#  Optimal quadrature errors and sampling numbers for Sobolev spaces with
logarithmic perturbation on spheres

Jiaxin Geng  School of Mathematical Sciences, Capital Normal University,
Beijing 100048, China<EMAIL_ADDRESS>,  Yun Ling  School of
Mathematical Sciences, Capital Normal University, Beijing 100048, China.
<EMAIL_ADDRESS>,  Jiansong Li  School of Mathematical Sciences, Capital
Normal University, Beijing 100048, China<EMAIL_ADDRESS>and  Heping
Wang  School of Mathematical Sciences, Capital Normal University, Beijing
100048, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study optimal quadrature errors, approximation numbers, and
sampling numbers in $L_{2}({\mathbb{S}}^{d})$ for Sobolev spaces ${\rm
H}^{\alpha,\beta}({\mathbb{S}}^{d})$ with logarithmic perturbation on the unit
sphere ${\mathbb{S}}^{d}$ in $\mathbb{R}^{d+1}$. First we obtain strong
equivalences of the approximation numbers for ${\rm
H}^{\alpha,\beta}({\mathbb{S}}^{d})$ with $\alpha>0$, which gives a clue to
Open problem 3 as posed by Krieg and Vybíral in [27]. Second, for the optimal
quadrature errors for ${\rm H}^{\alpha,\beta}({\mathbb{S}}^{d})$, we use the
“fooling” function technique to get lower bounds in the case $\alpha>d/2$, and
apply Hilbert space structure and Vybíral’s theorem about Schur product theory
to obtain lower bounds in the case $\alpha=d/2,\,\beta>1/2$ of small
smoothness, which confirms the conjecture as posed by Grabner and Stepanyukin
in [14] and solves Open problem 2 in [27]. Finally, we employ the weighted
least squares operators and the least squares quadrature rules to obtain
approximation theorems and quadrature errors for ${\rm
H}^{\alpha,\beta}({\mathbb{S}}^{d})$ with $\alpha>d/2$ or
$\alpha=d/2,\,\beta>1/2$, which are order optimal.

.

###### Key words and phrases:

Worst-case error, Numerical integration, Quadrature formulas, Reproducing
kernel Hilbert spaces

###### 2010 Mathematics Subject Classification:

41A63; 65C05; 65D15; 65Y20

##  1\. Introduction and main results

For $d\geq 2$, denote ${S}^{d}=\\{{\bf x}\in{R}^{d+1}|\,\|{\bf
x}\|=(x_{1}^{2}+\cdots+x_{d+1}^{2})^{1/2}=1\\}$ by the unit sphere. The
integral of a continuous function $f:{{S}}^{d}\to{R}$ is denoted by

${\rm INT}(f):=\int_{{{S}}^{d}}f({\bf{x}})\,d\sigma_{d}({\bf x}),$

where $\sigma_{d}(\bf x)$ is the normalized surface measure on ${{S}}^{d}$
(i.e., $\sigma_{d}({{S}}^{d})=1$). Denote by $L_{p}\equiv L_{p}({{S}}^{d})$
the usual Lebesgue space with norm $\|\cdot\|_{L_{p}}$. Then $L_{2}$ is a
Hilbert space with inner product

$\langle f,g\rangle=\int_{{{S}}^{d}}f({\bf x})g({\bf x})d\sigma_{d}({\bf x}).$

Let $H$ be a reproducing kernel Hilbert space on ${{S}}^{d}$ with norm
$\|\cdot\|_{H}$. We want to approximate ${\rm INT}(f)$ from $f\in H$ using
quadrature formulas

$Q_{{\bf c},X_{n}}=\sum_{j=1}^{n}c_{j}f({\bf x}_{j})$

with weights ${\bf c}\in{{R}}^{n}$ and knots $X_{n}=\\{{{\bf x}_{1}},{{\bf
x}_{2}},\cdot\cdot\cdot,{{\bf x}_{n}}\\}\subset{{S}}^{d}$.

The worst case error of $Q_{c,X_{n}}$ is defined by

$e(Q_{{\bf c},X_{n}},H,{\rm INT}):=\sup_{\|f\|_{H}\leq 1}|{\rm INT}(f)-Q_{{\bf
c},X_{n}}(f)|,$

and the $n$-th optimal quadrature error

$e_{n}(H,{\rm INT}):=\inf_{\begin{subarray}{c}{\bf c}\in{R}^{n}\\\
X_{n}\subset{S}^{d}\end{subarray}}e(Q_{{\bf c},X_{n}},H,{\rm
INT})=\inf_{\begin{subarray}{c}{\bf c}\in{R}^{n}\\\
X_{n}\subset{S}^{d}\end{subarray}}\sup_{\|f\|_{H}\leq 1}|{\rm INT}(f)-Q_{{\bf
c},X_{n}}(f)|.$

We write

$e_{n}(H)=e_{n}(H,{\rm INT}).$

We are also interested in approximation numbers and (linear) sampling numbers
in $L_{2}$. We define the approximation numbers $a_{n}(H)$ by

$a_{n}(H)\equiv a_{n}(H,L_{2}):=\inf_{\begin{subarray}{c}L_{1},...,L_{n}\in
H^{{}^{\prime}}\\\ g_{1},...,g_{n}\in L_{2}\end{subarray}}\sup_{\|f\|_{H}\leq
1}\|f-\sum_{i=1}^{n}L_{i}({f})g_{i}\|_{L_{2}},$

and the (linear) sampling numbers $g_{n}(H)$ by

$g_{n}(H)\equiv g_{n}^{\rm lin}(H,L_{2}):=\inf_{\begin{subarray}{c}{\bf
x}_{i},...,{\bf x}_{n}\in{{S}}^{d}\\\ g_{1},...,g_{n}\in
L_{2}\end{subarray}}\sup_{\|f\|_{H}\leq 1}\|f-\sum_{i=1}^{n}f({\bf
x}_{i})g_{i}\|_{L_{2}},$

where $H^{\prime}$ denotes the conjugate space of $H$ consisting of all
continuous linear functionals on $H$. The approximation numbers and the
(linear) sampling numbers reflect the errors of the best possible linear
algorithms for $L_{2}$ approximation using at most $n$ continuous linear
functionals on $H$ or $n$ function values, respectively. Clearly, we have

(1.1) $a_{n}(H)\leq g_{n}(H)\ \ \ {\rm and}\ \ \ e_{n}(H)\leq g_{n}(H).$

By (1.1) it suffices to discuss lower bounds for $e_{n}(H)$ and upper bounds
for $g_{n}(H)$.

This paper is devoted to discussing the optimal quadrature error and the
sampling numbers for the Sobolev spaces ${\rm H}^{\alpha,\beta}({{S}}^{d})$
with logarithmic perturbation (see the definition in Section 2). Note that
${\rm H}^{\alpha,\beta}({{S}}^{d})$ is a reproducing kernel Hilbert space if
and only if $\alpha>d/2,\beta\in{R}$ or $\alpha=d/2,\beta>1/2$. Moreover, this
space is just the Besov space $B_{2,2}^{\Omega}({{S}}^{d})$ (see [46, 47]) of
generalized smoothness with equivalent norms, where
$\Omega(t)=t^{\alpha}(1+(\ln\frac{1}{t})_{+})^{\beta}$ for $t>0$,
$a_{+}=\max\\{a,0\\}$, see Subsection 2.1.

For the classical Sobolev spaces ${\rm H}^{\alpha}({{S}}^{d})={\rm
H}^{\alpha,0}({{S}}^{d})$ on ${{S}}^{d}$, upper and lower bounds for the
optimal quadrature error were obtained in [3] and [17], asymptotics and strong
equivalences of the approximation numbers in [23] and [5], and asymptotics of
the sampling numbers in [47]. For the Sobolev spaces ${\rm
H}^{\alpha,\beta}({{S}}^{d})$ with logarithmic perturbation, asymptotics of
the approximation numbers were obtained in [46] for $\alpha>0$, asymptotics of
the sampling numbers in [47] for $\alpha>d/2$. Specifically, it was shown in
[46] that for $\alpha>0$,

(1.2) $a_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))\asymp
n^{-\alpha/d}\ln^{-\beta}n,$

and in [47] that for $\alpha>d/2$,

(1.3) $g_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))\asymp
n^{-\alpha/d}\ln^{-\beta}n,$

where for positive sequences $\\{a_{n}\\},\\{b_{n}\\}$, $a_{n}\lesssim b_{n}$
means that there is a constant $c>0$ independent of $n$ such that $a_{n}\leq
cb_{n}$ for all $n\geq 2$. We write $a_{n}\asymp b_{n}$ if $b_{n}\lesssim
a_{n}$ and $a_{n}\lesssim b_{n}$.

In the case $\alpha=d/2,\,\beta>1/2$ of small smoothness, Grabner and
Stepanyuk obtained in [14], among others, upper and lower bounds for the
optimal quadrature errors:

$n^{-1/2}\ln^{-\beta}n\lesssim e_{n}({\rm H}^{d/2,\beta}({{S}}^{d}))\lesssim
n^{-1/2}\ln^{-\beta+1/2}n.$

However, there is a logarithmic gap between the upper and lower bounds. They
conjectured in [14] that the correct order for $e_{n}({\rm
H}^{d/2,\beta}({{S}}^{d}))$ is just the upper bound. Also Krieg and Vybíral
posed Open problem 2 in [27], which stated that the existing lower bound can
be improved.

There has been an increased interest in the comparison of standard information
given by function values and general linear information for $L_{2}$
approximation problem. We refer to [9, 22, 25, 26, 34, 41] for recent upper
bounds and to [19, 20] for lower bounds. Specifically, applying random
information, weighted least squares method, and infinite-dimensional variant
of subsampling strategy, the authors in [9] obtained that there are two
absolute constants $c$, $C>0$, such that for every reproducing kernel Hilbert
space $H$ and every $n\in{N}$,

(1.4) $g_{n}(H)\leq C\Big{(}\sum_{k\geq cn}a_{n}(H)^{2}\Big{)}^{1/2}.$

By (1.2) and (1.4) we have for $\beta>1/2$,

(1.5) $g_{n}({\rm H}^{d/2,\beta}({{S}}^{d}))\lesssim
n^{-1/2}\ln^{-\beta+1/2}n.$

This give the upper bound for $g_{n}({\rm H}^{d/2,\beta}({{S}}^{d}))$.
However, the algorithm used in [9] is not constructive (see [9, Remark 7]).

For constructive algorithms, it was shown in [47] that the filtered
hyperinterpolation operators (see the definition in [47]) are asymptotically
optimal linear algorithms for $g_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$ with
$\alpha>d/2$. Lu and Wang showed in [28] that the weighted least squares
operators (see Subsection 2.3) are asymptotically optimal linear algorithms
for $g_{n}({\rm H}^{\alpha}({{S}}^{d}))$ with $\alpha>d/2$.

In this paper, we investigate the approximation numbers, the sampling numbers,
and the optimal quadrature error for the Sobolev spaces ${\rm
H}^{\alpha,\beta}({{S}}^{d}),\,\alpha>0$ with generalized smoothness. We make
special attention to the case $\alpha=d/2,\,\beta>1/2$.

First, we give strong equivalences of the approximation numbers $a_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d})),\ \alpha>0$ which generalized the result of [5,
Theorem 3.1]. Such result gives a clue to Open problem 3 posed in [27].

######  Theorem 1.1.

Let $\alpha>0,\beta\in{R}$. Then

(1.6) $\lim_{n\to\infty}n^{\alpha/d}(\ln n)^{\beta}a_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))=\Big{(}\frac{2}{d\,!}\Big{)}^{\alpha/d}d^{\beta}.$

######  Remark 1.

If $\beta=0$, then Theorem 1.1 reduces to [5, Theorem 3.1]. One can rephrase
(1.6) as strong equivalences

$a_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))\sim\Big{(}\frac{2}{d\,!}\Big{)}^{\alpha/d}d^{\beta}n^{-\alpha}(\ln
n)^{-\beta},$

as $n\to\infty$ for arbitrary fixed $d$ and $\alpha>0,\beta\in{R}$. The
novelty of Theorem 1.1 is that it gives strong equivalences of $a_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))$ and provides asymptotically optimal constants,
for arbitrary fixed $d$ and $\alpha>0,\beta\in{R}$.

Next, we discuss lower bounds for $e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$
with $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$. In the case of $\alpha>d/2$, we
can use the “fooling” function technique, i.e., for any quadrature rule
$Q_{{\bf c},X_{n}}$, we can construct a fooling function $f$ which vanishes in
all nodes $X_{n}$, and satisfies

$\|f\|_{H^{\alpha,\beta}({{S}}^{d})}\lesssim 1,\ \ \ \ \int_{{{S}}^{d}}f({\bf
x})d\sigma_{d}({\bf x})\gtrsim n^{-\alpha/d}\ln^{-\beta}n.$

See [8, 14, 44] for the details of this technique. We use such technique in
[8] to obtain the following theorem.

######  Theorem 1.2.

Let $\alpha>d/2$. Then

(1.7) $e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))\gtrsim
n^{-\alpha/d}\ln^{-\beta}n.$

However, in the case of $\alpha=d/2,\,\beta>1/2$, the “fooling” function
technique does not give optimal lower bounds (see [27]). We exploit the
Hilbert space structure and a variant of the Schur product theory to give
optimal lower bounds for $e_{n}({\rm H}^{d/2,\beta}({{S}}^{d}))$ with
$\beta>1/2$. Such technique was used in [19, 20, 27] on the $d$ dimensional
torus ${T}^{d}$. It is very surprising that the above technique seems not to
give optimal lower bounds for $e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$ with
$\alpha>d/2,\ d\geq 2$.

######  Theorem 1.3.

Let $\alpha=d/2,\,\beta>1/2$. Then

(1.8) $e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))\gtrsim
n^{-1/2}\ln^{-\beta+1/2}n.$

By (1.1), (1.7), (1.8), (1.3), and (1.5) we have the following corollary.

######  Corollary 1.4.

Let $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$. Then

(1.11) $\displaystyle e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))\asymp
g_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))\asymp\bigg{\\{}\begin{array}[]{ll}n^{-\alpha/d}\ln^{-\beta}n,&\alpha>d/2,\\\
n^{-1/2}\ln^{-\beta+1/2}n,&\alpha=d/2,\beta>1/2.\end{array}$

######  Remark 2.

By Corollary 1.4 and (1.2) we obtain for $\alpha=d/2,\,\beta>1/2$,

(1.12) $\displaystyle n^{-1/2}\ln^{-\beta+1/2}n\asymp e_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))\asymp g_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))\asymp a_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))\,\ln^{1/2}n.$

This confirms the conjecture in [14] and solves Open problem 2 posed in [27].
Of special interest is the logarithmic gap between the sampling numbers and
the approximation numbers, which has been shown in [20] on the torus ${T}$, in
[27] on the $d$-dimensional torus ${T}^{d}$, and is now extended to the sphere
${{S}}^{d}$.

Finally, we give constructive algorithms to obtain upper bounds for
$g_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$ with $\alpha>d/2$ or
$\alpha=d/2,\,\beta>1/2$. The algorithms are the weighted least squares
algorithms investigated in [15, 28].

An $L_{2}$ Marcinkiewicz-Zygmund (MZ) family $\mathcal{X}$ on a compact space
$\mathcal{M}$ is a double-indexed set of points $\\{{\bf x}_{n,k},\
n=1,2\dots,\,k=1,2,\dots,l_{n}\\}$ in $\mathcal{M}$ with a weight $\tau$ of
positive numbers $\\{\tau_{n,k}\\}$ such that the sampled $\ell_{2}$ norm of
the $n$-th layer $\left(\sum_{k=1}^{l_{n}}\tau_{n,k}|Q({\bf
x}_{n,k})|^{2}\right)^{1/2}$ is equivalent to the $L_{2}$ norm for all
“polynomials” $Q$ of degree at most $n$ with uniform constants. Given an
$L_{2}$ MZ family on $\mathcal{M}$, Gröchenig in [15] introduced weighted
least squares operators and least squares quadrature rules, and derived
approximation theorems and least squares quadrature errors for Sobolev spaces.
However, due to the generality of the compact space $\mathcal{M}$, the
obtained results are not optimal. Lu and Wang in [28] proved that the weighted
least squares operators $L_{N}^{{S}}$ and the least squares quadrature rules
$I_{N}^{{S}}$ on ${{S}}^{d}$ are asymptotical optimal for $g_{n}({\rm
H}^{\alpha}({{S}}^{d}))$ and $e_{n}({\rm H}^{\alpha}({{S}}^{d}))$ with
$\alpha>d/2$.

The following theorem shows that the weighted least squares operators
$L_{N}^{{S}}$ and the least squares quadrature rules $I_{N}^{{S}}$ on
${{S}}^{d}$ are asymptotical optimal for $g_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))$ and $e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$
with $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$.

######  Theorem 1.5.

Let $\mathcal{X}$ be an $L_{2}$ MZ family on ${{S}}^{d}$ with associated
weight $\tau$, global condition number $\kappa$, and $l_{N}\asymp N^{d}\asymp
n$, and let $L_{N}^{{S}}$ be the weighted least squares operator and
$I_{N}^{{S}}$ be least squares quadrature rule induced by $\mathcal{X}$. We
have for all $f\in{\rm H}^{\alpha,\beta}({{S}}^{d})$ with $\alpha>d/2$ or
$\alpha=d/2,\beta>1/2$,

(1.13) $\left|\int_{{S}^{d}}f({\bf x}){\rm d}\sigma_{d}({\bf
x})-I_{N}^{{S}}(f)\right|\leq\|f-L_{N}^{{S}}(f)\|_{L_{2}}\leq
c(1+\kappa^{1/2})C_{\alpha,\beta}(N)\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})},$

where $c>0$ is independent of $n$, $\kappa$, $f$, or $\mathcal{X}$, and

$C_{\alpha,\beta}(N):=\bigg{\\{}\begin{array}[]{ll}N^{-\alpha}\ln^{-\beta}N,&\alpha>d/2,\\\
N^{-\alpha}\ln^{-\beta+1/2}N,&\alpha=d/2,\beta>1/2.\end{array}$

######  Remark 3.

Let $\mathcal{X}$ be an $L_{2}$ MZ family on ${{S}}^{d}$ with $l_{N}\leq n$
and $l_{N}\asymp N^{d}\asymp n$. For such $L_{2}$ MZ family it follows from
(1.11) and (1.13) that

$g_{n}({{\rm H}^{\alpha,\beta}({{S}}^{d})})\leq\sup_{\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})}\leq 1}\|f-L_{N}^{{S}}(f)\|_{2}\asymp g_{n}({{\rm
H}^{\alpha,\beta}({{S}}^{d})})\asymp C_{\alpha,\beta}(N),$

which implies that the weighted least squares operators $L_{N}^{{S}}$ are
order optimal linear algorithms in the sense of the sampling numbers. Also, we
have

$e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))\leq\sup_{\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})}\leq 1}\left|\int_{\mathbb{S}^{d}}f({\bf
x})\,d\sigma_{d}({\bf x})-I_{N}^{{S}}(f)\right|\asymp e_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))\asymp C_{\alpha,\beta}(N),$

which means that the least squares quadrature rules $I_{N}^{{S}}$ are the
order optimal quadrature formulas.

The paper is structured as follows. In Subsection 2.1 we introduce Sobolev
spaces $H^{\alpha,\beta}$ with logarithmic perturbation. Then we delve into
lower bounds for optimal quadrature and positive definiteness and present some
auxiliary lemmas in Subsection 2.2. Definition and properties of the weighted
least squares operators are given in Subsection 2.3. After that, in Section 3
we give the proofs of Theorems 1.1 and 1.2. Section 4 is devoted to
elaborating the lower bounds for $e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$
with $\alpha=d/2,\,\beta>1/2$. We give the proof of Theorems 1.5 in Section 5.

##  2\. Preliminaries

###  2.1. Sobolev spaces ${\rm H}^{\alpha,\beta}({{S}}^{d})$ with logarithmic
perturbation

We denote by $\mathcal{H}_{\ell}^{d}$ the space of all spherical harmonics of
degree $\ell$ on ${{S}}^{d}$. Denote by $\Pi^{d+1}_{m}$ the set of spherical
polynomials on $\mathbb{S}^{d}$ of degree $\leq m$, which is just the set of
polynomials of total degree $\leq m$ on $\mathbb{R}^{d+1}$ restricted to
$\mathbb{S}^{d}$. It is well known (see [7, Corollaries 1.1.4 and 1.1.5]) that
the dimension of $\mathcal{H}_{\ell}^{d}$ is

(2.1) $N(d,\ell):={\rm
dim}\,\mathcal{H}_{\ell}^{d}=\left\\{\begin{array}[]{cl}1,&{\rm if}\ \
\ell=0,\\\ \frac{(2\ell+d-1)\,(\ell+d-2)!}{(d-1)!\ \ell!},&{\rm if}\ \
\ell=1,2,\dots,\end{array}\right.$

and

(2.2) $C(d,m):={\rm dim}\,\Pi_{m}^{d+1}=\frac{(2m+d)(m+d-1)!}{m!\,d!},\
m\in{N}.$

It is evident that

$N(d,\ell)\asymp\ell^{d-1}\ \ \ {\rm and}\ \ C(d,m)\asymp m^{d}.$

Let

$\\{Y_{\ell,k}\ |\ k=1,\dots,N(d,\ell)\\}$

be a collection of $L_{2}$-orthonormal real spherical harmonics of degree
$\ell$ (see, e.g., [33]). Each spherical harmonic $Y_{\ell,k}$ of exact degree
$\ell$ is an eigenfunction of the negative Laplace-Beltrami opertor
$-\Delta_{0}^{*}$ with eigenvalue

$\mathaccent 866{\lambda_{\ell}}=\ell(\ell+d-1).$

By the addition formula for spherical harmonic of exact degree $\ell$, we have
for ${\bf x},{\bf y}\in{{S}}^{d},$

(2.3) $\sum_{k=1}^{N(d,\ell)}Y_{\ell,k}({\bf x})Y_{\ell,k}({\bf
y})=\frac{n+\frac{d-1}{2}}{\frac{d-1}{2}}C^{\frac{d-1}{2}}_{n}({\bf
x}\cdot{\bf y})=N(d,\ell)G_{\ell}({\bf x}\cdot{\bf y}),$

where $C_{n}^{\frac{d-1}{2}}(t),\ n\geq 0,\,t\in[-1,1]$ are the usual
Gegenbauer polynomials defined as in [11, pp. 18-20 and Proposition 1.4.11],
and $G_{n},\,n=0,1,\dots,$ are the Gegenbauer polynomials of the index
$\frac{d-1}{2}$ normalized by

(2.4) $G_{n}(1)=1,\ \ \
C_{n}^{\frac{d-1}{2}}(1)\,G_{n}(t)=C_{n}^{\frac{d-1}{2}}(t).$

We fix $G_{n}$ in the sequel.

Let

$\\{Y_{\ell,k}\ |\ k=1,\dots,N(d,\ell)\\}$

be a fixed orthonormal basis for $\mathcal{H}_{\ell}^{d}$. Then

$\\{Y_{\ell,k}\ |\ \ell=0,1,2,\dots,\ k=1,\dots,N(d,\ell)\\}$

is an orthonormal basis for the Hilbert space $L_{2}({{S}}^{d})$. Thus any
$f\in L_{2}({{S}}^{d})$ can be expressed by its Fourier (or Laplace) series

$f=\sum_{\ell=0}^{\infty}H^{d}_{\ell}(f)=\sum_{\ell=0}^{\infty}{\sum_{k=1}^{N(d,\ell)}\langle
f,Y_{\ell,k}\rangle Y_{\ell,k}},$

where

$H_{\ell}^{d}(f)({\bf x})=\sum\limits_{k=1}^{N(d,\ell)}\langle
f,Y_{\ell,k}\rangle Y_{\ell,k}({\bf x})=N(d,\ell)\int_{{{S}}^{d}}f({\bf
y})G_{\ell}({\bf x}\cdot{\bf y})d\sigma_{d}({\bf y})$

denote the orthogonal projections of $f$ onto $\mathcal{H}_{\ell}^{d}$, and

$\langle f,Y_{\ell,k}\rangle=\int_{{{S}}^{d}}f({\bf x})Y_{\ell,k}({\bf
x})\,d\sigma_{d}({\bf x})$

are the Fourier coefficients of $f$. We also have the following Parseval
equality,

$\|f\|_{L_{2}}=\Big{(}\sum_{\ell=0}^{\infty}\sum_{k=1}^{N(d,\ell)}|\langle
f,Y_{\ell,k}\rangle|^{2}\Big{)}^{1/2}=\Big{(}\sum_{\ell=0}^{\infty}\|H_{\ell}^{d}(f)\|_{L_{2}}^{2}\Big{)}^{1/2}.$

Let $S_{n}$ denote the orthogonal projection onto $\Pi_{n}^{d+1}$. Then for
$f\in L_{1}({{S}}^{d})$,

(2.5) $S_{n}(f)=\sum_{j=0}^{n}H_{j}^{d}(f)=\int_{{{S}}^{d}}f({\bf
y})E_{n}({\bf x},{\bf y})d\sigma_{d}({\bf y}),$

where

$E_{n}({\bf x},{\bf y})=\sum_{\ell=0}^{n}\sum_{k=1}^{N(d,\ell)}Y_{\ell,k}({\bf
x})Y_{\ell,k}({\bf y})$

is the reproducing kernel of $\Pi_{n}^{d+1}$ with respect to the inner product
of $L_{2}$.

For $f\in L_{p},\ 1\leq p\leq\infty,$ we define

$E_{n}(f)_{L_{p}}=\inf_{g\in\Pi_{n}^{d+1}}\|f-g\|_{L_{p}}.$

It is well known that

(2.6) $E_{n}(f)_{L_{2}}=\|f-S_{n}(f)\|_{L_{2}}.$

Let $\Lambda=\\{\lambda_{\ell}\\}_{\ell=0}^{\infty}$ be a bounded sequence,
and let $T^{\Lambda}$ be a multiplier operator on $L_{2}({{S}}^{d})$ defined
by

$T^{\Lambda}(f)=\sum_{\ell=0}^{\infty}\lambda^{-1}_{\ell}H^{d}_{\ell}(f)=\sum_{\ell=0}^{\infty}\lambda^{-1}_{\ell}{\sum_{k=1}^{N(d,\ell)}\langle
f,Y_{\ell,k}\rangle Y_{\ell,k}}.$

######  Definition 2.1.

Let $\Lambda=\\{\lambda_{\ell}\\}_{\ell=0}^{\infty}$ be a non-increasing
positive sequence with $\lim\limits_{\ell\to\infty}\lambda_{\ell}=0$. We
define the multiplier space ${\rm H}^{\Lambda}({{S}}^{d})$ by

$\displaystyle{\rm H}^{\Lambda}({{S}}^{d})$
$\displaystyle:=\Big{\\{}T^{\Lambda}f\,\big{|}\,f\in L_{2}({{S}}^{d})\ ,\
\|T^{\Lambda}f\,\|_{{\rm
H}^{\Lambda}({{S}}^{d})}=\|f\|_{L_{2}({{S}}^{d})}<\infty\Big{\\}}$
$\displaystyle:=\Big{\\{}f\in L_{2}({{S}}^{d})\,\big{|}\,\|f\,\|_{{\rm
H}^{\Lambda}({{S}}^{d})}:=\Big{(}\sum_{\ell=0}^{\infty}\lambda_{\ell}^{-2}\sum_{k=1}^{N(d,\ell)}|\langle
f,Y_{\ell,k}\rangle|^{2}\Big{)}^{1/2}<\infty\Big{\\}}.$

Clearly, the multiplier space ${\rm H}^{\Lambda}({{S}}^{d})$ is a Hilbert
space with inner product

$\langle f,g\rangle_{{\rm
H}^{\Lambda}({{S}}^{d})}=\sum_{\ell=0}^{\infty}\lambda_{\ell}^{-2}\sum_{k=1}^{N(d,\ell)}\langle
f,Y_{\ell,k}\rangle\,\langle
g,Y_{\ell,k}\rangle=\sum_{\ell=0}^{\infty}\lambda_{\ell}^{-2}\langle
H_{\ell}^{d}(f),H_{\ell}^{d}(g)\rangle,$

and

(2.7) $E_{n}(f)_{L_{2}}=\|f-S_{n}(f)\|_{L_{2}}\leq\lambda_{n}\|f\|_{{\rm
H}^{\Lambda}({{S}}^{d})}.$

Moreover, we remark that the embedding ${\rm
H}^{\Lambda}({{S}}^{d})\hookrightarrow C({{S}}^{d})\ ({\rm or}\
L_{\infty}({{S}}^{d}))$ is continuous if

(2.8) $\sum_{\ell=0}^{\infty}\lambda^{2}_{\ell}N(d,\ell)<\infty,$

which ensures ${\rm H}^{\Lambda}({{S}}^{d})$ is a reproducing kernel Hilbert
space with the reproducing kernel

(2.9) $\displaystyle K^{\Lambda}({\bf x},{\bf y})$
$\displaystyle=\sum_{\ell=0}^{\infty}\lambda_{\ell}^{2}\sum_{k=1}^{N(d,\ell)}Y_{\ell,k}({\bf
x})Y_{\ell,k}({\bf
y})=\sum_{\ell=0}^{\infty}\lambda_{\ell}^{2}N(d,\ell)G_{\ell}({\bf x}\cdot{\bf
y}),$

where $G_{\ell}(t),\ t\in[-1,1]$ is given in (2.4). Indeed, if (2.8) holds,
then we have for any ${\bf x}\in{{S}}^{d}$,

$\|K^{\Lambda}({\bf x},\cdot)\|_{{\rm H}^{\Lambda}({{S}}^{d})}^{2}=\langle
K^{\Lambda}({\bf x},\cdot),K^{\Lambda}({\bf x},\cdot)\rangle_{{\rm
H}^{\Lambda}({{S}}^{d})}=K^{\Lambda}({\bf x},{\bf
x})=\sum_{\ell=0}^{\infty}\lambda_{\ell}^{2}N(d,\ell)<\infty,$

and for $f\in{\rm H}^{\Lambda}({{S}}^{d})$,

$\langle f,K^{\Lambda}({\bf x},\cdot)\rangle_{{\rm
H}^{\Lambda}({{S}}^{d})}=\sum_{\ell=0}^{\infty}\lambda_{\ell}^{2}\sum_{k=1}^{N(d,\ell)}\langle
f,Y_{\ell,k}\rangle\lambda_{\ell}^{-2}Y_{\ell,k}({\bf x})=f({\bf x}),$

which means that ${\rm H}^{\Lambda}({{S}}^{d})$ is a reproducing kernel
Hilbert space with the kernel $K^{\Lambda}(\cdot,\cdot)$. It follows from the
Cauchy inequality that for $f\in{\rm H}^{\Lambda}({{S}}^{d}),\ {\bf x},{\bf
y}\in{{S}}^{d},$

$\displaystyle|f({\bf x})-f({\bf y})|^{2}$ $\displaystyle=|\langle
f,K^{\Lambda}({\bf x},\cdot)-K^{\Lambda}({\bf y},\cdot)\rangle_{{\rm
H}^{\Lambda}({{S}}^{d})}|^{2}$ $\displaystyle\leq\|f\|_{{\rm
H}^{\Lambda}({{S}}^{d})}^{2}\ \|K^{\Lambda}({\bf x},\cdot)-K^{\Lambda}({\bf
y},\cdot)\|_{{\rm H}^{\Lambda}({{S}}^{d})}^{2}$ $\displaystyle=2\|f\|_{{\rm
H}^{\Lambda}({{S}}^{d})}^{2}\sum_{\ell=0}^{\infty}\lambda_{\ell}^{2}N(d,\ell)(1-G_{\ell}({\bf
x}\cdot{\bf y})).$

This implies that $f$ is continuous on ${{S}}^{d}$. Hence, the space ${\rm
H}^{\Lambda}({{S}}^{d})$ is continuously embedded in $C({{S}}^{d})$.

For $\Lambda=\\{\lambda_{\ell}\\}_{\ell=0}^{\infty}$, we set

$\tilde{\Lambda}=\\{\tilde{\lambda}_{\ell}\\}_{\ell=0}^{\infty},\ \
\tilde{\lambda}_{\ell}=\lambda_{\ell}^{2}N(d,\ell),\ \ell=0,1,\dots.$

If (2.8) holds, then $\tilde{\Lambda}\in\ell_{1}({N}_{0})$, i.e.,

$\|\tilde{\Lambda}\|_{\ell_{1}}\equiv\|\tilde{\Lambda}\|_{\ell_{1}({N}_{0})}:=\sum_{\ell=0}^{\infty}|\tilde{\lambda}_{\ell}|<\infty,$

and the reproducing kernel of ${\rm H}^{\Lambda}({{S}}^{d})$ can be written by

$K^{\Lambda}({\bf x},{\bf
y})=\sum_{\ell=0}^{\infty}\tilde{\lambda}_{\ell}G_{\ell}({\bf x}\cdot{\bf
y}),\ \ {\bf x},{\bf y}\in{{S}}^{d}.$

We write

${\rm H}_{\mathaccent 869{\Lambda}}:={\rm H}^{\Lambda}({{S}}^{d}),\ \ \
K_{\mathaccent 869{\Lambda}}({\bf x},{\bf y}):=K^{\Lambda}({\bf x},{\bf y}).$

######  Definition 2.2.

Let $\alpha>0,\,\beta\in{R}$. The Sobolev space ${\rm
H}^{\alpha,\beta}({{S}}^{d})$ with logarithmic perturbation is defined to be
the multiplier space ${\rm H}^{\Lambda^{(\alpha,\beta)}}({{S}}^{d})$, where
$\Lambda^{(\alpha,\beta)}=\\{\lambda_{\ell}^{(\alpha,\beta)}\\}_{\ell=0}^{\infty}$,
and

$\lambda_{\ell}^{(\alpha,\beta)}=(1+\ell(\ell+d-1))^{-\alpha/2}(\ln(3+\ell(\ell+d-1)))^{-\beta}.$

If $\beta=0$, then the space ${\rm H}^{\alpha,\beta}({{S}}^{d})$ reduces to
the Sobolev space ${\rm H}^{\alpha,+}$ as defined in [5, Definition 2.2].
Hence ${\rm H}^{\alpha,\beta}({{S}}^{d})$ is the Sobolev space with
logarithmic perturbation. Such spaces were investigated intensively (see [10,
14, 32, 47]). In the case $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$, the space
${\rm H}^{\alpha,\beta}({{S}}^{d})$ is continuously embedded into the space of
continuous functions $C({{S}}^{d})$, and is a reproducing kernel Hilbert
space.

For $f\in{\rm H}^{\alpha,\beta}({{S}}^{d}),\,\alpha>0$, we define

$\sigma_{1}(f)=S_{2}(f),\ \ \sigma_{j}(f)=S_{2^{j}}(f)-S_{2^{j-1}}(f),\ \ {\rm
for}\ j\geq 2.$

Then $\sigma_{j}(f)\in\Pi_{2^{j}}^{d+1}$ and

$f=\sum_{j=1}^{\infty}\sigma_{j}(f)$

converges in the $L_{2}$ norm. Note that $\sigma_{j}(f)$ satisfies the
Bernsetin inequality, i.e., for any $r>0$,

$\|(-\triangle_{0})^{r/2}(\sigma_{j}(f))\|_{L_{2}}\lesssim
2^{jr}\|\sigma_{j}(f)\|_{L_{2}}.$

Also, by (2.6) we obtain

$\displaystyle\|\sigma_{j}(f)\|_{L_{2}}\leq\|f-S_{2^{j}}(f)\|_{L_{2}}+\|f-S_{2^{j-1}}(f)\|_{L_{2}}\leq
2E_{2^{j-1}}(f)_{L_{2}}.$

Now set $\Omega(t)=t^{\alpha}(1+(\ln\frac{1}{t})_{+})^{\beta}$ for $t>0$.
Consider the Besov space $B_{2,2}^{\Omega}({{S}}^{d})$ of generalized
smoothness (see [46, 47]). It follows from [46, Remark 2] and the above two
inequalities that

$\|f\|_{B_{2,2}^{\Omega}({{S}}^{d})}\asymp\Big{(}\sum_{j=0}^{\infty}\frac{\|\sigma_{j}(f)\|_{L_{2}}^{2}}{\Omega(2^{-j})^{2}}\Big{)}^{1/2}.$

On the other hand, we have

$\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})}^{2}=\sum_{\ell=0}^{\infty}(\lambda_{\ell}^{(\alpha,\beta)})^{-2}\|H_{\ell}^{d}(f)\|_{L_{2}}^{2}=:\sum_{j=1}^{\infty}I_{j}.$

Here,

$I_{1}:=\sum_{\ell=0}^{2}(\lambda_{\ell}^{(\alpha,\beta)})^{-2}\|H_{\ell}^{d}(f)\|_{L_{2}}^{2}\asymp\sum_{\ell=0}^{2}\|H_{\ell}^{d}(f)\|_{L_{2}}^{2}=\|\sigma_{1}(f)\|_{L_{2}}^{2}\asymp\frac{\|\sigma_{1}(f)\|_{L_{2}}^{2}}{\Omega(2^{-1})^{2}},$

and for $j\geq 2$,

$I_{j}:=\sum_{\ell=2^{j-1}+1}^{2^{j}}(\lambda_{\ell}^{(\alpha,\beta)})^{-2}\|H_{\ell}^{d}(f)\|_{L_{2}}^{2}\asymp
2^{2j\alpha}j^{2\beta}\sum_{\ell=2^{j-1}+1}^{2^{j}}\|H_{\ell}^{d}(f)\|_{L_{2}}^{2}\asymp\frac{\|\sigma_{j}(f)\|_{L_{2}}^{2}}{\Omega(2^{-j})^{2}}.$

It follows that

$\displaystyle\|f\|_{{\rm H}^{\alpha,\beta}({{S}}^{d})}$
$\displaystyle\asymp\|f\|_{B_{2,2}^{\Omega}({{S}}^{d})}\asymp\Big{(}\sum_{j=1}^{\infty}\frac{\|\sigma_{j}(f)\|_{L_{2}}^{2}}{\Omega(2^{-j})^{2}}\Big{)}^{1/2}$
(2.10)
$\displaystyle\asymp\Big{(}\sum_{j=1}^{\infty}2^{2j\alpha}j^{2\beta}\|\sigma_{j}(f)\|_{L_{2}}^{2}\Big{)}^{1/2}.$

This means that the space ${\rm H}^{\alpha,\beta}({{S}}^{d})$ is just the
Besov space $B_{2,2}^{\Omega}({{S}}^{d})$ (see [46, 47]) of generalized
smoothness with above equivalent norms. Here,
$\Omega(t)=t^{\alpha}(1+(\ln\frac{1}{t})_{+})^{\beta}$ for $t>0$. Hence, from
[46, 47]) we have

(2.11) $\displaystyle\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})}\asymp\|f\|_{L_{2}}+\Big{(}\sum_{j=1}^{\infty}2^{2j\alpha}j^{2\beta}E_{2^{j}}(f)_{L_{2}}^{2}\Big{)}^{1/2}.$

###  2.2. Lower bounds for optimal quadrature and positive definiteness

Let $H\equiv H(K)$ be a reproducing kernel Hilbert space (RKHS) consisting of
real-valued functions on ${{S}}^{d}$ with reproducing kernel $K\in
C({{S}}^{d}\times{{S}}^{d})$ and inner product
$\langle\cdot,\cdot\rangle_{H}$. That is, $H$ is a Hilbert space of real-
valued functions on ${S}^{d}$ such that point evaluation

$\delta_{\bf x}:\,H(K)\to{R},\ \ \delta_{\bf x}(f):=f({\bf x})$

is a continuous functional on $H$ for all ${\bf x}\in{S}^{d}$, and the
reproducing kernel $K$ satisfies

(i) for all ${\bf x},{\bf y}\in{{S}}^{d}$, $K({\bf x},{\bf y})=K({\bf y},{\bf
x})$;

(ii) for all ${\bf x}\in{{S}}^{d}$, $K({\bf x},\cdot)\in H$;

(iii) for all ${\bf x}\in{{S}}^{d}$ and $f\in H$, $f({\bf x})=\langle f,K({\bf
x},\cdot)\rangle_{H}$.

We refer to [1, 2] for basics on RKHSs.

For $h\in H$, we define $S_{h}$ the continuous linear functional on $H$ by
$S_{h}(f)=\langle f,h\rangle_{H}$ for $f\in H$. Since $\rm INT$ is continuous
on $H$, by the Riesz representation theorem, there exists an $h_{0}\in H$ for
which

${\rm INT}(f)=\int_{{{S}}^{d}}f({\bf{x}})\,d\sigma_{d}({\bf{x}})=\langle
f,h_{0}\rangle_{H}=S_{h_{0}}(f).$

We have for ${\bf x}\in{{S}}^{d}$,

$h_{0}({\bf x})=\big{\langle}h_{0},K({\bf x},\cdot)\big{\rangle}_{H}={\rm
INT}(K({\bf x},\cdot))=\int_{{{S}}^{d}}K({\bf{x}},{\bf{y}})d\sigma_{d}({\bf
y}).$

We consider the space ${\rm H}^{\Lambda}({{S}}^{d})={\rm
H}_{\tilde{\Lambda}}$, where $\Lambda=\\{\lambda_{k}\\}_{k=0}^{\infty}$ is a
positive sequence, $\tilde{\Lambda}=\\{\tilde{\lambda}_{k}\\}_{k=0}^{\infty}$,
and $\tilde{\lambda}_{k}=\lambda_{k}^{2}N(d,k),\ k=0,1,\dots$. As seen in
Subsection 2.1, the space ${\rm H}^{\Lambda}({{S}}^{d})$ is RKHS on the sphere
if and only if

$\sum_{k=0}^{\infty}\lambda_{k}^{2}N(d,k)=\sum_{k=0}^{\infty}\tilde{\lambda}_{k}<+\infty.$

In this case, the reproduce kernel is

$K_{\tilde{\Lambda}}({\bf x},{\bf y}):=\sum_{\ell=0}^{\infty}\mathaccent
869{\lambda}_{\ell}G_{\ell}({\bf x}\cdot{\bf y}),\ {\bf x},{\bf y}\in{S}^{d},$

and the representer of the integration problem INT is

$h_{\mathaccent 869{\Lambda}}({\bf x})=\int_{{{S}}^{d}}K_{\mathaccent
869{\Lambda}}({\bf{x}},{\bf{y}})d\sigma_{d}({\bf
y})=\sum_{\ell=0}^{\infty}\mathaccent
869{\lambda}_{\ell}\int_{{S}^{d}}G_{\ell}({\bf x}\cdot{\bf y})d\sigma_{d}({\bf
y})=\mathaccent 869{\lambda}_{0}=\lambda_{0}^{2},$

for all ${\bf{x}}\in{{S}}^{d}$. This means that

${\rm INT}(f)=\int_{{{S}}^{d}}f({\bf{x}})\,d\sigma_{d}({\bf{x}})=\langle
f,h_{\mathaccent 869{\Lambda}}\rangle_{{\rm
H}_{\tilde{\Lambda}}}=S_{h_{\mathaccent 869{\Lambda}}}(f).$

Furthermore, we have

$\|h_{\mathaccent 869{\Lambda}}\|_{{\rm
H}_{\tilde{\Lambda}}}=(\tilde{\lambda}_{0})^{1/2}=\lambda_{0}.$

We say that a symmetric matrix $M\in{{R}}^{n\times n}$ is positive semi-
definite if $\mathbf{c}^{T}M\mathbf{c}\geq 0$ for all
$\mathbf{c}\in{{R}}^{n}$. If $M,N\in{{R}}^{n\times n}$, we denote by $M\circ
N$ their Hadamard (Schur) product, i.e., a matrix with entries $(M\circ
N)_{j,k}=M_{j,k}\cdot N_{j,k}$ for all $j,k=1,...,n$ (see [16]). Furthermore,
the partial ordering $M\succeq N$ of symmetric matrices means that $M-N$ is
positive semi-definite. We denote by ${\rm diag}M=(M_{1,1},...,M_{n,n})^{T}$
the vector of diagonal entries of $M$ whenever $M\in{{R}}^{n\times n}$.

In order to prove lower bounds for $e_{n}({\rm H}_{\tilde{\Lambda}})$ we need
the following lemmas. The first lemma is Schoenberg’s remarkable result about
positive definite functions on the sphere.

######  Lemma 2.3.

(See [37, 7].) Let
$\Gamma=\\{\gamma_{j}\\}_{j=0}^{\infty}\in\ell_{1}({N}_{0})$, and let

$f_{\Gamma}(t)=\sum_{j=1}^{\infty}\gamma_{j}G_{j}(t),\ \ \ \ K_{\Gamma}({\bf
x},{\bf y})=f_{\Gamma}({\bf x}\cdot{\bf y}),$

where $G_{j}$ are the normalized Gegenbauer polynomials given in (2.4),
$t\in[-1,1]$, ${\bf x},{\bf y}\in{{S}}^{d}$. Then for all $n\in{N}$ and ${\bf
x}_{1},{{\bf x}_{2}},\cdot\cdot\cdot,{{\bf x}_{n}}\in{{S}}^{d}$, the matrix

$\big{(}K_{\Gamma}({\bf x}_{j},{\bf x}_{k})\big{)}_{1\leq j,k\leq n}$

is positive semi-definite if and only if $\Gamma$ is nonnegative.

The second lemma gives a characterization of lower bounds for $e_{n}(H)$ via
the positive definiteness of certain matrices involving the reproducing kernel
$K$ and the representer $h_{0}$ of the integration problem INT.

######  Lemma 2.4.

(See [19, Proposition 1] and [20, 27].) Let $H$ be a RKHS on ${{S}}^{d}$ with
reproducing kernel $K\in C({{S}}^{d}\times{{S}}^{d})$, and let ${\rm
INT}=\langle\cdot,h_{0}\rangle_{H}$ for some $h_{0}\in H$, $\alpha>0$. For all
$n\in{N}$ and ${\bf x}_{1},{{\bf x}_{2}},\cdot\cdot\cdot,{{\bf
x}_{n}}\in{{S}}^{d}$, the matrix

$\big{(}K({\bf x}_{j},{\bf x}_{k})-\alpha h_{0}({\bf x}_{j})h_{0}({\bf
x}_{k})\big{)}_{1\leq j,k\leq n}$

is positive semi-definite, if and only if

$e_{n}^{2}(H)\geq\|h_{0}\|_{H}^{2}-\alpha^{-1}.$

The third lemma was obtained by J. Vybíral in [43] and shows that the Schur
product of a positive semi-definite matrix with itself is not only positive
semi-definite but also bounded from below by some rank-1 matrix.

######  Lemma 2.5.

(See [43, Theorem 1].) Let $M\in{{R}}^{n\times n}$ be a positive semi-definite
matrix. Then

$M\circ M\succeq\frac{1}{n}({\rm diag}\,M)({\rm diag}\,M)^{T}.$

###  2.3. Weighted least squares operators and least squares quadrature rules

First we give the definition of $L_{2}$ MZ families.

######  Definition 2.6.

Let $\mathcal{X}=\\{\mathcal{X}_{n}\\}=\\{{\bf x}_{n,k},\
n=1,2,\dots,k=1,\dots,l_{n}\\}$ be a doubly-indexed set of points in
${{S}}^{d}$ and $\tau=\\{\tau_{n,k}:n=1,2,\dots,\,k=1,\dots,l_{n}\\}$ be a
family of positive numbers. Then $\mathcal{X}$ is called an $L_{2}$
Marcinkiewicz-Zygmund (MZ) family with associated weight $\tau$, if there
exist constants $A,B>0$ independent of $n$ such that

(2.12) $A\|p\|_{2}^{2}\leq\sum_{k=1}^{l_{n}}|p({\bf
x}_{n,k})|^{2}\tau_{n,k}\leq B\|p\|_{2}^{2}\ \qquad{\rm for\ all}\
p\in\Pi_{n}^{d+1}.$

The ratio $\kappa=B/A$ is the global condition number of the $L_{2}$ MZ family
$\mathcal{X}$, and $\mathcal{X}_{n}=\\{{\bf x}_{n,k}:k=1,\dots,l_{n}\\}$ is
the $n$th layer of $\mathcal{X}$.

The existence of $L_{2}$ MZ families on ${{S}}^{d}$ (including the case
$\kappa=1$) has been established in [31, 4, 7, 29, 30]. Additionally,
necessary or sufficient density conditions for $L_{2}$ MZ families on
${{S}}^{d}$ have been discussed in [30, 33].

Let $\mathcal{X}$ be an $L_{2}$ MZ family with associated weight $\tau$. Given
the samples $\\{f({\bf x}_{N,k})\\}$ of a continuous function $f$ on
${{S}}^{d}$ on the $N$th layer $\mathcal{X}_{N}$, Gröchenig in [15] introduced
the weighted least squares polynomials $L_{N}^{{S}}(f)$ to approximate $f$
using only these samples, i.e., the solution of weighted least squares
problems:

(2.13) $L_{N}^{{S}}(f)=\arg\min_{p\in\Pi_{N}^{d+1}}\ \sum_{k=1}^{l_{N}}|f({\bf
x}_{N,k})-p({\bf x}_{N,k})|^{2}\tau_{N,k}.$

This procedure yields a sequence $\\{L_{N}^{{S}}(f)\\}$ of the best weighted
$\ell_{2}$-approximation of the data $\\{f({\bf x}_{N,k})\\}$ by a spherical
polynomial in $\Pi_{N}^{d+1}$ for every $f\in C({{S}}^{d})$. We call
$L_{N}^{{S}}$ the weighted least squares operator.

It follows from (2.12) that for $p\in\Pi_{N}^{d+1}$, $p=0$ whenever $p({\bf
x}_{N,k})=0$. This means that usually $\mathcal{X}_{N}$ contains more than
$d_{N}=\dim\Pi_{N}^{d+1}$ points, so that it is not an interpolating set for
$\Pi_{N}^{d+1}$. Therefore, $L_{N}^{{S}}(f)$ is usually a quasi-interpolant.

We use the weighted discretized inner product

(2.14) $\langle f,g\rangle_{(N)}:=\sum_{k=1}^{l_{N}}f({\bf x}_{N,k})g({\bf
x}_{N,k})\tau_{N,k}$

and the discretized norm

$\|f\|_{(N)}^{2}=\langle f,f\rangle_{(N)}.$

We consider the corresponding orthogonal polynomial projection $L_{N}^{{S}}$
onto $\Pi_{N}^{d+1}$ with respect to the weighted discretized inner product
$\langle\cdot,\cdot\rangle_{(N)}$, namely the weighted least squares
polynomial $L_{N}^{{S}}(f)$ defined by

$L_{N}^{{S}}(f)=\arg\min_{p\in\Pi_{N}^{d+1}}\|f-p\|_{(N)}^{2}=\arg\min_{p\in\Pi_{N}^{d+1}}\
\sum_{k=1}^{l_{N}}|f({\bf x}_{N,k})-p({\bf x}_{N,k})|^{2}\tau_{N,k}.$

We shall give a formal expression for $L_{N}^{{S}}(f)$ (see [28]). Let
$\varphi_{i},\ i=1,\dots,d_{N}$ be an orthonormal basis of $\Pi_{N}^{d+1}$
with respect to the weighted discrete scalar product (2.14), i.e.,

$\langle\varphi_{i},\varphi_{j}\rangle_{(N)}=\delta_{i,j}=\Big{\\{}\begin{array}[]{cl}1,&{\rm
if}\ \ i=j,\\\ 0,&{\rm if}\ \ i\neq j,\end{array}\ \ \ 1\leq i,j\leq d_{N}.$

We set

$D_{N}({\bf x},{\bf y})=\sum_{k=1}^{d_{N}}\varphi_{k}({\bf x})\varphi_{k}({\bf
y}).$

Clearly, $D_{N}({\bf x},{\bf y})$ is the reproducing kernel of $\Pi_{N}^{d+1}$
with respect to the weighted discrete scalar product (2.14), and the weighted
least squares polynomial $L_{N}^{{S}}(f)$ is just the orthogonal projection of
$f$ onto $\Pi_{N}^{d+1}$ with respect to the weighted discrete scalar product
(2.14), i. e.,

(2.15) $L_{N}^{{S}}(f)({\bf x})=\langle f,D_{N}(\cdot,{\bf
x})\rangle_{(N)}=\sum_{k=1}^{l_{N}}f({\bf x}_{N,k})D_{N}({\bf x}_{N,k},{\bf
x})\tau_{N,k}.$

Obviously, $L_{N}^{{S}}$ is a linear operator satisfying

(2.16) $\displaystyle\|f-L_{N}^{{S}}(f)\|_{(N)}\leq\|f-p\|_{(N)},\ \ {\rm for\
all}\ p\in\Pi_{N}^{d+1}.$

Also, Gröchenig in [15] used the frame theory to construct _least squares
quadrature rules_

$I_{N}^{{S}}(f)=\sum_{k=1}^{l_{N}}w_{N,k}f({\bf x}_{N,k})$

induced by an $L_{2}$ MZ family. Lu and Wang in [28] proved that

$w_{N,k}=\tau_{N,k}\int_{{{S}}^{d}}D_{N}({\bf x},{\bf
x}_{n,k})d\sigma_{d}({\bf x}),\ \ \
I_{N}^{{S}}(f)=\int_{{{S}}^{d}}L_{N}^{{S}}(f)({\bf x}){d}\sigma_{d}({\bf x}),$

and

(2.17) $\left|\int_{{{S}}^{d}}f({\bf x}){d}\sigma_{d}({\bf
x})-I_{N}^{{S}}(f)\right|\leq\left\|f-L_{N}^{{S}}(f)\right\|_{L_{2}}.$

Finally, we give the relation between weighted least squares polynomials and
hyperinterpolation (see [28]). Hyperinterpolation was originally introduced by
I. H. Sloan (see [38]). It uses the Fourier orthogonal projection of a
function which can be expressed in the form of integrals, but approximates the
integrals used in the expansion by means of a positive quadrature formula. In
recent years, hyperinterpolation and its generalization on ${{S}}^{d}$ has
attracted much interest, and a great number of interesting results have been
obtained (see [6, 13, 18, 35, 36, 38, 39, 40, 44, 45]).

Assume that $Q_{N}(f)=\sum\limits_{k=1}^{l_{N}}\tau_{N,k}f({\bf x}_{N,k}),\
N=1,2,\dots$ is a sequence of positive quadrature formulas on ${{S}}^{d}$
which are exact for $\Pi_{2N}^{d+1}$, i.e., $\tau_{N,k}>0$, and for all
$f\in\Pi_{2N}^{d+1}$,

$\int_{{{S}}^{d}}f({\bf x})d\sigma_{d}({\bf
x})=Q_{N}(f)=\sum_{k=1}^{l_{N}}\tau_{N,k}f({\bf x}_{N,k}).$

Then the family $\mathcal{X}=\\{{\bf x}_{N,k}\\}$ with weight $\tau$ is an
$L_{2}$ MZ family with the constants $A=B=1$ and the global condition number
$\kappa$ is equal to $1$. On the other hand, if the global condition number
$\kappa$ of an $L_{2}$ MZ family $\mathcal{X}=\\{{\bf x}_{N,k}\\}$ with weight
$\tau$ is equal to $1$, then $\mathcal{X}$ determines a sequence of positive
quadrature formulas
$Q_{N}(f)=\frac{1}{A}\sum\limits_{k=1}^{l_{N}}\tau_{N,k}f({\bf x}_{N,k})$ on
${{S}}^{d}$ which are exact for $\Pi_{2N}^{d+1}$.

Let $Q_{N}(f)=\sum\limits_{k=1}^{l_{N}}\tau_{N,k}f({\bf x}_{N,k}),\
N=1,2,\dots$ be a sequence of positive quadrature formulas on ${{S}}^{d}$
which are exact for $\Pi_{2N}^{d+1}$. The hyperinterpolation operator
$H_{N}^{{S}}$ on ${{S}}^{d}$ is defined by

$H_{N}^{{S}}(f)({\bf x})=\sum\limits_{k=1}^{l_{N}}\tau_{N,k}f({\bf
x}_{N,k})E_{N}({\bf x},{\bf x}_{N,k}),\ \ \ f\in C({{S}}^{d}),$

where $E_{N}({\bf x},{\bf y})$ is the reproducing kernel of $\Pi_{N}^{d+1}$
with respect to the inner product of $L_{2}$. We note that for all
$f,g\in\Pi_{N}^{d+1}$,

$\langle f,g\rangle=\langle f,g\rangle_{(N)},$

where $\langle\cdot,\cdot\rangle_{(N)}$ is defined by (2.14). This means that

$D_{N}({\bf x},{\bf y})=E_{N}({\bf x},{\bf y}),\ \ \
H_{N}^{{S}}(f)=L_{N}^{{S}}(f),$

and

$I_{N}^{{S}}(f)=Q_{N}(f).$

Hence, the hyperinterpolation is just the weighted least squares polynomial
for an $L_{2}$ MZ family with the global condition number $\kappa=1$, and a
sequence of positive quadrature formulas $Q_{N}(f)$ on ${{S}}^{d}$ which are
exact for $\Pi_{2N}^{d+1}$ is just the least squares quadrature for an $L_{2}$
MZ family with the global condition number $\kappa=1$, and the weighted least
squares polynomial and the least squares quadrature may be viewed as a
generalization of the hyperinterpolation and a sequence of positive quadrature
formulas $Q_{N}(f)$ which are exact for $\Pi_{2N}^{d+1}$, respectively.

##  3\. Proofs of Theorems 1.1 and 1.2

Theorem 1.1 follows from the following theorem immediately.

######  Theorem 3.1.

Let $\Lambda=\\{\lambda_{\ell,d}\\}_{\ell=0}^{\infty}$ be a nonnegative non-
increasing sequence with
$\lim\limits_{\ell\to\infty}\lambda_{\ell,d}\,\ell^{\alpha}\ln^{\beta}\ell=1$
for some $\alpha>0$ and $\beta\in{R}$. We have

(3.1) $\lim_{n\to\infty}n^{\alpha/d}(\ln n)^{\beta}a_{n}({\rm
H}^{\Lambda}({{S}}^{d}))=\Big{(}\frac{2}{d\,!}\Big{)}^{\alpha/d}d^{\beta}.$

In order to prove Theorem 3.1 we need the following two lemmas.

######  Lemma 3.2.

(See [5, Theorem 2.8].) Let $\Lambda=\\{\lambda_{\ell,d}\\}_{\ell=0}^{\infty}$
be a nonnegative non-increasing sequence with
$\lim\limits_{\ell\to\infty}\lambda_{\ell,d}=0$. For $C(d,\ell-1)<n\leq
C(d,\ell),\ \ell=0,1,2,\dots,$ we have

$a_{n}({\rm H}^{\Lambda}({{S}}^{d})))=\lambda_{\ell,d},$

where we set $C(d,-1)=0$.

######  Lemma 3.3.

Let $\\{a_{n}\\},\ \\{b_{n}\\}$ be two positive sequences such that

$\lim_{n\to\infty}\frac{b_{n}}{a_{n}}=A\neq 0\ \ {\rm and}\ \
\lim_{n\to\infty}a_{n}=+\infty.$

Then we have

$\lim_{n\to\infty}\frac{\ln b_{n}}{\ln a_{n}}=1.$

###### Proof.

According to $\lim\limits_{n\to\infty}\frac{b_{n}}{a_{n}}=A$, we can get for
arbitrary $\varepsilon\in(0,A)$, there is a positive integer $N$ such that
$n>N$,

$\big{|}\frac{b_{n}}{a_{n}}-A\big{|}<\varepsilon.$

This means that

$\ln a_{n}+\ln(A-\varepsilon)<\ln{b_{n}}<\ln a_{n}+\ln(A+\varepsilon),$

which yields that

$1+\frac{\ln((A-\varepsilon))}{\ln a_{n}}<\frac{\ln b_{n}}{\ln
a_{n}}<1+\frac{\ln(A+\varepsilon))}{\ln a_{n}}.$

Since $\lim\limits_{n\to\infty}a_{n}=+\infty$, we obtain that

$\lim_{n\to\infty}\frac{\ln b_{n}}{\ln a_{n}}=1.$

∎

Proof of Theorem 3.1

By Lemma 3.2, we have for $C(d,\ell-1)<n\leq C(d,\ell),\ \ell=0,1,2,\dots,$

$a_{n}({\rm H}^{\Lambda}({{S}}^{d}))=\lambda_{\ell,d}.$

Since the function $x^{\alpha/d}(\ln x)^{\beta}\ (\alpha>0)$ is increasing for
sufficiently large $x$, it follows that for sufficiently large $\ell$,

$\displaystyle(C(d,\ell-1))^{\alpha/d}(\ln(C(d,\ell-1)))^{\beta}\lambda_{\ell,d}$
$\displaystyle<n^{\alpha/d}(\ln n)^{\beta}a_{n}({\rm H}^{\Lambda}({{S}}^{d}))$
(3.2)
$\displaystyle\leq(C(d,\ell))^{\alpha/d}(\ln(C(d,\ell)))^{\beta}\lambda_{\ell,d}.$

We recall from (2.2) that

$C(d,\ell)=\frac{(2\ell+d)(\ell+d-1)\,!}{\ell\,!\,d\,!},$

which yields that

$\lim_{\ell\to\infty}\frac{C(d,\ell)}{\ell^{d}}=\lim_{\ell\to\infty}\frac{C(d,\ell-1)}{\ell^{d}}=\frac{2}{d\,!}.$

It follows from Lemma 3.3 that

$\lim_{\ell\to\infty}\frac{\ln(C(d,\ell))}{\ln(\ell^{d})}=\lim_{\ell\to\infty}\frac{\ln(C(d,\ell-1))}{\ln(\ell^{d})}=1.$

We obtain

$\displaystyle\quad\
\lim_{\ell\to\infty}(C(d,\ell-1))^{\alpha/d}(\ln(C(d,\ell-1)))^{\beta}\lambda_{\ell,d}$
$\displaystyle=\lim_{\ell\to\infty}(\frac{C(d,\ell)}{\ell^{d}})^{\alpha/d}\Big{(}\frac{\ln(C(d,\ell-1))}{\ln(\ell^{d})}\Big{)}^{\beta}d^{\beta}\ell^{\alpha}(\ln\ell)^{\beta}\lambda_{\ell,d}$
(3.3) $\displaystyle=\Big{(}\frac{2}{d\,!}\Big{)}^{\alpha/d}d^{\beta},$

and

(3.4)
$\lim_{\ell\to\infty}(C(d,\ell))^{\alpha/d}(\ln(C(d,\ell)))^{\beta}\lambda_{\ell,d}=\Big{(}\frac{2}{d\,!}\Big{)}^{\alpha/d}d^{\beta}.$

Clearly, $n\to\infty$ if and only if $\ell\to\infty$. Equation (3.1) follows
from (3.2), (3.3) and (3.4) immediately. This completes the proof of Theorem
3.1. $\hfill\Box$

######  Corollary 3.4.

Let $\Lambda=\\{\lambda_{\ell}\\}_{\ell=0}^{\infty}$ be a nonnegative non-
increasing sequence with $\lambda_{\ell}\asymp\ell^{-\alpha}\ln^{-\beta}\ell$
for some $\alpha>0$ and $\beta\in{R}$. We have

$a_{n}({\rm H}^{\Lambda}({{S}}^{d}))\asymp n^{-\alpha/d}\ln^{-\beta}n.$

Next we show Theorem 1.2. In order to prove Theorem 1.2 we need the following
lemma.

######  Lemma 3.5.

(See [8, Proposition 3.5].) Let $X$ be a linear subspace of $\Pi_{n}^{d+1}$
with $\dim X\geq\varepsilon\dim\Pi_{n}^{d+1}$ for some $\varepsilon\in(0,1)$.
Then there exists a function $f\in X$ such that

$\|f\|_{L_{p}}\asymp\|f\|_{L_{\infty}}\asymp 1,\ \ \ 1\leq p<\infty,$

where

$\|f\|_{L_{\infty}}=\max_{{\bf x}\in{{S}}^{d}}|f({\bf x})|,\ \ {\rm and}\ \
\|f\|_{L_{p}}^{p}=\int_{{{S}}^{d}}|f({\bf x})|^{p}d\sigma_{d}({\bf x}),\ 1\leq
p<\infty.$

Proof of Theorem 1.2.

Let ${\bf x}_{1},\dots,{\bf x}_{n}$ be any $n$ distinct points on ${S}^{d}$.
Take a positive integer $N$ such that $2n\leq d_{N}\leq Cn$ and denote

$X_{0}:=\big{\\{}g\in\Pi_{N}^{d+1}\ \big{|}\ g({\bf x}_{j})=0\ {\rm for\ all}\
j=1,\dots,n\big{\\}},$

where $d_{N}={\rm dim}\,\Pi_{N}^{d+1}$. Thus, $n\asymp N^{d}$, and $X_{0}$ is
a linear subspace of $\Pi_{N}^{d+1}$ with

${\rm dim}\,X_{0}\geq d_{N}-n\geq d_{N}/2.$

It follows from Lemma 3.5 that there exists a function $g_{0}\in X_{0}$ such
that

$\|g_{0}\|_{L_{p}}\asymp 1,\ {\rm}\ {\rm for\ all}\ 1\leq p\leq\infty.$

Let

$f_{0}({\bf x})=N^{-\alpha}\ln^{-\beta}N(g_{0}({\bf x}))^{2},\ {\bf
x}\in{{S}}^{d}.$

Then

$f_{0}\in\Pi_{2N}^{d+1},\ \ f_{0}({\bf x}_{1})=\dots=f_{0}({\bf x}_{n})=0,\ \
f_{0}({\bf x})\geq 0\ {\rm for}\ {\bf x}\in{{S}}^{d},$

and

$\int_{{{S}}^{d}}f_{0}({\bf x})d\sigma_{d}({\bf
x})=N^{-\alpha}\ln^{-\beta}N\|g_{0}\|_{L_{2}}^{2}\asymp
N^{-\alpha}\ln^{-\beta}N.$

Now we show $\|f_{0}\|_{{\rm H}^{\alpha,\beta}({{S}}^{d})}\lesssim 1$. Take
$m\in{N}$ such that $2^{m-1}\leq N<2^{m}$. Then by (2.11) and the fact that
$E_{2^{j}}(f_{0})_{L_{2}}\leq\|f_{0}\|_{L_{2}}$ for $j\leq m+1$ and
$E_{2^{j}}(f_{0})_{L_{2}}=0$ for $j>m+1$, we have

$\displaystyle\|f_{0}\|_{{\rm H}^{\alpha,\beta}({{S}}^{d})}$
$\displaystyle\asymp\|f_{0}\|_{L_{2}}+\Big{(}\sum_{j=1}^{\infty}2^{2\alpha
j}j^{2\beta}E_{2^{j}}(f)_{L_{2}}^{2}\Big{)}^{1/2}$
$\displaystyle\lesssim\|f_{0}\|_{L_{2}}+\Big{(}\sum_{j=1}^{m+1}2^{2\alpha
j}j^{2\beta}\|f_{0}\|_{L_{2}}^{2}\Big{)}^{1/2}$ $\displaystyle\lesssim
2^{\alpha m}m^{\beta}\|f_{0}\|_{L_{2}}\lesssim\|g_{0}\|_{L_{4}}^{2}\lesssim 1$

It follows from [42, Section 5, Chapter 4] that

$\displaystyle e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$
$\displaystyle\geq\inf_{\begin{subarray}{c}{\bf x}_{1},\dots,{\bf
x}_{n}\in{S}^{d}\end{subarray}}\sup_{\begin{subarray}{c}\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})}\leq 1\\\ f({\bf x}_{1})=\dots=f({\bf
x}_{n})=0\end{subarray}}\left|\int_{{S}^{d}}f({\bf x}){\rm d}\sigma_{d}({\bf
x})\right|$ $\displaystyle\gtrsim\inf_{\begin{subarray}{c}{\bf
x}_{1},\dots,{\bf
x}_{n}\in{S}^{d}\end{subarray}}\left|\int_{{S}^{d}}f_{0}({\bf x}){\rm
d}\sigma_{d}({\bf x})\right|$ $\displaystyle\gtrsim
N^{-\alpha}\ln^{-\beta}N\gtrsim n^{-\alpha/d}\ln^{-\beta}n.$

The proof of Theorem 1.2 is finished. $\hfill\Box$

##  4\. Lower bounds for $e_{n}({\rm H}^{\alpha,\beta}({{S}}^{d}))$ with
$\alpha=d/2,\,\beta>1/2$

We start with a definition of convolution of two sequences and obtain lower
bounds for $e_{n}({\rm H}_{\mathaccent 869{\Lambda}})$ for a sequence
$\mathaccent 869{\Lambda}$ which is given as a sum of convolution squares.
Then we give lower bounds for $e_{n}({\rm H}_{\mathaccent 869{\Lambda}})$ for
a nonnegative non-increasing sequence $\mathaccent 869{\Lambda}$. Finally, we
prove Theorem 1.3.

Let $w_{0}(t):=c_{0}(1-t^{2})^{(d-2)/2},\ d\geq 2$ is a weight function on
$[-1,1]$, where $c_{0}=(\int_{-1}^{1}(1-t^{2})^{(d-2)/2}{\rm d}t)^{-1}$. For
any $f\in L_{1}([-1,1],w_{0})$, we define

$\langle f\rangle:=\int_{-1}^{1}f(t)w_{0}(t){\rm d}t.$

Thus, $L_{2}([-1,1],w_{0})$ is a Hilbert space with inner product

$\langle f,g\rangle_{2,w_{0}}=\int_{-1}^{1}f(t)g(t)w_{0}(t){\rm d}t=\langle
fg\rangle,$

and norm

$\|f\|_{2,w_{0}}=\left(\int_{-1}^{1}|f(t)|^{2}w_{0}(t){\rm
d}t\right)^{1/2}=\sqrt{\langle f^{2}\rangle}.$

Let $G_{\ell},\,\ell=0,1,\dots,$ be the normalized Gegenbauer polynomials
given in (2.4). Then $G_{\ell},\ \ell=0,1,\dots$, satisfy

(4.1) ${\rm deg}\,G_{\ell}=\ell\ \ {\rm and}\ \ \langle fG_{\ell}\rangle=0\
{\rm for\ all}\ f\in\mathcal{P}_{\ell-1},$

where $\mathcal{P}_{k}$ is the space of algebraic polynomials of degree at
most $k$. It follows from [11] that

(4.2)
$h_{\ell}^{2}:=\|G_{\ell}\|^{2}_{2,w_{0}}=\frac{\ell!(\ell+d-1)}{(d)_{\ell}(2\ell+d+1)}\,\,\asymp\ell^{-d+1},\
\ (d)_{\ell}=d(d+1)\cdots(d+\ell-1).$

We have

$\displaystyle\frac{h_{\ell}^{2}}{h_{\ell+1}^{2}}$
$\displaystyle=\frac{\ell!(\ell+d-1)(d)_{\ell+1}(2\ell+d+3)}{(\ell+1)!(\ell+d)(d)_{\ell}(2\ell+d+1)}=\frac{(\ell+d-1)(2\ell+d+3)}{(\ell+1)(2\ell+d+1)}>1,$

which means that $h_{\ell}^{2}$ is decreasing. Obviously,
$\\{G_{\ell}/h_{\ell}\\}_{\ell=0}^{\infty}$ forms an orthonormal basis of
$L_{2}([-1,1],w_{0})$. It follows that

(4.3) $\displaystyle
G_{\ell}G_{s}=\sum_{k=0}^{\infty}\big{\langle}G_{\ell}G_{s},\frac{G_{k}}{h_{k}}\big{\rangle}_{2,w_{0}}\frac{G_{k}}{h_{k}}=\sum_{k=0}^{\infty}\frac{\langle
G_{\ell}G_{s}G_{k}\rangle}{h^{2}_{k}}G_{k}=\sum_{k=0}^{\infty}C_{k}^{\ell,s}G_{k},$

where $C_{k}^{\ell,s}:={\langle G_{\ell}G_{s}G_{k}\rangle}/h_{k}^{2}$.
Moreover, $C_{k}^{\ell,s}$ have the following properties.

######  Proposition 4.1.

For any $\ell,s,k=0,1,\dots$, we have

  1. (1)

$C_{k}^{\ell,s}=C_{k}^{s,\ell}$;

  2. (2)

$C_{k}^{\ell,s}\geq 0$ if $|\ell-s|\leq k\leq\ell+s$;

  3. (3)

$C_{k}^{\ell,s}=0$ if $k>\ell+s$ and $k<|\ell-s|$;

  4. (4)

$\sum_{k=|\ell-s|}^{\ell+s}C_{k}^{\ell,s}=1$;

  5. (5)

$\sum_{k=|\ell-s|}^{\ell+s}\langle
G_{\ell}G_{s}G_{k}\rangle^{2}/h_{k}^{2}=\langle G_{\ell}^{2}G_{s}^{2}\rangle$.

###### Proof.

Property (1) follows from the definition of $C_{k}^{\ell,s}$. Property (2)
follows from [12, 21]. Property (3) is due to (4.1) and the fact that
$G_{\ell}G_{s}\in\mathcal{P}_{\ell+s}$. Property (4) comes from (4.3),
Property (3), and $G_{\ell}(1)=1$. Property (5) follows from (4.3), Property
(3), and the Parseval equality. ∎

Let $\mathaccent 869{\mu}$ be a nonnegative sequence in $\ell_{1}({N}_{0})$.
We set

$f_{\mathaccent 869{\mu}}(t)=\sum_{j=0}^{\infty}\mathaccent
869{\mu}_{j}G_{j}(t),\ \ \ {\rm and}\ \ \ K_{\mathaccent 869{\mu}}({\bf
x},{\bf y})=f_{\mathaccent 869{\mu}}({\bf x}\cdot{\bf y}),\ {\bf x},{\bf
y}\in{{S}}^{d}.$

Then ${\rm H}_{\mathaccent 869{\mu}}\equiv{\rm H}_{\mathaccent
869{\mu}}({{S}}^{d})$ is a RKHS with the reproducing kernel $K_{\mathaccent
869{\mu}}$, and for all ${\bf x}\in{{S}}^{d}$,

$K_{\mathaccent 869{\mu}}({\bf x},{\bf x})=\|\mathaccent
869{\mu}\|_{\ell_{1}},\ \ \ h_{\mathaccent 869{\mu}}({\bf x})=\mathaccent
869{\mu}_{0}$

where $h_{\mathaccent 869{\mu}}\in{\rm H}_{\mathaccent 869{\mu}}$ is the
representer of the integration problem INT for ${\rm H}_{\mathaccent
869{\mu}}$.

For two nonnegative sequences $\mathaccent 869{\mu},\mathaccent 869{\nu}$ in
$\ell_{1}({N}_{0})$, we define the formal convolution $\mathaccent
869{\mu}*\mathaccent 869{\nu}=\\{(\mathaccent 869{\mu}*\mathaccent
869{\nu})_{k}\\}_{k=0}^{\infty}$ by

$(\mathaccent 869{\mu}\ast\mathaccent
869{\nu})_{k}=\sum_{\ell=0}^{\infty}\sum_{s=0}^{\infty}C_{k}^{\ell,s}\mathaccent
869{\mu}_{\ell}\mathaccent 869{\nu}_{s}.$

It follows from (4.3) that

$f_{\mathaccent 869{\mu}\ast\mathaccent 869{\nu}}(t)=f_{\mathaccent
869{\mu}}(t)f_{\mathaccent 869{\nu}}(t),\ \ {\rm and}\ \ K_{\mathaccent
869{\mu}\ast\mathaccent 869{\nu}}({\bf x},{\bf y})=K_{\mathaccent
869{\mu}}({\bf x},{\bf y})K_{\mathaccent 869{\nu}}({\bf x},{\bf y})$

for all $t\in[-1,1]$ and ${\bf x},{\bf y}\in{{S}}^{d}.$ Specifically, we have

$K_{\mathaccent 869{\mu}\ast\mathaccent 869{\mu}}({\bf x},{\bf
y})=K_{\mathaccent 869{\mu}}({\bf x},{\bf y})^{2}.$

######  Theorem 4.2.

Let $\mathaccent 869{\mu},\mathaccent 869{\nu}$ be two nonnegative sequences
in $\ell_{1}({N}_{0})$, and let $\mathaccent 869{\Lambda}=\\{\mathaccent
869{\lambda}_{k}\\}_{k=0}^{\infty}$ satisfy $\mathaccent
869{\Lambda}=\mathaccent 869{\mu}*\mathaccent 869{\mu}+\mathaccent
869{\nu}*\mathaccent 869{\nu}$. Then we have

$e_{n}({\rm H}_{\mathaccent 869{\Lambda}})^{2}\geq\mathaccent
869{\lambda}_{0}\left(1-\frac{n\mathaccent
869{\lambda}_{0}}{\sum_{\ell=0}^{\infty}\mathaccent
869{\lambda}_{\ell}}\right).$

###### Proof.

For $n=0$, we have the initial error satisfies

$e_{0}({\rm H}_{\mathaccent 869{\Lambda}})^{2}=\|h_{\mathaccent
869{\Lambda}}\|_{{\rm H}_{\mathaccent 869{\Lambda}}}^{2}={\mathaccent
869{\lambda}_{0}},$

where $h_{\mathaccent 869{\Lambda}}({\bf x})=\mathaccent 869{\lambda}_{0}$ is
the representer of the integration problem INT for ${\rm H}_{\mathaccent
869{\Lambda}}$.

For $n=1,2,\dots$, it follows from Lemma 2.3 that for all ${\bf
x}_{1},...,{\bf x}_{n}\in{{S}}^{d}$, we know the matrixes $(K_{\mathaccent
869{\mu}}({\bf x}_{j},{\bf x}_{k}))^{n}_{j,k=1}$ and $(K_{\mathaccent
869{\nu}}({\bf x}_{j},{\bf x}_{k}))^{n}_{j,k=1}$ are positive semi-definite,
and by Lemma 2.5, the matrix with the entries

$K_{\mathaccent 869{\mu}}({\bf x}_{j},{\bf x}_{k})^{2}-\frac{K_{\mathaccent
869{\mu}}({\bf x}_{j},{\bf x}_{j})K_{\mathaccent 869{\mu}}({\bf x}_{k},{\bf
x}_{k})}{n}=K_{\mathaccent 869{\mu}*\mathaccent 869{\mu}}({\bf x}_{j},{\bf
x}_{k})-\frac{\|\mathaccent 869{\mu}\|_{\ell_{1}}^{2}}{n}$

is positive semi-definite. Similarly, the matrix with the entries

$K_{\mathaccent 869{\nu}*\mathaccent 869{\nu}}({\bf x}_{j},{\bf
x}_{k})-\frac{\|\mathaccent 869{\nu}\|_{\ell_{1}}^{2}}{n}$

is positive semi-definite. It follows that the matrix with the entries

$K_{\mathaccent 869{\mu}*\mathaccent 869{\mu}}({\bf x}_{j},{\bf
x}_{k})+K_{\mathaccent 869{\nu}*\mathaccent 869{\nu}}({\bf x}_{j},{\bf
x}_{k})-\frac{\|\mathaccent 869{\mu}\|_{\ell_{1}}^{2}}{n}-\frac{\|\mathaccent
869{\nu}\|_{\ell_{1}}^{2}}{n}$

is also positive semi-definite. We note that

$K_{\mathaccent 869{\Lambda}}({\bf x},{\bf y})=K_{\mathaccent
869{\mu}*\mathaccent 869{\mu}}({\bf x},{\bf y})+K_{\mathaccent
869{\nu}*\mathaccent 869{\nu}}({\bf x},{\bf y})\ \ {\rm and}\ \ \|\mathaccent
869{\Lambda}\|_{\ell_{1}}=\|\mathaccent
869{\mu}\|_{\ell_{1}}^{2}+\|\mathaccent 869{\nu}\|_{\ell_{1}}^{2}.$

Hence, the matrix with the entries

$K_{\mathaccent 869{\Lambda}}({\bf x}_{j},{\bf x}_{k})-\frac{\|\mathaccent
869{\Lambda}\|_{\ell_{1}}}{n}=K_{\mathaccent 869{\Lambda}}({\bf x}_{j},{\bf
x}_{k})-\frac{\|\mathaccent 869{\Lambda}\|_{\ell_{1}}}{n\mathaccent
869{\lambda}_{0}^{2}}h_{\mathaccent 869{\Lambda}}({\bf x}_{j})h_{\mathaccent
869{\Lambda}}({\bf x}_{k})$

is also positive semi-definite. Thus applying Lemma 2.4 and the fact that
$\|h_{\mathaccent 869{\Lambda}}\|_{{\rm H}_{\mathaccent
869{\Lambda}}}^{2}=\mathaccent 869{\lambda}_{0}$, we have

$e_{n}({\rm H}_{\mathaccent 869{\Lambda}})^{2}\geq\|h_{\mathaccent
869{\Lambda}}\|_{{\rm H}_{\mathaccent 869{\Lambda}}}^{2}-\frac{n\mathaccent
869{\lambda}^{2}_{0}}{\|\mathaccent 869{\Lambda}\|_{\ell_{1}}}=\mathaccent
869{\lambda}_{0}\left(1-\frac{n\mathaccent
869{\lambda}_{0}}{\sum_{\ell=0}^{\infty}\mathaccent
869{\lambda}_{\ell}}\right).$

This completes the proof of Theorem 4.2. ∎

######  Theorem 4.3.

Let $\mathaccent 869{\Lambda}=\\{\mathaccent
869{\lambda}_{\ell}\\}_{\ell=0}^{\infty}$ be a nonnegative non-increasing
sequence in $\ell_{1}({N}_{0})$. Then

$e_{n}({\rm H}_{\mathaccent 869{\Lambda}})^{2}\geq\mathaccent
869{\lambda}_{0}\left(1-\frac{2cn\mathaccent
869{\lambda}_{0}}{\sum_{\ell=1}^{\infty}\mathaccent
869{\lambda}_{2\ell}+2c\mathaccent 869{\lambda}_{0}}\right),$

where $c>1$ is an absolute constant such that
$h_{\lfloor\ell/2\rfloor}^{2}\leq ch_{\ell}^{2}$ for all $\ell\in{N}_{0}$,
$\lfloor a\rfloor$ denotes the largest integer not exceeding $a$, and
$h_{\ell}^{2}$ is given in (4.2).

###### Proof.

We set

$\mathaccent 869{\mu}=\\{\mathaccent 869{\mu}_{\ell}\\}_{\ell=0}^{\infty},\ \
\ \mathaccent 869{\mu}_{\ell}=(2cA)^{-1/2}\mathaccent 869{\lambda}_{2\ell},\
A=\sum_{\ell=0}^{\infty}{\mathaccent 869{\lambda}}_{2\ell},$

and

$\delta_{0}=\\{\delta_{0\ell}\\}_{\ell=0}^{\infty},\ \ \ \delta_{00}=1,\ {\rm
and}\ \delta_{0\ell}=0\ {\rm for}\ \ell\geq 1,$

where $c>1$ is an absolute constant such that
$h_{\lfloor\ell/2\rfloor}^{2}\leq ch_{\ell}^{2}$. Such constant $c$ exists
since $h_{\ell}^{2}\asymp(\ell+1)^{-d+1}$. We note that

$\delta_{0}*\delta_{0}=\delta_{0}.$

We have for $k\geq 1$,

$\displaystyle(\mathaccent 869{\mu}\ast\mathaccent 869{\mu})_{k}$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{s=0}^{\infty}C_{k}^{\ell,s}\mathaccent
869{\mu}_{\ell}\mathaccent
869{\mu}_{s}=(2cA)^{-1}\sum_{\ell=0}^{\infty}\sum_{s=0}^{\infty}C_{k}^{\ell,s}{\mathaccent
869{\lambda}}_{2\ell}{\mathaccent 869{\lambda}}_{2s}$
$\displaystyle\leq\frac{(2cA)^{-1}}{h_{k}^{2}}\left\\{\sum_{\ell=0}^{\infty}\sum_{s\geq
k/2}C_{s}^{\ell,k}{\mathaccent 869{\lambda}}_{2\ell}\mathaccent
869{\lambda}_{2s}h_{s}^{2}+\sum_{s=0}^{\infty}\sum_{\ell\geq
k/2}C_{\ell}^{s,k}{\mathaccent 869{\lambda}}_{2s}\mathaccent
869{\lambda}_{2\ell}h_{\ell}^{2}\right\\}$
$\displaystyle\leq\frac{(2cA)^{-1}}{h_{k}^{2}}\left\\{\sum_{\ell=0}^{\infty}\left(\sum_{s\geq
k/2}C_{s}^{\ell,k}\right){\mathaccent
869{\lambda}}_{2\ell}+\sum_{s=0}^{\infty}\left(\sum_{\ell\geq
k/2}C_{\ell}^{s,k}\right){\mathaccent 869{\lambda}}_{2s}\right\\}{\mathaccent
869{\lambda}}_{k}h_{\lfloor k/2\rfloor}^{2}$
$\displaystyle\leq(2cA)^{-1}2\frac{h_{\lfloor
k/2\rfloor}^{2}}{h_{k}^{2}}\left(\sum_{\ell=0}^{\infty}{\mathaccent
869{\lambda}}_{2\ell}\right)\mathaccent 869{\lambda}_{k}\leq\mathaccent
869{\lambda}_{k},$

where in the first inequality we used the fact that $C_{k}^{\ell,s}\neq 0$
implies $s\geq k/2$ or ${\ell\geq k/2}$ and that
$C_{k}^{\ell,s}h_{k}^{2}=\langle G_{k}G_{\ell}G_{s}\rangle$, in the second
inequality we used $\mathaccent 869{\Lambda}$ is a nonnegative non-increasing
sequence, in the last second inequality we used Proposition 4.1 (4). And for
$k=0$ we have

$\displaystyle(\mathaccent 869{\mu}\ast\mathaccent 869{\mu})_{0}$
$\displaystyle=\sum_{\ell=0}^{\infty}\frac{h_{\ell}^{2}}{h^{2}_{0}}\tilde{\mu}^{2}_{\ell}=(2cA)^{-1}\sum_{\ell=0}^{\infty}\frac{h_{\ell}^{2}}{h^{2}_{0}}\mathaccent
869{\lambda}_{2\ell}^{2}$
$\displaystyle\leq(2cA)^{-1}\left(\sum_{\ell=0}^{\infty}{\mathaccent
869{\lambda}}_{2\ell}\right)\mathaccent 869{\lambda}_{0}=(2c)^{-1}\mathaccent
869{\lambda}_{0}.$

Now we put $\mathaccent 869{\nu}=\mathaccent 869{\mu}\ast\mathaccent
869{\mu}+t\,\delta_{0}*\delta_{0}$ and choose $t\geq(1-(2c)^{-1})\mathaccent
869{\lambda}_{0}$ such that $\mathaccent 869{\nu}_{0}=\mathaccent
869{\lambda}_{0}$. Then $\mathaccent 869{\nu}$ is a sum of two convolution
squares, and $\mathaccent 869{\lambda}_{\ell}\geq\mathaccent 869{\mu}_{\ell}$
for all $\ell\in{N}_{0}$. It follows from Theorem 4.2 that

(4.4) $e_{n}({\rm H}_{\mathaccent 869{\Lambda}})^{2}\geq e_{n}({\rm
H}_{\mathaccent 869{\nu}})^{2}\geq\mathaccent
869{\nu}_{0}\left(1-\frac{n\mathaccent 869{\nu}_{0}}{\|\mathaccent
869{\nu}\|_{\ell_{1}}}\right)=\mathaccent
869{\lambda}_{0}\left(1-\frac{n\mathaccent 869{\lambda}_{0}}{\|\mathaccent
869{\nu}\|_{\ell_{1}}}\right),$

where

$\displaystyle\|\mathaccent 869{\nu}\|_{\ell_{1}}$
$\displaystyle=\|\mathaccent
869{\mu}\|_{\ell_{1}}^{2}+t\|\delta_{0}\|_{\ell_{1}}^{2}\geq(2cA)^{-1}\Bigl{(}\sum_{\ell=0}^{\infty}{\mathaccent
869{\lambda}}_{2\ell}\Bigr{)}^{2}+(1-(2c)^{-1})\mathaccent 869{\lambda}_{0}$
$\displaystyle=(2c)^{-1}\sum_{\ell=0}^{\infty}{\mathaccent
869{\lambda}}_{2\ell}+(1-(2c)^{-1})\mathaccent
869{\lambda}_{0}=(2c)^{-1}\sum_{\ell=1}^{\infty}{\mathaccent
869{\lambda}}_{2\ell}+\mathaccent 869{\lambda}_{0}.$

By (4.4) we obtain

$e_{n}({\rm H}_{\mathaccent 869{\Lambda}})^{2}\geq\mathaccent
869{\lambda}_{0}\left(1-\frac{2cn\mathaccent
869{\lambda}_{0}}{\sum_{\ell=1}^{\infty}{\mathaccent
869{\lambda}}_{2\ell}+2c\mathaccent 869{\lambda}_{0}}\right).$

This completes the proof of Theorem 4.3. ∎

######  Theorem 4.4.

Let $\mathaccent 869{\Lambda}=\\{\mathaccent
869{\lambda}_{\ell}\\}_{\ell=0}^{\infty}$ be a nonnegative non-increasing
sequence in $\ell_{1}({N}_{0})$. Then

$e_{n}({\rm H}_{\mathaccent
869{\Lambda}})^{2}\geq\min\left\\{\frac{1}{2}\mathaccent
869{\lambda}_{0},\frac{1}{4cn}\sum_{\ell>2cn}\mathaccent
869{\lambda}_{2\ell}\right\\},$

where $c>1$ is an absolute constant such that
$h_{\lfloor\ell/2\rfloor}^{2}\leq ch_{\ell}^{2}$ for all $\ell\in{N}_{0}$.

###### Proof.

Let $\tau=\\{\tau_{\ell}\\}_{\ell=0}^{\infty}$ defined by

$\tau_{0}=\min\left\\{\mathaccent
869{\lambda}_{0},\,\max\left\\{\frac{1}{2}\mathaccent 869{\lambda}_{\lfloor
4cn\rfloor},\frac{1}{2cn}\sum_{\ell>2cn}\mathaccent
869{\lambda}_{2\ell}\right\\}\right\\}$

and

$\tau_{\ell}:=\left\\{\begin{aligned} &\mathaccent 869{\lambda}_{\ell},&&{\rm
for}\ \ \ell>4cn,\\\ &\mathaccent 869{\lambda}_{\lfloor 4cn\rfloor},&&{\rm
for}\ \ 0<\ell\leq 4cn.\end{aligned}\right.$

Then $\tau$ is a nonnegative non-increasing sequence in $\ell_{1}({N}_{0})$.
Moreover, we have $\mathaccent 869{\lambda}_{\ell}>\tau_{\ell}$ for all
$\ell\in{N}_{0}$, and

$\displaystyle\sum^{\infty}_{\ell=1}\tau_{2\ell}+2c\tau_{0}$
$\displaystyle\geq\sum_{2\ell>4cn}\mathaccent
869{\lambda}_{2\ell}+\sum_{0<2\ell\leq 4cn}\mathaccent 869{\lambda}_{\lfloor
4cn\rfloor}+2c\tau_{0}$ $\displaystyle\geq\sum_{\ell>2cn}\mathaccent
869{\lambda}_{2\ell}+cn\mathaccent 869{\lambda}_{\lfloor 4cn\rfloor}\geq
4cn\tau_{0}.$

Applying Theorem 4.3, we have

$\displaystyle e_{n}({\rm H}_{\mathaccent 869{\Lambda}})^{2}\geq e_{n}({\rm
H}_{\tau})^{2}$
$\displaystyle\geq\tau_{0}\left(1-\frac{2cn\tau_{0}}{\sum^{\infty}_{\ell=1}\tau_{2\ell}+2c\tau_{0}}\right)\geq\frac{1}{2}\tau_{0}$
$\displaystyle\geq\min\left\\{\frac{1}{2}\mathaccent
869{\lambda}_{0},\frac{1}{4cn}\sum_{\ell>2cn}\mathaccent
869{\lambda}_{2\ell}\right\\}.$

This completes the proof of Theorem 4.4. ∎

Proof of Theorem 1.3

We consider the space ${\rm H}^{\alpha,\beta}({{S}}^{d})={\rm
H}^{\Lambda^{(\alpha,\beta)}}({{S}}^{d})={\rm H}_{\mathaccent
869{\Lambda}^{(\alpha,\beta)}}$ with $\alpha=d/2,\,\beta>1/2$, where

$\Lambda^{(\alpha,\beta)}=\\{\lambda_{\ell}^{(\alpha,\beta)}\\}_{\ell=0}^{\infty},\
\ \ \mathaccent 869{\Lambda}^{(\alpha,\beta)}=\\{\mathaccent
869{\lambda}_{\ell}^{(\alpha,\beta)}\\}_{\ell=0}^{\infty},$
$\lambda_{\ell}^{(\alpha,\beta)}=(1+\ell(\ell+d-1))^{-d/4}(\ln(3+\ell(\ell+d-1)))^{-\beta},$

and

$\mathaccent
869{\lambda}_{\ell}^{(\alpha,\beta)}=(\lambda_{\ell}^{(\alpha,\beta)})^{2}N(d,\ell)\asymp\ell^{-1}\ln^{-2\beta}\ell.$

It follows from Theorem 4.4 that

$e_{n}({\rm
H}^{\alpha,\beta}({{S}}^{d}))^{2}\gtrsim\frac{1}{4cn}\sum_{\ell>2cn}\mathaccent
869{\lambda}^{(\alpha,\beta)}_{2\ell}\asymp
n^{-1}\sum_{\ell>2cn}\ell^{-1}\ln^{-2\beta}\ell\asymp
n^{-1}\ln^{-2\beta+1}\ell.$

This completes the proof of Theorem 1.3. $\hfill\Box$

##  5\. Proof of Theorems 1.5

In order to prove Theorem 1.5, we shall use the following lemma.

######  Lemma 5.1.

(See [6, Theorem 2.1].) Let $\Gamma$ be a finite subset of $\,\mathbb{S}^{d}$,
and let $\\{\mu_{\boldsymbol{\omega}}:\ \ \boldsymbol{\omega}\in\Gamma\\}$ be
a set of positive numbers satisfying

$\sum_{\boldsymbol{\omega}\in\Gamma}\mu_{\boldsymbol{\omega}}|f(\boldsymbol{\omega})|^{p_{0}}\leq
C_{1}\int_{\mathbb{S}^{d}}|f({\bf x})|^{p_{0}}\,d\sigma_{d}({\bf x}),\ \
\forall f\in\Pi_{N}^{d+1},$

for some $0<p_{0}<\infty$ and some positive integer $N$. If $0<q<\infty$,
$M\geq N$ and $f\in\Pi_{M}^{d+1}$, then

$\sum_{\boldsymbol{\omega}\in\Gamma}\mu_{\boldsymbol{\omega}}|f(\boldsymbol{\omega})|^{q}\leq
CC_{1}\Bigl{(}\frac{M}{N}\Bigr{)}^{d}\int_{\mathbb{S}^{d}}|f({\bf
y})|^{q}\,d\sigma_{d}({\bf y}),$

where $C>0$ depends only on $d$ and $q$.

Proof of Theorem 1.5

By (2.17) it suffices to give the upper bounds for
$\|f-L_{N}^{{S}}(f)\|_{L_{2}}$. For $f\in{\rm H}^{\alpha,\beta}({{S}}^{d})$
with $\alpha>d/2$ or $\alpha=d/2,\,\beta>1/2$, we define

$\sigma_{1}(f)=S_{2}(f),\ \ \sigma_{j}(f)=S_{2^{j}}(f)-S_{2^{j-1}}(f),\ \ {\rm
for}\ j\geq 2.$

Then $\sigma_{j}(f)\in\Pi_{2^{j}}^{d+1}$ and

$f=\sum_{j=1}^{\infty}\sigma_{j}(f)$

converges in the uniform norm.

For a positive integer $N$, let $m$ be an integer such that $2^{m}\leq
N<2^{m+1}$. Observe that for any $f\in{\rm H}^{\alpha,\beta}({{S}}^{d})$

$\displaystyle\|f-L_{N}^{{S}}(f)\|_{L_{2}}$
$\displaystyle\leq\|f-S_{2^{m}}(f)\|_{L_{2}}+\|L_{N}^{{S}}(f)-S_{2^{m}}(f)\|_{L_{2}}$
(5.1) $\displaystyle=:{\rm I}+{\rm II}.$

To estimate ${\rm I}=\|f-S_{2^{m}}(f)\|_{L_{2}}$, we can use (2.7) to obtain

$\displaystyle{\rm I}\ $
$\displaystyle\lesssim\left(2^{m}\right)^{-\alpha}\ln^{-\beta}(1+2^{m})\,\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})}$ (5.2) $\displaystyle\lesssim
N^{-\alpha}\ln^{-\beta}N\,\|f\|_{{\rm H}^{\alpha,\beta}({{S}}^{d})}.$

To estimate $\text{II}=\|L_{N}^{{S}}(f)-S_{2^{m}}(f)\|_{L_{2}}$, we can use
(2.12) and (2.16) with $S_{2^{m}}(f)\in\Pi_{N}^{d+1}$ to obtain that

$\displaystyle{\rm II}\ $ $\displaystyle\leq
A^{-1/2}\|L_{N}^{{S}}(f)-S_{2^{m}}(f)\|_{(N)}$ $\displaystyle\leq
A^{-1/2}\\{\|L_{N}^{{S}}(f)-f\|_{(N)}+\|f-S_{2^{m}}(f)\|_{(N)}\\}$ (5.3)
$\displaystyle\leq 2A^{-1/2}\|f-S_{2^{m}}(f)\|_{(N)},$

where we use the fact that $L_{N}^{{S}}(f)-S_{2^{m}}(f)\in\Pi_{N}^{d+1}$.
Since for all $f\in{\rm H}^{\alpha,\beta}({{S}}^{d})$, the series
$\sum_{j=m+1}^{\infty}\sigma_{j}(f)$ converges to $f-S_{2^{m}}(f)$ in the
uniform norm, we have

(5.4)
$\|f-S_{2^{m}}(f)\|_{(N)}\leq\sum\limits_{j=m+1}^{\infty}\|\sigma_{j}(f)\|_{(N)}.$

For $j\geq m+1$, using Lemma 5.1 with $p_{0}=q=2$ we get

(5.5) $\displaystyle\|\sigma_{j}(f)\|_{(N)}^{2}$
$\displaystyle=\sum_{k=1}^{l_{N}}\tau_{N,k}|\sigma_{j}(f)(\mathbf{x}_{N,k})|^{2}\lesssim
B{2^{jd}}N^{-d}\|\sigma_{j}(f)\|_{L_{2}}^{2}.$

Applying (5.3), (5.4), (5.5), the Hölder inequality, and (2.10) we have

$\displaystyle{\rm II}$ $\displaystyle\leq
A^{-1/2}\sum_{j=m+1}^{\infty}\|\sigma_{j}(f)\|_{(N)}$
$\displaystyle\lesssim\kappa^{1/2}N^{-d/2}\sum_{j=m+1}^{\infty}{2^{jd/2}}\|\sigma_{j}(f)\|_{L_{2}}$
$\displaystyle\lesssim\kappa^{1/2}N^{-d/2}\left(\sum_{j=m+1}^{\infty}2^{2j\alpha}j^{2\beta}\|\sigma_{j}(f)\|_{L_{2}}^{2}\right)^{1/2}\
\left(\sum_{j=m+1}^{\infty}2^{-2j\alpha+jd}j^{-2\beta}\right)^{1/2}$
$\displaystyle\lesssim\kappa^{1/2}N^{-d/2}\left(\sum_{j=m+1}^{\infty}2^{-2j\alpha+jd}j^{-2\beta}\right)^{1/2}\
\|f\|_{{\rm H}^{\alpha,\beta}({{S}}^{d})}$
$\displaystyle\lesssim\left\\{\begin{aligned}
&\kappa^{1/2}N^{-d/2}m^{-\beta+1/2}\ \|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})},&&\alpha=d/2,\,\beta>1,\\\
&\kappa^{1/2}N^{-d/2}m^{-\beta}2^{-(\alpha-d/2)m}\ \|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})},&&\alpha>d/2,\ \beta\in{R},\end{aligned}\right.$
(5.6) $\displaystyle\lesssim\kappa^{1/2}C_{\alpha,\beta}(N)\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})}$

Therefore, it follows from (5), (5), and (5) that

$\|f-L_{N}^{{S}}(f)\|_{L_{2}}\lesssim(1+\kappa^{1/2})C_{\alpha,\beta}(N)\|f\|_{{\rm
H}^{\alpha,\beta}({{S}}^{d})},$

where

$C_{\alpha,\beta}(N):=\bigg{\\{}\begin{array}[]{ll}N^{-\alpha}\ln^{-\beta}N,&\alpha>d/2,\\\
N^{-\alpha}\ln^{-\beta+1/2}N,&\alpha=d/2,\beta>1/2.\end{array}$

This completes the proof of Theorem 1.5. $\hfill\Box$

Acknowledgment The authors were supported by the National Natural Science
Foundation of China (Project no. 12371098).

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