#  Entropy numbers of Reproducing Hilbert Space of zonal positive definite
kernels on compact two-point homogeneous spaces

Karina Gonzalez & Thaís Jordão

Abstract: We present estimates for the covering numbers of the unit ball of
Reproducing Kernel Hilbert Spaces (RKHSs) of functions on $\mathbb{M}^{d}$ a
$d$-dimensional compact two-point homogeneous space. The RKHS is generated by
a continuous zonal/isotropic positive definite kernel. We employ the
representation in terms of the Schoenberg/Fourier series expansion for
continuous isotropic positive definite kernels, given in terms of a family of
orthogonal polynomials on $\mathbb{M}^{d}$. The bounds we present carry
accurate information about the asymptotic constants depending on the dimension
of the manifold and the decay or growth rate of the coefficients of the
kernel. The results we present extend the estimates previously known for
continuous isotropic positive definite kernels on the $d$-dimensional unit
sphere. We present the weak asymptotic equivalence for the order of the growth
of covering numbers associated to kernels on $\mathbb{M}^{d}$ with a
convergent geometric sequence of coefficients. We apply our estimates in order
to present a bound for the covering numbers of the spherical Gaussian kernel,
and to present bounds for formal examples on $\mathbb{M}^{d}$.

##  1 Introduction

In this paper we present estimates for the covering numbers of the unit ball
of Reproducing Kernel Hilbert Spaces (RKHSs) on a compact two-point
homogeneous space $\mathbb{M}^{d}$ of dimension $d\geq 1$. This releases a
follow up paper with the extension of the main results in [13], obtained on
the unit sphere $\mathbb{S}^{d}$ of $\mathbb{R}^{d+1}$, for a more general
class of manifolds that includes the spherical setting.

The covering number or Kolmogorov $\epsilon$-entropy ([17]) on manifolds are
popular concepts that play important role in applied areas such as kernel-
based learning algorithms and Gaussian process ([19, 20, 26]). The bounds for
the covering numbers is a useful and powerful tool in order to estimate the
probabilistic error of the statistical nature of the observations from which
the algorithms designed are learning from ([8, 27]). Estimates with explicit
constants and sharpness included in the asymptotic analysis of finding the
bounds for the covering numbers are also very important as illustrated in [18,
24].

Isotropic positive definite kernels on the spherical setting play important
roles in many applications, like kernel methods and spatial statistics ([4,
11, 14]). The series representation in terms of orthogonal polynomials for
isotropic positive definite kernels, known as Schoenberg representation ([12,
23]). This representation is widely considered on applications and closely
related with the Fourier series expansion of the kernels we work with. The
Gaussian Kernel is the main and useful example of spherical isotropic positive
kernel ([20]).

The framework of the paper is a compact two-point homogeneous space
$\mathbb{M}^{d}$ ([5]). This space is a Riemannian manifold, a compact
symmetric space of rank 1, and possess an invariant Riemannian normalized
metric $d(\cdot,\cdot)$ such that all geodesics on $\mathbb{M}^{d}$ are closed
and have length $2\pi$ ([7, 15]). Let $d\sigma_{d}$ be the volume element on
$\mathbb{M}^{d}$, we write
$L^{1}(\mathbb{M}^{d}):=L^{1}(\mathbb{M}^{d},\sigma_{d})$ for the Banach space
of all measurable functions $f$ on $\mathbb{M}^{d}$ such that

$\|f\|_{1}:=\frac{1}{\sigma_{d}(\mathbb{M}^{d})}\int_{\mathbb{M}^{d}}|f(x)|d\sigma_{d}x<\infty.$

The group motions $SO(\mathbb{M}^{d})$ of the manifold $\mathbb{M}^{d}$ is
compact and acts transitively on $\mathbb{M}^{d}$. Let $p$ be a fixed point (a
pole) of the manifold $\mathbb{M}^{d}$ and consider $SO_{p}$ be the stationary
subgroup of $p$ in $SO(\mathbb{M}^{d})$. We denote by $d\alpha$ and
$d\alpha_{p}$ the elements of the normalized Haar measures on the groups
$SO(\mathbb{M}^{d})$ and $SO_{p}$, respectively. In this case,
$\mathbb{M}^{d}=SO(\mathbb{M}^{d})/SO_{p}$ is the orbit space of the compact
subgroup $SO_{p}$ of the orthogonal group $SO(\mathbb{M}^{d})$. Due the fact
the groups $SO(\mathbb{M}^{d})$ and $SO_{p}$ are compact, these Haar measures
are bi-invariant, that is, they are invariant with respect to left and right
translations. Moreover, on $\mathbb{M}^{d}$ the measure $\sigma_{d}$ is
induced by a normalized left Haar measure and the following relation holds

$\frac{1}{\sigma_{d}(\mathbb{M}^{d})}\int_{\mathbb{M}^{d}}f(x)d\sigma_{d}x=\int_{SO(\mathbb{M}^{d})}f(\alpha
p)d\alpha,\quad f\in L^{1}(\mathbb{M}^{d}).$

There are four classes of these spaces and one example. A compact two-point
homogeneous space of dimension $d\geq 1$ is the 16-dimensional Cayley’s
elliptic plane $\mathbb{P}^{16}$ or one of the following manifolds: the
$d$-dimensional unit sphere $\mathbb{S}^{d}$, the $(d+1)$-dimensional real
projective space $\mathbb{P}^{d+1}(\mathbb{R})$, the $2d$-dimensional complex
projective space $\mathbb{P}^{2d}(\mathbb{C})$, or the $4d$-dimensional
quaternion projective space $\mathbb{P}^{4d}(\mathbb{H})$, for
$d=1,2,3,\ldots$.

Let $L^{2}(\mathbb{M}^{d}):=L^{2}\left(\mathbb{M}^{d},\sigma_{d}\right)$ be
the Hilbert space of square integrable functions
$f:\mathbb{M}^{d}\longrightarrow\mathbb{C}$ with the norm $\|\cdot\|_{2}$
induced by the inner product

$\langle
f,g\rangle_{2}=\int_{\mathbb{M}^{d}}f(x)\overline{g(x)}d\sigma_{d}x,\quad
f,g\in L^{2}(\mathbb{M}^{d}).$

The Laplace-Beltrami operator $\mathcal{B}_{d}$ on $\mathbb{M}^{d}$ has the
spectrum discrete formed by real and non-positive numbers. The radial part
$\mathcal{B}$ of the Laplace-Beltrami operator can be written as

$\mathcal{B}=(1-x)^{-\alpha}(1+x)^{-\beta}\frac{d}{dx}(1-x)^{1+\alpha}(1+x)^{1+\beta}\frac{d}{dx},$

for $\alpha=(d-2)/2$, and $\beta$ is given as follows: for
$\mathbb{M}^{d}=P^{16}$, $\beta=3$, for the case
$\mathbb{M}^{d}=\mathbb{S}^{d}$ or $\mathbb{M}^{d}=P^{d}(\mathbb{R})$, then
$\beta=\alpha=(d-2)/2$, if $\mathbb{M}^{d}=P^{d}(\mathbb{C})$, then $\beta=0$,
and for the remaining case $\mathbb{M}^{d}=P^{d}(\mathbb{H})$, we have
$\beta=1$. The eigenfunctions of $\mathcal{B}$ are the Jacobi polynomials
$P_{k}^{(\alpha,\beta)}$ with the corresponding eigenvalues
$-k(k+\alpha+\beta+1)$, $k=0,1,\ldots$. The eigenvalues of
$\Delta=-\mathcal{B}$ are arranged in an increasing order, with each
eigenvalue associated to the finite dimensional space $\mathcal{H}_{k}^{d}$.
The dimension $\tau_{k}^{d}:=\dim\mathcal{H}_{k}^{d}$ is zero if
$\mathbb{M}^{d}=P^{d}(\mathbb{R})$ and $k$ is odd, otherwise is given by

$\tau_{k}^{d}=\frac{\Gamma(\beta+1)(2k+\alpha+\beta+1)\Gamma(k+\alpha+1)\Gamma(k+\alpha+\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)\Gamma(k+1)\Gamma(k+\beta+1)},\quad
k=0,1,\ldots.$ (1.1)

The spaces are mutually orthogonal in $L^{2}(\mathbb{M}^{d})$ and it holds
$L^{2}(\mathbb{M}^{d})=\bigoplus_{k=0}^{\infty}\mathcal{H}_{k}^{d}$. If we set
$\beta_{0}=\\{1\\}$, and
$\beta_{k}=\left\\{S_{k,j}:j=1,2,\ldots,\tau_{k}^{d}\right\\}$, for
$k=1,2,\ldots$, an orthonormal basis of $\mathcal{H}_{k}^{d}$ , then
$\\{\beta_{n}\\}$ is an orthonormal complete system of
$L^{2}(\mathbb{M}^{d})$. This allows to consider the Fourier series expansion
of $f\in L^{2}(\mathbb{M}^{d})$ as follows

$f\sim\sum_{k=0}^{\infty}\sum_{j=1}^{\tau_{k}^{d}}\left\langle
f,S_{k,j}\right\rangle_{2}S_{k,j}.$ (1.2)

The relation between the Fourier series expansions of isotropic functions and
the so-called Schoenberg series representation is given in terms of the Jacobi
polynomials. We present just the necessary facts bellow, more details can be
found in [3, Section 2].

For $\alpha,\beta>-1$, we write
$P_{k}^{(\alpha,\beta)}:[-1,1]\longrightarrow\mathbb{R}$ for the Jacobi
polynomial of degree $k=0,1,\ldots$ associated to the weight
$w(x)=(1-x)^{\alpha}(1+x)^{\beta}$. Let
$L_{1}^{\alpha,\beta}:=L_{1}([-1,1],\omega(t)dt)$ be the space of the
measurable functions $f:[-1,1]\longrightarrow\mathbb{R}$ such that

$\int_{-1}^{1}|f(t)|\omega(t)dt<\infty.$

The family of orthogonal polynomials $\\{P_{n}^{(\alpha,\beta)}\\}$ is a
complete orthogonal system on the interval $[-1,1]$ with the weighted measure
as bellow

$\int_{-1}^{1}P_{k}^{(\alpha,\beta)}(t)P_{n}^{(\alpha,\beta)}(t)\omega(t)dt=\delta_{kn}h_{k}^{\alpha,\beta},\qquad
h_{k}^{\alpha,\beta}=\frac{2^{\alpha+\beta+1}}{(2k+\alpha+\beta+1)}\frac{\Gamma(k+\alpha+1)\Gamma(k+\beta+1)}{\Gamma(k+1)\Gamma(k+\alpha+\beta+1)},$

$k=1,2,\ldots$ with $\delta_{kn}$ representing the Kronecker delta. If $f\in
L_{1}^{\alpha,\beta}$, then it has the series representation given by

$f(t)\sim\sum_{k=0}^{\infty}a_{k}^{\alpha,\beta}\frac{P_{k}^{(\alpha,\beta)}(t)}{P_{k}^{(\alpha,\beta)}(1)},\qquad
a_{k}^{\alpha,\beta}=\frac{P_{k}^{(\alpha,\beta)}(1)}{h_{k}^{\alpha,\beta}}\int_{-1}^{1}f(t)P_{k}^{(\alpha,\beta)}(t)(1-t)^{\alpha}(1+t)^{\beta}dx,$
(1.3)

with

$P_{k}^{(\alpha,\beta)}(1)=\max_{t\in[-1,1]}P_{k}^{(\alpha,\beta)}(t)=\frac{\Gamma(k+\alpha+1)}{k!\Gamma(\alpha+1)},\quad
k=0,1,\ldots.$ (1.4)

We will deal with a kernel $K$ on $\mathbb{M}^{d}$, it means a function
$K:\mathbb{M}^{d}\times\mathbb{M}^{d}\longrightarrow\mathbb{R}$, that is
continuous, isotropic, and positive definite. The kernel $K$ is zonal (or
isotropic), if

$K(x,y)=C(\alpha x,\alpha y),\quad x,y\in\mathbb{M}^{d},$

all $\alpha\in SO(\mathbb{M}^{d})$. It holds that $K$ is an isotropic
continuous kernel if it can be written as $K(x,y)=f(\cos(d(x,y)))$,
$x,y\in\mathbb{M}^{d}$, for a continuous function
$f:[-1,1]\longrightarrow\mathbb{R}$. This equivalence is carefully proved in
[9] for function on the spherical setting, it is standard the adaptation of
the proofs for kernels on a compact two-point homogeneous spaces. The kernel
$K$ is _positive definite_ ([25, Chapter 3]) if $K$ is symmetric and for any
$n\in\mathbb{N}$, $x_{1},\ldots,x_{n}\in\mathbb{M}^{d}$ and
$c_{1},\ldots,c_{n}\in\mathbb{R}$ it holds

$\sum_{i=1}^{n}\sum_{j=1}^{n}c_{i}c_{j}K(x_{i},x_{j})\geq 0.$

The characterization of the Schoenberg type (see [3, Theorem 1.1] and [4]))
asserts that an continuous isotropic positive definite kernel $K$ can be
written as a series expansion in terms of the Jacobi polynomial, with summable
sequence of non-negative real numbers $\\{a_{n}^{\alpha,\beta}\\}$ called the
sequence of Schoenberg coefficients. Precisely, let
$K:\mathbb{M}^{d}\times\mathbb{M}^{d}\longrightarrow\mathbb{R}$ be a kernel
such that $K(x,y)=f(\cos(d(x,y)))$, $x,y\in\mathbb{M}^{d}$, for a continuous
function $f:[-1,1]\longrightarrow\mathbb{R}$. The kernel $K$ is positive
definite if and only if

$K(x,y)=\sum_{k=0}^{\infty}a_{k}^{\alpha,\beta}\frac{P_{k}^{(\alpha,\beta)}(\cos(d(x,y)))}{P_{k}^{(\alpha,\beta)}(1)},\quad
x,y\in\mathbb{M}^{d},$ (1.5)

with $\\{a_{n}^{\alpha,\beta}\\}$ given in (1.3) a summable sequence of non-
negative real numbers.

In this paper $K$ is a continuous isotropic positive definite kernel
represented by the series expansion (1.5). We write
$J_{k}^{\alpha,\beta}(s)=P_{k}^{\alpha,\beta}(s)/P_{k}^{\alpha,\beta}(1)$ for
the normalized Jacobi polynomial of degree $k$ associated to the pair
$(\alpha,\beta)$, for $k=0,1\ldots$. The Addition Formula, given by the
following identity (see [5, p.405] or [21, Section 2.3], for the spherical
setting)

$\sum_{j=1}^{\tau_{k}^{d}}S_{k,j}(x)S_{k,j}(y)=\tau_{k}^{d}J_{k}^{\alpha,\beta}(\cos(d(x,y))),\quad
x,y\in\mathbb{M}^{d},\quad k=0,1,\ldots,$ (1.6)

implies the Fourier series representation of $K$ as follows

$K(x,y)=\sum_{k=0}^{\infty}a_{k}^{\alpha,\beta}J_{k}^{\alpha,\beta}(\cos(d(x,y)))=\sum_{k=0}^{\infty}\frac{a_{k}^{\alpha,\beta}}{\tau_{k}^{d}}\sum_{j=1}^{\tau_{k}^{d}}S_{k,j}(x)S_{k,j}(y),\quad
x,y\in\mathbb{M}^{d}.$ (1.7)

We assume that $a_{k}^{\alpha,\beta}\downarrow 0$, as $k\to\infty$.

For a positive definite kernel $K$ the classical Aronszajn theory of RKHSs
([1]) asserts that there exists a unique reproducing Hilbert space
$(\mathcal{H}_{K},\langle\,\cdot\,,\cdot\,\rangle_{K})$ of functions on
$\mathbb{M}^{d}$, satisfying:
i. $K(x,\,\cdot\,)\in\mathcal{H}_{K}$ for all $x\in\mathbb{M}^{d}$;
ii. span$\\{K(\,\cdot\,,x):x\in X\\}$ is dense in $\mathcal{H}_{K}$;
iii. (Reproducing property) $f(x)=\langle{f},{K(\cdot,x)}\rangle_{K}$, for all
$x\in\mathbb{M}^{d}$ and $f\in\mathcal{H}_{K}$.
From the reproducing property and the Cauchy-Schwarz inequality, it is not
hard to see that it holds

$|f(x)-f(y)|\leq\|f\|_{K}\|K(\cdot,x)-K(\cdot,y)\|_{K}=\|f\|_{K}\left(K(x,x)+K(y,y)-2K(x,y)\right)^{1/2}$

for any $f\in\mathcal{H}_{K}$, and for all $x,y\in\mathbb{M}^{d}$. If $K$ is a
continuous kernel, then $\mathcal{H}_{K}\hookrightarrow C(\mathbb{M}^{d})$,
i.e., we have a continuous embedding of the RKHS $\mathcal{H}_{K}$ into
$C(\mathbb{M}^{d})$, the space of continuous functions endowed with the
supremum norm $\|\,\cdot\,\|_{\infty}$. In this case, if $K$ is represented as
in formula (1.7), then the embedding $I_{K}:\mathcal{H}_{K}\longrightarrow
C(\mathbb{M}^{d})$ satisfies
$\|I_{K}\|\leq\sum_{k=0}^{\infty}a_{k}^{\alpha,\beta}$, also due to the
reproducing property. Precisely, the interest is on this embedding and the
estimates for the related covering numbers. The covering numbers of the
operator $I_{K}$ are defined in terms of the concept of the covering numbers
for metric spaces. If $A$ is a subset of a metric space $X$ and $\epsilon>0$,
then the _covering numbers_ $\mathcal{C}(\epsilon,A)$ are the minimal number
of balls of $X$ with radius $\epsilon$ which covers $A$. We denote by $B_{K}$
the unit ball in $\mathcal{H}_{K}$ and the _covering numbers of the kernel
$K$_ (or the covering numbers of the RKHS of the kernel $K$) are defined in
terms of the covering numbers of the embedding $I_{K}$, given as follows

$\mathcal{C}(\epsilon,I_{K}):=\mathcal{C}(\epsilon,I_{K}(B_{K})),\quad\epsilon>0.$

The content of the paper is organized as follows. In Section 2, we present
some general facts about the RKHS of isotropic positive definite kernels $K$
represented as in (1.7), and general properties for covering numbers. In
Section 3, the bounds for the covering numbers of kernels with the sequence of
coefficients possessing rapid decay or decay slower than a geometric
progression are presented. As a consequence of the main theorems we present
the weak equivalence for order of the growth of the covering numbers for
kernels with a geometric progression as sequence of coefficients, and we
obtain bounds for the covering numbers of the Gaussian kernel on the
$d$-dimensional unit sphere. In Subsection 3.1, we present the $d$-dimensional
spherical kernel of the Gaussian type, and we apply the estimates obtained
previously in order to obtain accurate upper and the lower bounds for the
covering numbers of those kernels. In Section 4, we present the bounds for the
covering numbers of kernels with the sequence of coefficients possessing decay
or growth comparable to an harmonic sequence. We finish the section with a
formal example of kernel on $\mathbb{M}^{d}$ fulfilling the assumptions of the
main theorems, and we present the final comments and the acknowledgments.

##  2 Basic theoretical background

The proofs of the theorems apply general estimates for the covering numbers
given em terms of operator norm and for operators with finite dimensional
range.

The following proposition presents the characterization of the RKHS in terms
of Fourier series expansions, the proof easily follows from the unicity of the
RKHS and it is standard in theory of RKHSs. If $K$ is an isotropic positive
definite kernel represented by the series expansion (1.7), then we will employ
the following notation

$\eta_{k}^{d}:=\sqrt{\frac{a_{k}^{\alpha,\beta}}{\tau_{k}^{d}}},\quad
k=0,1,\ldots,$

######  Lemma 2.1.

If $K$ is an isotropic positive definite kernel represented by the series
expansion (1.7), then the RKHS
$(\mathcal{H}_{K},\langle\,\cdot\,,\cdot\,\rangle_{K})$ associated with $K$ is

$\mathcal{H}_{K}=\left\\{g:\mathbb{M}^{d}\longrightarrow\mathbb{R}:g(x)=\sum_{k=0}^{\infty}\eta_{k}^{d}\sum_{j=1}^{\tau_{k}^{d}}c_{k}^{j}S_{k,j}(x),\quad
x\in\mathbb{M}^{d},\,\\{c_{k}^{j}\\}\in\ell^{2},\,j=1,..,\tau_{k}^{d}\right\\},$

endowed the inner product

$\langle
g,h\rangle_{K}=\sum_{k=0}^{\infty}\sum_{j=1}^{\tau_{k}^{d}}c_{k}^{j}d_{k}^{j},\quad
g,h\in\mathcal{H}_{K},$

where

$g(x)=\sum_{k=0}^{\infty}\eta_{k}^{d}\sum_{j=1}^{\tau_{k}^{d}}c_{k}^{j}S_{k,j}(x)\quad\mbox{and}\quad
h(x)=\sum_{k=0}^{\infty}\eta_{k}^{d}\sum_{j=1}^{\tau_{k}^{d}}d_{k}^{j}S_{k,j}(x),\quad
x\in\mathbb{M}^{d}.$

We consider $P_{m}$ the orthogonal projections onto the subspace
$\mathcal{V}_{m}$ of $\mathcal{H}_{K}$ and $P_{m}^{s}$ the orthogonal
projection onto the subspace and $\mathcal{V}_{m}^{s}$, the orthogonal
complement of $\mathcal{V}_{m}$, with the basis

$\left\\{\eta_{k}^{d}S_{k,j}:\quad
j=1,..,\tau_{k}^{d},\,\,k=0,1,\ldots,m\right\\}\quad\mbox{and}\quad\left\\{\eta_{k}^{d}S_{k,j}:\quad
j=1,..,\tau_{k}^{d},\,\,k=m+1,m+2,\ldots\right\\},$

respectively, for each $m=1,2,\ldots$. Then, we can write

$f=P_{m}(f)+P_{m}^{s}(f),\quad
f\in\mathcal{H}_{K}=\bigoplus_{k=0}^{\infty}\mathcal{V}_{k},\quad
m=1,2,\ldots,$

and we consider the natural embeddings $\mathcal{V}_{m}\hookrightarrow
C(\mathbb{M}^{d})$ and $\mathcal{V}_{m}^{s}\hookrightarrow C(\mathbb{M}^{d})$,
keeping the notation $P_{m}$ and $P_{m}^{s}$ for them, respectively. The
operator norm of the embeddings, in terms of the Fourier coefficients of the
kernel, is given in the following lemma. The proof is standard in the theory
and we omitted it here. It can be find in the previous article [13,
Proposition 2.3] for the spherical setting and the same proof fits here.

######  Lemma 2.2.

Let $K$ an isotropic positive definite kernel represented by the series
expansion (1.7). For $m=1,2,\ldots$, the following embeddings

$I_{K}:\mathcal{H}_{K}\longrightarrow C(\mathbb{M}^{d}),\quad
P_{m}:\mathcal{V}_{m}\longrightarrow C(\mathbb{M}^{d}),\quad\mbox{and}\quad
P_{m}^{s}:\mathcal{V}_{m}^{s}\longrightarrow C(\mathbb{M}^{d}),$

satisfy respectively,

$\|I_{K}\|^{2}=\sum_{k=0}^{\infty}a_{k},\quad\|P_{m}\|^{2}=\sum_{k=0}^{m}a_{k},\quad\mbox{and}\quad\|P_{m}^{s}\|^{2}=\sum_{k=m+1}^{\infty}a_{k}.$

We will employ the notation: for two given functions
$f,g:(0,\infty)\longrightarrow\mathbb{R_{+}}$, $f(\epsilon)\asymp g(\epsilon)$
means the weak equivalence and it stands for $f(\epsilon)=O(g(\epsilon))$ and
$g(\epsilon)=O(f(\epsilon))$, as $\epsilon\to 0$, we write $f(\epsilon)\approx
g(\epsilon)$ for the strong equivalence, that is, $f(\epsilon)/g(\epsilon)\to
1$, as $\epsilon\to 0$.

The Stirling’s approximation formula ([2, p. 24])

$\Gamma(x)\approx\sqrt{2\pi}x^{x-1/2}e^{-x},\quad\mbox{as $x\to\infty$},$

raises from the identity

$\Gamma(x)=\sqrt{2\pi}x^{x-1/2}e^{-x+\mu(x)},\quad x>0.$

where

$\mu(x)=\sum_{n=0}^{\infty}(x+n+1/2)\log\left(1+\frac{1}{x+n}\right)-1=\frac{\theta}{12x},\quad
0<\theta<1.$

The formula deliveres the following strong equivalence for the dimension
defined in (1.1)

$\tau_{n}^{d}\approx\frac{2\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}n^{2\alpha+1}=\frac{2\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}n^{d-1},\quad\mbox{as
$n\to\infty$}.$ (2.8)

Immediate consequence of the previous considerations is the following result.

######  Proposition 2.3.

Let $K$ be an isotropic positive definite kernel represented by the series
expansion (1.7). If $a_{k}>0$, for $k=0,1,2,\ldots$, then

$\dim\mathcal{V}_{m}\approx\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}m^{d},\quad\mbox{as
$m\to\infty$}.$

Proof. From identity (1.1), for $m=1,2,\ldots$, it is clear that

$\dim\mathcal{V}_{m}=\sum_{k=0}^{m}\tau_{k}^{d}=\frac{\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}\sum_{k=0}^{m}\frac{(2k+\alpha+\beta+1)\Gamma(k+\alpha+1)\Gamma(k+\alpha+\beta+1)}{\Gamma(k+1)\Gamma(k+\beta+1)},$

and we obtain

$\dim\mathcal{V}_{m}=\frac{\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}\frac{(m+1)(\beta+m+1)\Gamma(m+\alpha+2)\Gamma(m+\alpha+\beta+2)}{(\alpha+1)\Gamma(m+2)\Gamma(m+\beta+2)}.$

The Stirling’s approximation formula now implies

$\displaystyle\dim\mathcal{V}_{m}\approx\frac{\Gamma(\beta+1)}{e^{2\alpha}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)(\alpha+1)}b_{m},\quad\mbox{as
$m\to\infty$},$

with

$\displaystyle
b_{m}=\frac{(m+1)(\beta+m+1)(m+\alpha+2)^{m+\alpha+3/2}(m+\alpha+\beta+2)^{(m+\alpha+\beta+3/2)}}{(m+2)^{m+3/2}(m+\beta+2)^{m+\beta+3/2}},$

for $m=1,2,\ldots$. Due the fact that the sequence $\\{b_{m}\\}_{m}$ satisfies
$b_{m}\approx e^{2\alpha}m^{2\alpha+2}$, as $m\to\infty$, we gain

$\displaystyle\dim\mathcal{V}_{m}\approx\frac{\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)(\alpha+1)}m^{2\alpha+2},\quad\mbox{as
$m\to\infty$},$

since $\alpha=(d-2)/2$ and $(\alpha+1)\Gamma(\alpha+1)=\Gamma(\alpha+2)$, the
strong equivalence stated follows.

We present the basic and general properties of the covering numbers we are
going to use in the following sections. The covering numbers as defined in
Section 1 is the particular case of the following. Consider
$(X,\|\cdot\|_{X}),(Y,\|\cdot\|_{Y})$ two Banach spaces and $B_{X}$ the unit
ball in $X$, then the _covering numbers_ of an operator $T:X\longrightarrow Y$
are given by

$\mathcal{C}(\epsilon,T):=\mathcal{C}(\epsilon,T(B_{X})),\quad\epsilon>0.$

It is not hard to see that if $\|T\|\leq\epsilon$, then
$\mathcal{C}(\epsilon,T)=1$, for any $\epsilon>0$, and that $C(\epsilon,T)$ is
non-increasing in $\epsilon$. The following properties can be found in [6,
22]. If $S,\,T:X\longrightarrow Y$ and $R:Z\longrightarrow X$ are operators on
real Banach spaces, then

  1. c1.

$\mathcal{C}(\epsilon+\delta,T+S)\leq\mathcal{C}(\epsilon,T)\hskip
2.84544pt\mathcal{C}(\delta,S)$, and

  2. c2.

$\mathcal{C}(\epsilon\delta,TR)\leq\mathcal{C}(\epsilon,T)\,\mathcal{C}(\delta,R)$,
for any $\epsilon,\delta>0$;

  3. c3.

if $n:=\mbox{rank}(T)<\infty$, then
$\mathcal{C}(\epsilon,T)\leq\left(1+2\|T\|/\epsilon\right)^{n}$, for any
$\epsilon>0$.

For the lower estimates we will need the following inequality for the covering
numbers for two $n$-dimensional Hilbert spaces $X$ and $Y$,

$\sqrt{\det{T^{*}T}}\left(\frac{1}{\epsilon}\right)^{n}\leq\mathcal{C}(\epsilon,T),\quad\epsilon>0.$
(2.9)

##  3 Estimates of kernels with rapid decay of the coefficients

In this section present the bounds for the covering numbers of the RKHS of
kernels on $\mathbb{M^{d}}$ represented by the series expansion (1.7) with
sequence of coefficients with decay comparable to a geometric progression. We
apply them in order to estimate the covering numbers of the Gaussian kernel on
the unit sphere and the covering numbers of spherical kernels the Gaussian
type.

In order to present the results we fix the following short notation for the
quantities given in Lemma 2.2,

$\kappa=\|I_{K}\|,\quad\kappa_{m}=\|P_{m}\|,\quad\mbox{and}\quad\kappa_{m}^{s}=\|P_{m}^{s}\|,\quad\mbox{for
$m=1,2,\ldots$}.$ (3.10)

Let $K$ be a kernel represented by the series expansion (1.7) on
$\mathbb{M}^{d}$, we will write $a_{k}:=a_{k}^{\alpha,\beta}$, $k=0,1,\ldots$,
for the sequence of coefficients of the kernel. In this case, we observe that
if $a_{0}\neq 0$ and there exists $0<\theta<1$ such that
$a_{k}\leq\theta\,a_{k-1}$, for $k=1,2\ldots$, then

$0<(k_{m}^{s})^{2}\leq\frac{\theta^{m+1}}{1-\theta}\,a_{0},\quad
m=1,2,\ldots.$ (3.11)

The next theorem is the extension of the Proposition 3.1 of the reference
[13].

######  Theorem 3.1.

Let $K$ a continuous isotropic positive definite kernel given by the series
expansion (1.7). If there exists $0<\theta<1$ such that
$a_{k}\leq\theta\,a_{k-1}$, for $k=1,2\ldots$, then

$\limsup_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{[\ln(1/\epsilon)]^{d+1}}\leq\frac{2^{d+1}\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\frac{1}{[\ln({1}/{\theta})]^{d}}.$

Proof. With no loss of generality we can consider $a_{0}\neq 0$. Otherwise, in
the proof bellow, we just replace $a_{0}$ by the first positive term of the
sequence $\\{a_{n}\\}$.

For $\epsilon>0$, we consider $m:=m(\epsilon)\in\mathbb{Z}_{+}$ as follows

$\left(\frac{\theta^{m+1}}{1-\theta}\,a_{0}\right)^{1/2}<\frac{\epsilon}{2}<\left(\frac{\theta^{m}}{1-\theta}\,a_{0}\right)^{1/2}.$
(3.12)

Applying the logarithm above, we have that

$m\approx\frac{\ln(\epsilon^{2}(1-\theta)/4\,a_{0})}{\ln\theta}=\frac{2\ln(2a_{0}^{1/2}/\epsilon(1-\theta)^{1/2})}{\ln(1/\theta)},\quad\mbox{as
$\epsilon\to 0^{+}$}.$ (3.13)

Considerations above, inequality (3.11), and inequality (3.12) imply

$\mathcal{C}\left(\epsilon,P_{m}^{s}\right)=1,\quad m=1,2,\ldots.$

Thus, properties c1. and c3. for covering numbers, respectively, lead us to
the estimate

$\mathcal{C}(\epsilon,I_{K}=P_{m}+P_{m}^{s})\leq\left(1+4\|P_{m}\|/\epsilon\right)^{\mbox{rank}(P_{m})}\leq\left(1+4\kappa_{m}/\epsilon\right)^{\dim{\mathcal{V}_{m}}},$

that implies

$\displaystyle\ln(\mathcal{C}(\epsilon,I_{K}))\leq\dim{\mathcal{V}_{m}}\ln\left(8\kappa/\epsilon\right),\quad
m=1,2,\ldots,$

since $\kappa_{m}\leq\kappa$, for any $m$. From Proposition 2.3, we have

$\mathcal{C}(\epsilon,I_{K})\leq\dim{\mathcal{V}_{m}}\ln\left(8\kappa/\epsilon\right)<2\frac{\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)(\alpha+1)}m^{d}\ln\left(8\kappa/\epsilon\right),$

for $m=m_{0},m_{0}+1,\ldots$, and some $m_{0}\in\mathbb{N}$. With the
application of the strong equivalences (3.13) and $\ln t\approx\ln 8\kappa t$,
we obtain

$\displaystyle\limsup_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{[\ln(1/\epsilon)]^{d+1}}=\limsup_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{[\ln(2a_{0}^{1/2}/\epsilon(1-\theta)^{1/2})]^{d+1}}$
$\displaystyle\leq$
$\displaystyle\frac{2^{d+1}\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\frac{1}{[\ln(1/\theta)]^{d}}.$

We present an application of Theorem 3.1 to the spherical Gaussian Kernel. The
background material we will employ can be found in the references [16, 20].

######  Example 3.2.

The Gaussian kernel on the unit sphere $\mathbb{S}^{d}$, for $d\geq 2$, is
given by

$K_{\varrho}(x,y)=\exp\left(-2\varrho^{-2}(1-x\cdot y)\right),\quad
x,y\in\mathbb{S}^{d},$ (3.14)

for $\varrho>0$, where $x\cdot y$ is the dot product in $\mathbb{R}^{d+1}$. It
is well known (see [20, Theorem 8] and [16, Example 2.5]) that the Gaussian
kernel is an isotropic (continous) positive definite kernel represented by the
series expansion (1.7), as follows

$K_{\varrho}(x,y)=\sum_{k=0}^{\infty}\lambda_{k}^{\varrho}\tau^{d}_{k}G_{k}^{(d-2)/2}(x\cdot
y)=\sum_{k=0}^{\infty}\lambda_{k}^{\varrho}\sum_{j=1}^{\tau^{d}_{k}}S_{k,j}(x)S_{k,j}(y),\quad
x,y\in\mathbb{S}^{d},$ (3.15)

where $G_{k}^{(d-2)/2}(s)=J_{k}^{\alpha,\alpha}(s)$, for $\alpha=(d-2)/2$, is
the normalized Gegenbauer polynomial of degree $k$ associated to the dimension
$d$, and

$\lambda_{k}^{\varrho}=e^{-2/\varrho^{2}}\varrho^{d-1}\Gamma\left(\frac{d+1}{2}\right)I_{k+(d-1)/2}\left(\frac{2}{\varrho^{2}}\right),\quad
k=0,1,2,\ldots,$ (3.16)

with $I_{\nu}(\cdot)$ denoting the modified Bessel function of first kind
associated with the parameter $\nu$. The modified Bessel function of first
kind is defined in terms of the Bessel function $J_{\nu}$ with the parameter
$\nu>-1/2$ ([10, p. 5, formula (12)]) and given by

$I_{\nu}(z)=e^{-i\nu\pi/2}J_{\nu}(ze^{i\pi/2})=\sum_{j=0}^{\infty}\left(\frac{z}{2}\right)^{\nu+2j}\frac{1}{j!\Gamma(j+\nu+1)}.$

The definition of the Bessel function above implies the following

$\displaystyle
I_{k+(d-1)/2}\left(\frac{2}{\varrho^{2}}\right)<\frac{1}{\varrho^{2}(k+(d-1)/2)}I_{k-1+(d-1)/2}\left(\frac{2}{\varrho^{2}}\right),\quad
k=0,1,2,\ldots,$

this means that

$\varrho^{2}(k+(d+1)/2)\lambda_{k+1}^{\varrho}<\lambda_{k}^{\varrho},\quad
k=0,1,\ldots.$ (3.17)

######  Corollary 3.3.

Let $K_{\varrho}$ be the Gaussian kernel on $\mathbb{S}^{d}$ defined in
(3.14). If $\varrho^{2}>2$, then

$\limsup_{\epsilon\rightarrow
0^{+}}\frac{\ln\left(\mathcal{C}\left(\epsilon,I_{K}\right)\right)}{[\ln(1/\epsilon)]^{d+1}}\leq\frac{2^{d+2}}{d!\,[{\ln\left(\varrho^{2}/2\right)}]^{d}}=\frac{4}{d!}\frac{1}{[\ln(\varrho/\sqrt{2})]^{d}}.$

Proof. In this spherical case, ($\alpha=\beta=(d-2)/2$ in formula (1.1)) we
have that

$\tau_{k}^{d}=\frac{(2k+d-1)(k+d-2)!}{k!(d-1)!},$

then the identity

$\tau_{k+1}^{d}=\frac{(2k+d+1)(k+d-1)}{(2k+d-1)(k+1)}\tau_{k}^{d},\quad\quad
k=0,1,2,\ldots,$ (3.18)

and inequality (3.17), imply

$\displaystyle\lambda_{k}^{\varrho}\tau_{k}^{d}>\varrho^{2}(k+(d+1)/2)\lambda_{k+1}^{\varrho}\tau_{k}^{d}$
$\displaystyle=$
$\displaystyle\frac{\varrho^{2}}{2}\frac{(2k+d-1)(k+1)}{(k+d-1)}\lambda_{k+1}^{\varrho}\tau_{k+1}^{d},$

for $k=0,1,2,\ldots$. Therefore, it holds the following

$\lambda_{k+1}^{\varrho}\tau_{k+1}^{d}<\frac{2}{\varrho^{2}}\frac{(k+d-1)}{(2k+d-1)(k+1)}\lambda_{k}^{\varrho}\tau_{k}^{d}\quad\mbox{for
$k=0,1,2,\ldots$}.$ (3.19)

It is easy to see that $\\{a_{k}=\lambda_{k}^{\varrho}\tau_{k}^{d}\\}$ is the
sequence of coefficients of the kernel $K_{\varrho}$ in the series
representation given in formula (1.7). Then, the inequality (3.19) promptly
implies that the sequence of coefficients of the kernel $K_{\varrho}$
satisfies $a_{k+1}<(2/\rho^{2})a_{k}$, $k=1,2,\ldots$. An application of
Theorem 3.1 with $\alpha=\beta=(d-2)/2$ and for $\theta=2/\rho^{2}$ leads to
the estimate stated.

The lower estimate for the covering numbers of kernels on $\mathbb{M}^{d}$
with decay not faster than a rapid decay is is presented in the next theorem.
This extends Proposition 3.2 of [13].

######  Theorem 3.4.

Let $K$ a continuous isotropic positive definite kernel given by the series
expansion (1.7). If there exists $0<\delta<1$ such that
$a_{k}\geq\delta\,a_{k-1}>0$, for $k=1,2\ldots$, then

$\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\,\frac{2^{d-1}d^{d}}{[\ln(1/\delta)+d-1]^{d}(d+1)^{d+1}}\leq\liminf_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{[\ln(1/\epsilon)]^{d+1}}.$

Proof. For $m=1,2,\ldots$ we consider the following composition operator

$T_{m}:\mathcal{V}_{m}\stackrel{{\scriptstyle
J_{m}}}{{\longrightarrow}}\mathcal{H}_{K}\stackrel{{\scriptstyle
I_{K}}}{{\longrightarrow}}C(\mathbb{M}^{d})\stackrel{{\scriptstyle{L_{m}}}}{{\longrightarrow}}L^{2}(\mathbb{M}^{d})\stackrel{{\scriptstyle
P_{m}}}{{\longrightarrow}}L_{m}\circ I_{K}\circ J_{m}(\mathcal{V}_{m}),$

where $J_{m}$ stands for the identity operator given by
$\mathcal{V}_{m}\hookrightarrow\mathcal{H}_{K}$, $L_{m}$ and $P_{m}$ are the
orthogonal projections on $L_{m}I_{K}J_{m}(\mathcal{V}_{m})$. It is clear that
$\|P_{m}\|=\|J_{m}\|=\|L_{m}\|=1$, and

$\det(T_{m}^{*}T_{m})=\prod_{k=0}^{m}\left(\frac{a_{k}}{\tau_{k}^{d}}\right)^{\tau^{d}_{k}}.$
(3.20)

For any $\epsilon>0$, properties c1. and c2, for the covering numbers, imply

$\displaystyle\mathcal{C}(\,\epsilon,T_{m})=\mathcal{C}(\,\epsilon,P_{m}L_{m}I_{K}J_{m})$
$\displaystyle\leq$
$\displaystyle\mathcal{C}(1,P_{m})\mathcal{C}(1,L_{m})\mathcal{C}(\epsilon,I_{K})\mathcal{C}(1,J_{m})=\mathcal{C}(\epsilon,I_{K}),$

for $m=1,2,\ldots$. The application of the lower bound for the covering number
given in inequality (2.9), leads us to

$\sqrt{\det(T_{m}^{*}T_{m})}\left(\frac{1}{{}\,\epsilon}\right)^{\dim\mathcal{V}_{m}}\leq\mathcal{C}(\,\epsilon,T_{m})\leq\mathcal{C}(\epsilon,I_{K}),$

and due the formula (3.20), we have

$\prod_{k=0}^{m}\left(\frac{a_{k}}{\tau^{d}_{k}}\right)^{\tau^{d}_{k}/2}\left(\frac{1}{\,\epsilon}\right)^{\dim\mathcal{V}_{m}}\leq\mathcal{C}(\epsilon,I_{K}),\quad
m=1,2,\ldots.$

For $\epsilon>0$, consider

$J_{m}:=\frac{1}{2}\sum_{k=0}^{m}\tau_{k}^{d}\ln\left(a_{k}/\tau_{k}^{d}\right)-\dim\mathcal{V}_{m}\ln({\epsilon})\leq\ln(\mathcal{C}(\epsilon,I_{K})),\quad
m=1,2,\ldots,$ (3.21)

We observe that, for $m=1,2,\ldots$, the estimate

$\delta^{m}a_{0}/\tau_{m}^{d}\leq a_{m}/\tau_{m}^{d}\leq
a_{k}/\tau_{k}^{d},\quad k=0,1,\ldots m,$

implies

$\frac{\sum_{k=0}^{m}\tau_{k}^{d}}{2}\left(\ln(\delta^{m}a_{0})-\ln(\tau_{m}^{d})\right)\leq\frac{1}{2}\sum_{k=0}^{m}\tau_{d}^{k}\ln\left(a_{k}/\tau_{k}^{d}\right),\quad
m=1,2,\ldots.$ (3.22)

If we write

$I_{m}:=\frac{\sum_{k=0}^{m}\tau_{k}^{d}}{2}\left(\ln(\delta^{m}a_{0})-\ln(\tau_{m}^{d})\right),\quad
m=1,2,\ldots,$

then,

$\displaystyle J_{m}=I_{m}-\dim
V_{m}\ln(\epsilon)\leq\ln(\mathcal{C}(\epsilon,I_{K})),\quad m=1,2,\ldots.$

Formula (2.8) implies that

$\tau^{d}_{k}\leq\tau^{d}_{m}<\frac{3\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}m^{d-1},$

for $m=m_{0},m_{0}+1,\ldots$ and some $m_{0}\in\mathbb{Z}_{+}$. By an
application of the Proposition 2.3, we can consider that

$\frac{\Gamma(\beta+1)}{2\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}m^{d}<\dim
V_{m}=\sum_{k=0}^{m}\tau_{k}^{d},\quad m=m_{0},m_{0}+1,\ldots.$

Therefore, for any $\epsilon>0$ and $m=m_{0},m_{0}+1,\ldots$, it holds

$A_{\alpha,\beta}\,m^{d}\left[m\ln\delta+\ln
a_{0}-(d-1)m-\ln\left(\frac{3\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}\right)-2\ln(\epsilon)\right]<J_{m},$

with

$A_{\alpha,\beta}:=\frac{\Gamma(\beta+1)}{4\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}.$

We consider

$\varphi_{\epsilon}(m):=-m^{d}\ln{\left(\frac{3\Gamma(\beta+1)\,\epsilon^{2}}{a_{0}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}\right)}-m^{d+1}\left[\ln(1/\delta)+d-1\right],$

with $\epsilon>0$ and $m=m(\epsilon)$ such that $1/(m+1)\leq\epsilon<1/m$.
The final steps of the proof is to solve the optimization problem given by
$\varphi_{\epsilon}$. Thereby, we obtain

$A_{\alpha,\beta}\varphi_{\epsilon}(m)<\ln(\mathcal{C}(\epsilon,I_{K})),\quad
m=m_{0},m_{0}+1,\ldots,$ (3.23)

for some $m_{0}\in\mathbb{Z}_{+}$. The critical point of $\varphi_{\epsilon}$,
as a function on $\mathbb{R}$, is given by

$c=-\frac{d}{(d+1)[\ln(1/\delta)+d-1]}\ln\left(\frac{3\Gamma(\beta+1)\epsilon^{2}}{a_{0}\Gamma(\alpha+\beta+2)\Gamma(\alpha+1)}\right).$

Since $\lfloor c\rfloor\approx c\approx\lceil c\rceil$, we can estimate (3.23)
in terms of $\varphi_{\epsilon}(c)$, even if $c$ is not an integer. A simple
calculation leads us to

$\displaystyle
A_{\alpha,\beta}\varphi_{\epsilon}(c)=A_{\alpha,\beta}\frac{[-\ln(3\,\Gamma(\beta+1)\,\epsilon^{2}/a_{0}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)]^{d+1}}{[\ln(1/\delta)+d-1]^{d}}\frac{d^{d}}{(d+1)^{d+1}},$

we can consider $a_{0}$ such that
$a_{0}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)/3\,\Gamma(\beta+1)\geq 1$, and
then inequality (3.23) implies

$\frac{2^{d+1}d^{d}A_{\alpha,\beta}}{[\ln(1/\delta)+d-1]^{d}(d+1)^{d+1}}\left[\ln\left(\frac{\sqrt{a_{0}\Gamma(\alpha+\beta+2)\Gamma(\alpha+1)}}{\sqrt{3\Gamma(\beta+1)}\epsilon}\right)\right]^{d+1}<\ln(\mathcal{C}(\epsilon,I_{K})),$

for $1/(m+1)\leq\epsilon<1/m$ and $m=m_{0},m_{0}+1,\ldots$. The proof follows
by observing that

$\ln\left(\frac{\sqrt{a_{0}\Gamma(\alpha+\beta+2)\Gamma(\alpha+1)}}{\sqrt{3\Gamma(\beta+1)}\epsilon}\right)\approx\ln(1/\epsilon),\quad\mbox{as
$\epsilon\to 0^{+}$.}$

The next consequence of the Theorem 3.1 and the Theorem 3.4 is the weak
equivalence for the asymptotic behavior of the covering numbers of kernels
with a convergent geometric sequence of coeffcients. This equivalence is well
known for the spherical kernels with a convergent geometric sequence of
coeffcients ([13, Theorem 1.1]) and it deliveres us the exact order of the
growth of the covering numbers.

######  Corollary 3.5.

Let $K$ a continuous isotropic positive definite kernel given by the series
expansion (1.7), with $a_{k}=\delta^{k}$, for $k=1,2\ldots$, for some
$0<\delta<1$. Then,

$\ln(\mathcal{C}(\epsilon,I_{K}))\asymp[\ln(1/\epsilon)]^{d+1},\quad\mbox{as
$\epsilon\to 0^{+}$}.$

###  3.1 Estimates of spherical kernels of the Gaussian type

We close this section with the application of the Theorem 3.1 and the Theorem
3.4 to estimate the covering numbers of the RKHS associated to the spherical
kernel of the Gaussian type.

A kernel of the Gaussian type on the unit sphere $\mathbb{S}^{d}$, for $d\geq
1$, is defined by

$K_{\delta}(x,y)=\sum_{k=0}^{\infty}\lambda_{k}\tau^{d}_{k}G_{k}^{(d-2)/2}(x\cdot
y)=\sum_{k=0}^{\infty}\lambda_{k}\sum_{j=1}^{\tau^{d}_{k}}S_{k,j}(x)S_{k,j}(y),\quad
x,y\in\mathbb{S}^{d},$ (3.24)

with $\\{\lambda_{k}:=\delta^{k}\\}$ for some $0<\delta<1$.

Concerning about the estimates for the covering numbers of the RKHS of kernels
of the Gaussian type we have the following upper and lower estimates.

######  Proposition 3.6.

Let $K_{\delta}$ be a kernel of the Gaussian type represented by the series
expansion (3.24) on $\mathbb{S}^{d}$. For any $0<\delta<1/(d+3)$, if $d\geq
2$, and for any $0<\delta<1$, if $d=1$, it holds

$\frac{1}{2(d+1)!}\left[\frac{d/(d+1)}{\ln(e^{d-1}/\delta)}\right]^{d}\leq\liminf_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{[2\ln(1/\epsilon)]^{d+1}}\leq\limsup_{\epsilon\rightarrow
0^{+}}\frac{\ln\left(\mathcal{C}\left(\epsilon,I_{K}\right)\right)}{[2\ln(1/\epsilon)]^{d+1}}\leq\frac{2}{d!}\frac{1}{[{\ln\left({1}/{\delta(d+3)}\right)}]^{d}}.$

Proof. For the case $d\geq 2$, the sequence of coefficients
$\\{a_{k}=\delta^{k}\tau_{k}^{d}\\}$ fits in the assumptions of Theorem 3.1
and Theorem 3.4, for the given $0<\delta<1/(d+3)$ and $\theta=\delta(d+3)$,
respectively. In fact, the application of identity (3.18) leads us to

$\displaystyle
a_{k+1}=\delta^{k+1}\tau^{d}_{k+1}=\delta\frac{(2k+d+1)(k+d-1)}{(2k+d-1)(k+1)}a_{k},\quad
k=0,1,\ldots.$

The inequality

$1\leq\frac{(2k+d+1)(k+d-1)}{(2k+d-1)(k+1)}\leq\frac{2k+d+1}{k+1}\leq
d+3,\quad k=0,1,\ldots,$

implies that

$\displaystyle\delta a_{k}\leq a_{k+1}\leq\delta(d+3)a_{k},\quad
k=0,1,\ldots.$ (3.25)

We remember that for $\mathbb{M}^{d}=\mathbb{S}^{d}$, $d\geq 1$, we have
$\alpha=\beta=(d-2)/2$. Then, we obtain

$\frac{2^{d}d^{d}}{[\ln(1/\delta)+d-1]^{d}d!(d+1)^{d+1}}\leq\liminf_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{[\ln(1/\epsilon)]^{d+1}},$

and

$\limsup_{\epsilon\rightarrow
0^{+}}\frac{\ln\left(\mathcal{C}\left(\epsilon,I_{K}\right)\right)}{[\ln(1/\epsilon)]^{d+1}}\leq\frac{2^{d+2}}{d!}\frac{1}{[{\ln\left(1/\delta(d+3)\right)}]^{d}}.$

A simple rearrangement of the terms in the inequalities above leads us to the
inequality stated in the proposition.

For $d=1$ the sequence of coefficients $\\{a_{k}=\delta^{k}\tau_{k}^{1}\\}$
fits in the assumptions of Theorem 3.1 and Theorem 3.4, for $\theta=\delta$.
In fact, we know that $\tau_{k}^{1}=2$, for $k=1,2,\ldots$, then

$\displaystyle
a_{k+1}=\delta^{k+1}\tau^{1}_{k+1}=2\delta^{k+1}=\delta\delta^{k}\tau_{k}^{1}=\delta
a_{k},\quad k=0,1,\ldots.$ (3.26)

We remember that $\alpha=\beta=-1/2$, then we obtain

$\frac{1}{2}\,\frac{1}{\ln(1/\theta)}\leq\liminf_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{[\ln(1/\epsilon)]^{2}}\leq\limsup_{\epsilon\rightarrow
0^{+}}\frac{\ln\left(\mathcal{C}\left(\epsilon,I_{K}\right)\right)}{[\ln(1/\epsilon)]^{2}}\leq\frac{8}{{\ln\left({1}/{\theta}\right)}}\leq\frac{8}{{\ln\left({1}/{4\theta}\right)}}.$

This is the inequality stated.

##  4 Estimates of kernels with general decay of the coefficients

In this section present the bounds for the covering numbers of kernels on
$\mathbb{M}^{d}$, represented by the series expansion (1.7), with sequence of
coefficients with decay comparable to an harmonic progression.

We observe that the kernel $K$ is represented by the series (1.7) with
sequence of coefficients $\\{a_{k}\\}$ satisfying

$\limsup_{n\to\infty}a_{n}/n^{-\gamma}\leq c,$

for some $\gamma>1$ and $c>0$, then there exists $m_{0}\in\mathbb{N}$ such
that

$0<(k_{m}^{s})^{2}\leq\frac{c}{\gamma-1}{(m+1)}^{-\gamma+1},\quad
m=m_{0},m_{0}+1,\ldots,$ (4.27)

with $\kappa_{m}^{s}$ as introduced in (3.10). Supposing only $K$ given as
(1.7) on $\mathbb{M}^{d}$ with summable sequence of coefficients
$\\{a_{k}\\}$, then $a_{k}/\tau_{k}^{d}=O(k^{-d})$ (as on the spherical
setting [13, Section 5]) and due the formula (2.8), it holds that
$a_{k}=O(k^{-1})$. We consider this basic decay for our assumption in the next
result.

The next theorem extends Proposition 3.3 of the reference [13]. The proofs are
similar to the previous case (Section 3) and we will short the presentation
for the similar details.

######  Theorem 4.1.

Let $K$ a continuous isotropic positive definite kernel given by the series
expansion (1.7). If there exists $\gamma>0$ and $c_{1}>0$ such that
$a_{k}/k^{-d-\gamma}\leq c_{1}$, for $k=n_{0},n_{0}+1,\ldots$, then

$\limsup_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{(1/\epsilon)^{2d/(\gamma+d-1)}\ln(1/\epsilon)}\leq\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\left(\frac{4c_{1}}{\gamma+d-1}\right)^{d/(\gamma+d-1)}.$

Proof. For $\epsilon>0$, we consider $m:=m(\epsilon)\in\mathbb{N}$ such that

$\left[\frac{c_{1}}{\gamma+d-1}(m+1)^{-\gamma-d+1}\right]^{1/2}<\frac{\epsilon}{2}<\left[\frac{c_{1}}{\gamma+d-1}m^{-\gamma-d+1}\right]^{1/2}.$
(4.28)

In this case, we have

$m(\epsilon)\approx\left(\frac{4c_{1}}{\gamma+d-1}\right)^{1/(\gamma+d-1)}\left(\frac{1}{\epsilon}\right)^{2/(\gamma+d-1)},\quad\mbox{as
$\epsilon\to 0^{+}.$}$ (4.29)

Repeating exactly the same steps in the proof of Theorem 3.1, we obtain

$\mathcal{C}(\epsilon,I_{K})\leq\dim{\mathcal{V}_{m}}\ln\left(8\kappa/\epsilon\right)\approx\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}m^{d}\ln\left(8\kappa/\epsilon\right),$

as $m\to\infty$. With the application of the strong equivalences (4.29), we
obtain

$\limsup_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{(1/\epsilon)^{2d/(\gamma+d-1)}\ln(1/\epsilon)}\leq\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\left(\frac{4\,c_{1}}{\gamma+d-1}\right)^{d/(\gamma+d-1)}.$

The next result is the extension of the Proposition 3.4 of the reference [13].

######  Theorem 4.2.

Let $K$ a continuous isotropic positive definite kernel given by the series
expansion (1.7). If there exist $\rho>1$ and $c_{2}>0$ such that
$a_{k}/k^{-\rho}\geq c_{2}$, for $k=1,2,\ldots$, then

$\ln{\sqrt{a_{0}}}+\frac{e^{-1}(\rho+d-1)\Gamma(\beta+1)}{d\,\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\left(\frac{c_{2}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}{3\Gamma(\beta+1)}\right)^{d/(\rho+d-1)}\leq\liminf_{\epsilon\to
0}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{(1/\epsilon)^{d/2(\rho+d-1)}}.$

Proof. Following the same steps of the proof of Proposition 3.4, from formula
(3.20) to formula (3.21), for any $\epsilon>0$, we attain the following lower
bound

$J_{m}:=\frac{1}{2}\sum_{k=0}^{m}\tau_{k}^{d}\ln\left(a_{k}/\tau_{k}^{d}\right)-\dim\mathcal{V}_{m}\ln\epsilon\leq\ln(\mathcal{C}(\epsilon,I_{K})),\quad
m=1,2,\ldots.$ (4.30)

For $m=1,2,\ldots$, we write

$I_{m}:=\frac{1}{2}\left(\ln a_{0}+\sum_{k=1}^{m}\tau_{k}^{d}\ln
a_{k}+\sum_{k=1}^{m}\tau_{k}^{d}\ln\left(1/\tau_{k}^{d}\right)\right)=\frac{1}{2}\sum_{k=0}^{m}\tau_{k}^{d}\ln\left(a_{k}/\tau_{k}^{d}\right),$
(4.31)

and if $a_{0}\geq 1$ we observe that, for $m=1,2,\ldots$,

$\displaystyle I_{m}$ $\displaystyle\geq$ $\displaystyle\frac{1}{2}\left(\ln
a_{0}+(\dim\mathcal{V}_{m}-1)\ln c_{2}-\rho(\dim\mathcal{V}_{m}-1)\ln
m+\sum_{k=1}^{m}\tau_{k}^{d}\ln(1/\tau_{k}^{d})\right).$

Since $\tau_{k}^{d}\leq\tau_{m}^{d}$ for $k=1,2\ldots,m$ we have
$\ln(1/\tau_{k}^{d})\geq\ln(1/\tau_{m}^{d}),$ and, consequently

$\displaystyle I_{m}$ $\displaystyle\geq$ $\displaystyle\frac{1}{2}\left[\ln
a_{0}+(\dim\mathcal{V}_{m}-1)\ln c_{2}-\rho(\dim\mathcal{V}_{m}-1)\ln
m+(\dim\mathcal{V}_{m}-1)\ln(1/\tau_{m}^{d})\right]$ $\displaystyle=$
$\displaystyle\ln\sqrt{a_{0}}+\frac{(\dim\mathcal{V}_{m}-1)}{2}\left[\ln
c_{2}-\rho\ln m+\ln(1/\tau_{m}^{d})\right].$

From inequality (4.30), we have

$J_{m}=I_{m}-\dim\mathcal{V}_{m}\ln(\epsilon)\leq\ln(\mathcal{C}(\epsilon,I_{K})),\quad
m=1,2,\ldots.$ (4.32)

An application of Proposition 2.3 implies

$\frac{\dim\mathcal{V}_{m}-1}{2}<\frac{\Gamma(\beta+1)m^{d}}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)},\quad\quad-\dim\mathcal{V}_{m}<-\frac{\Gamma(\beta+1)m^{d}}{2\,\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)},$

and

$\ln\left(\frac{1}{\tau_{m}^{d}}\right)\geq\ln\left(\frac{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}{3\Gamma(\beta+1)m^{d-1}}\right),$

for $m=m_{0},m_{0}+1,\ldots.$ Therefore,

$\ln\sqrt{a_{0}}+\frac{\Gamma(\beta+1)m^{d}}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\left[\ln\left(\frac{c_{2}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}{3\Gamma(\beta+1)\,\epsilon^{1/2}}\right)-(\rho+d-1)\ln{m}\right]$
(4.33)

is less than $J_{m}$ for $m=m_{0},m_{0}+1,\ldots$.

In order to proceed analogously to the final steps of the proof of the Theorem
3.4, we consider $\epsilon>0$ and $m=m(\epsilon)$ such that
$1/(m+1)\leq\epsilon<1/m$, and

$\varphi_{\epsilon}(m):=m^{d}\left[\ln\left(\frac{c_{2}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}{3\Gamma(\beta+1)\,\epsilon^{1/2}}\right)-(\rho+d-1)\ln
m\right].$

Due inequality (4.32), it holds

$\ln{\sqrt{a_{0}}}+\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\,\varphi_{\epsilon}(m)<\ln(\mathcal{C}(\epsilon,I_{K})),\quad
m=m_{0},m_{0}+1,\ldots,$ (4.34)

We observe that from inequality (4.34), we will have

$\ln{\sqrt{a_{0}}}+\frac{\Gamma(\beta+1)}{\Gamma(\alpha+2)\Gamma(\alpha+\beta+2)}\,\varphi_{\epsilon}(c)<\ln(\mathcal{C}(\epsilon,I_{K})),$
(4.35)

for the critical point $c$ of $\varphi_{\epsilon}$. In this case the critical
point is

$c=\left(\frac{c_{2}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}{3\Gamma(\beta+1)\,\epsilon^{1/2}}\right)^{1/(\rho+d-1)}e^{-1/d},$

and we have

$\displaystyle\varphi_{\epsilon}(c)=\frac{\rho+d-1}{d\,e}\left(\frac{c_{2}\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}{3\Gamma(\beta+1)}\right)^{d/(\rho+d-1)}\left(\frac{1}{\epsilon}\right)^{d/2(\rho+d-1)},$

and the estimate follows.

We close the paper by presenting an example of kernel fulfilling the
assumptions of the theorems of this section. In what follows, for
$\alpha,\beta>-1$ will write

$c(\alpha,\beta):=\frac{\Gamma(\beta+1)}{\Gamma(\alpha+1)\Gamma(\alpha+\beta+2)}.$

We observe that $P_{0}^{\alpha,\beta}(s)=P_{0}^{\alpha,\beta}(1)=1$ implies
that $J_{0}^{\alpha,\beta}(s)=1$. Then, a kernel $K$ that is an isotropic
positive definite kernel given by the series expansion (1.7) with $a_{0}=1$,
is written as

$K(x,y)=1+\sum_{k=1}^{\infty}a_{k}J_{k}^{(\alpha,\beta)}(\cos(d(x,y))),\quad
x,y\in\mathbb{M}^{d}.$

######  Example 4.3.

Let $\gamma>0$ and $K$ be the isotropic positive definite kernel given by the
series expansion

$K(x,y)=1+\sum_{k=1}^{\infty}k^{-d-\gamma}J_{k}^{(\alpha,\beta)}(\cos(d(x,y))),\quad
x,y\in\mathbb{M}^{d},$ (4.36)

Then,

$\displaystyle\limsup_{\epsilon\to
0^{+}}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{(1/\epsilon)^{2d/(d+\gamma-1)}\ln(1/\epsilon)}\leq\frac{c(\alpha,\beta)}{(\alpha+1)}\frac{4^{d/(\gamma+d-1)}}{(\gamma+d-1)^{d/(\gamma+d-1)}}.$

and

$\displaystyle\frac{(\gamma+2d-1)}{ed[3c(\alpha,\beta)]^{d/(\gamma+2d-1)}}\frac{c(\alpha,\beta)}{(\alpha+1)}\leq\liminf_{\epsilon\to
0}\frac{\ln(\mathcal{C}(\epsilon,I_{K}))}{(1/\epsilon)^{d/2(2d+\gamma-1)}},$

In particular, if $\mathbb{M}^{d}=\mathbb{S}^{d}$, then

$\displaystyle\limsup_{\epsilon\to
0^{+}}\frac{d!\ln(\mathcal{C}(\epsilon,I_{K}))}{2(1/\epsilon)^{2d/(d+\gamma-1)}\ln(1/\epsilon)}\leq\left(\frac{4}{\gamma+d-1}\right)^{d/(\gamma+d-1)},$

and

$\displaystyle\frac{(\gamma+2d-1)}{ed}\left(\frac{(d-1)!}{3}\right)^{d/(2d+\gamma-1)}$
$\displaystyle\leq$ $\displaystyle\liminf_{\epsilon\to
0}\frac{d!\ln(\mathcal{C}(\epsilon,I_{K}))}{2(1/\epsilon)^{d/2(2d+\gamma-1)}}.$

In fact, the first inequality is a an direct application of Theorem 4.1 and
the second stated inequality follows from the direct application of Theorem
4.2, for $\rho=\gamma+d$. For the particular case
$\mathbb{M}^{d}=\mathbb{S}^{d}$, we have

$c(\alpha,\beta)=c((d-2)/2,(d-2)/2)=\frac{1}{(d-1)!}\quad\mbox{and}\quad\frac{c(\alpha,\beta)}{(\alpha+1)}=\frac{2}{d!},$

and the estimate follows.

An interesting theoretical question related to bound covering numbers is to
decide if the asymptotic limit $\lim_{\epsilon\to
0^{+}}\ln(\mathcal{C}(\epsilon,I_{K}))/\varphi(\epsilon)$, exists for some
function $\varphi:(0,\infty)\rightarrow\mathbb{R}$. In the affirmative case,
the asymptotic limit deliveries us the best bounds for the numbers, and
reciprocally precise bounds for the covering numbers could deliver the
existence of the asymptotic limit. The estimates we present in Theorem 3.4 and
in Theorem 3.1 are accurate but we are not able to ensure that the asymptotic
limit exists from the bounds even on the spherical setting (Subsection 3.1).
The theme is explored in [18] for approximation numbers of infinitely many
times differentiable functions on $\mathbb{R}^{d}$ satisfying a size condition
on the order of the derivatives on the compacts.

Acknowledgements. This research is supported by the São Paulo Research
Foundation (FAPESP) grant no. 2022/11032-0 and was financed in part by the
Higher Education and Science Committee of the Republic of Armenia grant no.
23RL-1A027.

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Karina Gonzalez
Faculty of Mathematics and Mechanics
Yerevan State University
0025 Yerevan, Republic of Armenia
Email<EMAIL_ADDRESS>
Thaís Jordão
Departament of Mathematics
Universidade de São Paulo
13566-590 São Paulo, Brazil
Email<EMAIL_ADDRESS>

Keywords: covering numbers; positive definite kernel, Reproducing Hilbert
Space; isotropic kernel

MSC: 47B06; 42C10; 46E22; 41A60