknowledge high-
performance computing support from Derecho (https://doi.org/10.5065/qx9a-pg09)
provided by NCAR’s Computational and Information Systems Laboratory, sponsored
by the National Science Foundation Computational and Information Systems
Laboratory (2024). We wish to thank Marco Velli for helpful discussions.

## References

  * Iroshnikov (1964) P. S. Iroshnikov, Soviet Astronomy 7, 566 (1964).
  * Kraichnan (1965) R. H. Kraichnan, Physics of Fluids 8, 1385 (1965).
  * Goldreich and Sridhar (1997) P. Goldreich and S. Sridhar, Astrophys. J. 485, 680 (1997), astro-ph/9612243 .
  * Boldyrev (2006) S. Boldyrev, Phys. Rev. Lett. 96, 115002 (2006).
  * Howes  _et al._ (2012) G. Howes, D. Drake, K. Nielson, T. Carter, C. Kletzing, and F. Skiff, Physical review letters 109, 255001 (2012).
  * Hollweg (1971) J. V. Hollweg, Journal of Geophysical Research 76, 5155 (1971).
  * Medvedev  _et al._ (1998) M. Medvedev, P. Diamond, M. Rosenbluth, and V. Shevchenko, Physical review letters 81, 5824 (1998).
  * Araneda  _et al._ (2008) J. A. Araneda, E. Marsch, F. Adolfo, _et al._ , Physical review letters 100, 125003 (2008).
  * Matteini  _et al._ (2010) L. Matteini, S. Landi, M. Velli, and P. Hellinger, Journal of Geophysical Research: Space Physics 115 (2010).
  * Valentini  _et al._ (2008) F. Valentini, P. Veltri, F. Califano, and A. Mangeney, Physical review letters 101, 025006 (2008).
  * Valentini and Veltri (2009) F. Valentini and P. Veltri, Physical review letters 102, 225001 (2009).
  * Valentini  _et al._ (2011) F. Valentini, D. Perrone, and P. Veltri, The Astrophysical Journal 739, 54 (2011).
  * Valentini  _et al._ (2014) F. Valentini, A. Vecchio, S. Donato, V. Carbone, C. Briand, J. Bougeret, and P. Veltri, The Astrophysical Journal Letters 788, L16 (2014).
  * Graham  _et al._ (2021) D. B. Graham, Y. V. Khotyaintsev, A. Vaivads, N. J. Edberg, A. I. Eriksson, E. P. Johansson, L. Sorriso-Valvo, M. Maksimovic, J. Souček, D. Píša, _et al._ , Astronomy & Astrophysics 656, A23 (2021).
  * Mozer  _et al._ (2020) F. Mozer, J. Bonnell, T. Bowen, G. Schumm, and I. Vasko, The Astrophysical Journal 901, 107 (2020).
  * Malaspina  _et al._ (2023) D. Malaspina, L. Kromyda, R. Ergun, and R. Livi, AGU23 (2023).
  * Malaspina  _et al._ (2024) D. M. Malaspina, R. E. Ergun, I. H. Cairns, B. Short, J. L. Verniero, C. Cattell, and R. Livi, arXiv preprint arXiv:2404.14559 (2024).
  * Squire  _et al._ (2020) J. Squire, B. D. Chandran, and R. Meyrand, The Astrophysical Journal Letters 891, L2 (2020).
  * Mallet  _et al._ (2021) A. Mallet, J. Squire, B. D. Chandran, T. Bowen, and S. D. Bale, The Astrophysical Journal 918, 62 (2021).
  * Tenerani  _et al._ (2021) A. Tenerani, N. Sioulas, L. Matteini, O. Panasenco, C. Shi, and M. Velli, The Astrophysical Journal Letters 919, L31 (2021).
  * Mozer  _et al._ (2022) F. Mozer, S. Bale, C. Cattell, J. Halekas, I. Vasko, J. Verniero, and P. Kellogg, The Astrophysical Journal Letters 927, L15 (2022).
  * Kellogg  _et al._ (2024) P. Kellogg, F. Mozer, M. Moncuquet, D. Malaspina, J. Halekas, S. Bale, and K. Goetz, The Astrophysical Journal 964, 68 (2024).
  * Woodham  _et al._ (2021) L. Woodham, T. Horbury, L. Matteini, T. Woolley, R. Laker, S. Bale, G. Nicolaou, J. Stawarz, D. Stansby, H. Hietala, _et al._ , Astronomy & Astrophysics 650, L1 (2021).
  * Gorelenkov  _et al._ (2009) N. Gorelenkov, M. Van Zeeland, H. Berk, N. Crocker, D. Darrow, E. Fredrickson, G.-Y. Fu, W. Heidbrink, J. Menard, and R. Nazikian, Physics of Plasmas 16 (2009).
  * Curran  _et al._ (2012) D. Curran, P. Lauber, P. Mc Carthy, S. da Graca, V. Igochine, A. U. Team, _et al._ , Plasma Physics and Controlled Fusion 54, 055001 (2012).
  * Bierwage  _et al._ (2015) A. Bierwage, N. Aiba, and K. Shinohara, Physical Review Letters 114, 015002 (2015).
  * Chen and Zonca (2016) L. Chen and F. Zonca, Reviews of Modern Physics 88, 015008 (2016).
  * Burch  _et al._ (2016) J. L. Burch, T. E. Moore, R. B. Torbert, and B. L. Giles,  Space Sci. Rev. 199, 5 (2016).
  * Eriksson  _et al._ (2016) S. Eriksson, B. Lavraud, F. Wilder, J. Stawarz, B. Giles, J. Burch, W. Baumjohann, R. Ergun, P.-A. Lindqvist, W. Magnes, _et al._ , Geophysical Research Letters 43, 5606 (2016).
  * Stawarz  _et al._ (2016) J. Stawarz, S. Eriksson, F. Wilder, R. Ergun, S. Schwartz, A. Pouquet, J. Burch, B. Giles, Y. Khotyaintsev, O. L. Contel, _et al._ , Journal of Geophysical Research: Space Physics 121, 11 (2016).
  * Sorriso-Valvo  _et al._ (2019) L. Sorriso-Valvo, F. Catapano, A. Retinò, O. Le Contel, D. Perrone, O. W. Roberts, J. T. Coburn, V. Panebianco, F. Valentini, S. Perri, _et al._ , Physical Review Letters 122, 035102 (2019).
  * Wilder  _et al._ (2016) F. Wilder, R. Ergun, S. Schwartz, D. Newman, S. Eriksson, J. Stawarz, M. Goldman, K. Goodrich, D. Gershman, D. Malaspina, _et al._ , Geophysical Research Letters 43, 8859 (2016).
  * (33) See Supplemental Materials for the repetitive crossings of the magnetopause boundary layer over the $2.5$ hours interval.
  * (34) See the four-spacecraft interferometry analysis of Alfvén waves in Supplemental Materials.
  * Paschmann and Schwartz (2000) G. Paschmann and S. Schwartz, in  _Cluster-II workshop multiscale/multipoint plasma measurements_ , Vol. 449 (2000) p. 99.
  * Russell  _et al._ (2016) C. T. Russell, B. J. Anderson, W. Baumjohann, K. R. Bromund, D. Dearborn, D. Fischer, G. Le, H. K. Leinweber, D. Leneman, W. Magnes, J. D. Means, M. B. Moldwin, R. Nakamura, D. Pierce, F. Plaschke, K. M. Rowe, J. A. Slavin, R. J. Strangeway, R. Torbert, C. Hagen, I. Jernej, A. Valavanoglou, and I. Richter,  Space Sci. Rev. 199, 189 (2016).
  * Pollock  _et al._ (2016) C. Pollock, T. Moore, A. Jacques, J. Burch, U. Gliese, Y. Saito, T. Omoto, L. Avanov, A. Barrie, V. Coffey, J. Dorelli, D. Gershman, B. Giles, T. Rosnack, C. Salo, S. Yokota, M. Adrian, C. Aoustin, C. Auletti, S. Aung, V. Bigio, N. Cao, M. Chandler, D. Chornay, K. Christian, G. Clark, G. Collinson, T. Corris, A. De Los Santos, R. Devlin, T. Diaz, T. Dickerson, C. Dickson, A. Diekmann, F. Diggs, C. Duncan, A. Figueroa-Vinas, C. Firman, M. Freeman, N. Galassi, K. Garcia, G. Goodhart, D. Guererro, J. Hageman, J. Hanley, E. Hemminger, M. Holland, M. Hutchins, T. James, W. Jones, S. Kreisler, J. Kujawski, V. Lavu, J. Lobell, E. LeCompte, A. Lukemire, E. MacDonald, A. Mariano, T. Mukai, K. Narayanan, Q. Nguyan, M. Onizuka, W. Paterson, S. Persyn, B. Piepgrass, F. Cheney, A. Rager, T. Raghuram, A. Ramil, L. Reichenthal, H. Rodriguez, J. Rouzaud, A. Rucker, Y. Saito, M. Samara, J.-A. Sauvaud, D. Schuster, M. Shappirio, K. Shelton, D. Sher, D. Smith, K. Smith, S. Smith, D. Steinfeld, R. Szymkiewicz, K. Tanimoto, J. Taylor, C. Tucker, K. Tull, A. Uhl, J. Vloet, P. Walpole, S. Weidner, D. White, G. Winkert, P.-S. Yeh, and M. Zeuch,  Space Sci. Rev. 199, 331 (2016).
  * Lindqvist  _et al._ (2016) P.-A. Lindqvist, G. Olsson, R. B. Torbert, B. King, M. Granoff, D. Rau, G. Needell, S. Turco, I. Dors, P. Beckman, J. Macri, C. Frost, J. Salwen, A. Eriksson, L. Åhlén, Y. V. Khotyaintsev, J. Porter, K. Lappalainen, R. E. Ergun, W. Wermeer, and S. Tucker,  Space Sci. Rev. 199, 137 (2016).
  * Ergun  _et al._ (2016) R. E. Ergun, S. Tucker, J. Westfall, K. A. Goodrich, D. M. Malaspina, D. Summers, J. Wallace, M. Karlsson, J. Mack, N. Brennan, B. Pyke, P. Withnell, R. Torbert, J. Macri, D. Rau, I. Dors, J. Needell, P.-A. Lindqvist, G. Olsson, and C. M. Cully,  Space Sci. Rev. 199, 167 (2016).
  * (40) See Supplemental Materials for the detailed inter-antenna interferometry analysis.
  * Vasko  _et al._ (2018) I. Vasko, F. Mozer, V. Krasnoselskikh, A. Artemyev, O. Agapitov, S. Bale, L. Avanov, R. Ergun, B. Giles, P.-A. Lindqvist, _et al._ , Geophysical Research Letters 45, 5809 (2018).
  * Vasko  _et al._ (2020) I. Y. Vasko, R. Wang, F. S. Mozer, S. D. Bale, and A. V. Artemyev, Frontiers in Physics 8, 156 (2020).
  * Le  _et al._ (2023) A. Le, A. Stanier, L. Yin, B. Wetherton, B. Keenan, and B. Albright, Physics of Plasmas 30 (2023).
  * Cohen and Kulsrud (1974) R. H. Cohen and R. M. Kulsrud, The Physics of Fluids 17, 2215 (1974).
  * González  _et al._ (2021) C. González, A. Tenerani, L. Matteini, P. Hellinger, and M. Velli, The Astrophysical Journal Letters 914, L36 (2021).
  * Buneman (1959) O. Buneman, Physical Review 115, 503 (1959).
  * Davidson  _et al._ (1970) R. Davidson, N. Krall, K. Papadopoulos, and R. Shanny, Physical Review Letters 24, 579 (1970).
  * (48) IAWs are allowed to propagate in the hybrid-kinetic scheme. However, the Debye-scale IAWs cannot be resolved in this scheme, because there is no other intrinsic, physical spatial scale in the system smaller than the ion inertial length.
  * Gurnett  _et al._ (1979) D. Gurnett, E. Marsch, W. Pilipp, R. Schwenn, and H. Rosenbauer, Journal of Geophysical Research: Space Physics 84, 2029 (1979).
  * Kamaletdinov  _et al._ (2022) S. Kamaletdinov, I. Vasko, A. Artemyev, R. Wang, and F. Mozer, Physics of Plasmas 29 (2022).
  * Kamaletdinov  _et al._ (2024) S. R. Kamaletdinov, I. Y. Vasko, and A. V. Artemyev, Journal of Plasma Physics 90, 905900201 (2024).
  * Coroniti and Eviatar (1977) F. Coroniti and A. Eviatar, Astrophysical Journal Supplement Series, vol. 33, Feb. 1977, p. 189-210. 33, 189 (1977).
  * Sagdeev (1979) R. Z. Sagdeev, Reviews of Modern Physics 51, 1 (1979).
  * Smith and Priest (1972) D. F. Smith and E. Priest, Astrophysical Journal, vol. 176, p. 487 176, 487 (1972).
  * Coppi and Friedland (1971) B. Coppi and A. B. Friedland, Astrophysical Journal, vol. 169, p. 379 169, 379 (1971).
  * Angelopoulos  _et al._ (2019) V. Angelopoulos, P. Cruce, A. Drozdov, E. W. Grimes, N. Hatzigeorgiu, D. A. King, D. Larson, J. W. Lewis, J. M. McTiernan, D. A. Roberts, C. L. Russell, T. Hori, Y. Kasahara, A. Kumamoto, A. Matsuoka, Y. Miyashita, Y. Miyoshi, I. Shinohara, M. Teramoto, J. B. Faden, A. J. Halford, M. McCarthy, R. M. Millan, J. G. Sample, D. M. Smith, L. A. Woodger, A. Masson, A. A. Narock, K. Asamura, T. F. Chang, C.-Y. Chiang, Y. Kazama, K. Keika, S. Matsuda, T. Segawa, K. Seki, M. Shoji, S. W. Y. Tam, N. Umemura, B.-J. Wang, S.-Y. Wang, R. Redmon, J. V. Rodriguez, H. J. Singer, J. Vandegriff, S. Abe, M. Nose, A. Shinbori, Y.-M. Tanaka, S. UeNo, L. Andersson, P. Dunn, C. Fowler, J. S. Halekas, T. Hara, Y. Harada, C. O. Lee, R. Lillis, D. L. Mitchell, M. R. Argall, K. Bromund, J. L. Burch, I. J. Cohen, M. Galloy, B. Giles, A. N. Jaynes, O. Le Contel, M. Oka, T. D. Phan, B. M. Walsh, J. Westlake, F. D. Wilder, S. D. Bale, R. Livi, M. Pulupa, P. Whittlesey, A. DeWolfe, B. Harter, E. Lucas, U. Auster, J. W. Bonnell, C. M. Cully, E. Donovan, R. E. Ergun, H. U. Frey, B. Jackel, A. Keiling, H. Korth, J. P. McFadden, Y. Nishimura, F. Plaschke, P. Robert, D. L. Turner, J. M. Weygand, R. M. Candey, R. C. Johnson, T. Kovalick, M. H. Liu, R. E. McGuire, A. Breneman, K. Kersten, and P. Schroeder,  Space Sci. Rev. 215, 9 (2019).
  * Computational and Information Systems Laboratory (2024) Computational and Information Systems Laboratory, “Derecho: HPE Cray EX System,” Boulder, CO: National Center for Atmospheric Research (2024).