# Approximation and quadrature by weighted least squares polynomials on the
sphere

Wanting Lu  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.

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_{2}$ on a usual
compact space $\mathcal{M}$, Gröchenig in [11] introduced the weighted least
squares polynomials and the least squares quadrature from pointwise samples of
a function, and obtained approximation theorems and quadrature errors. In this
paper we obtain approximation theorems and quadrature errors on the sphere
which are optimal. We also give upper bounds of the operator norms of the
weighted least squares operators.

###### Key words and phrases:

Marcinkiewicz-Zygmund inequalities; Sampling; Sobolev spaces; Least squares
problem; operator norms

###### 2010 Mathematics Subject Classification:

41A17; 42A10; 65D30.

##  1\. Introduction

Let

${{S}}^{d}:=\\{x=(x_{1},\dots,x_{d+1})\in\mathbb{R}^{d+1}\,\big{|}\
|x|^{2}=\sum_{k=1}^{d+1}|x_{k}|^{2}=1\\}\ \ \ (d\geq 2)$

be the unit sphere in $\mathbb{R}^{d+1}$ endowed with the rotationally
invariant measure $\sigma$ normalized by $\int_{{{S}}^{d}}d\sigma=1$. We
denote by $L_{2}({{S}}^{d})$ the usual Hilbert space of square-integrable
functions on ${{S}}^{d}$ with the inner product

(1.1) $\langle f,g\rangle:=\int_{{{S}}^{d}}f(x)g(x)d\sigma(x)$

and the norm $\|f\|_{2}=\langle f,f\rangle^{1/2}$, and by $C({{S}}^{d})$ the
space of continuous functions on ${{S}}^{d}$ with supremum norm

$\|f\|:=\|f\|_{\infty}:=\sup_{x\in{{S}}^{d}}|f(x)|.$

The space $\Pi_{n}^{d}$ of spherical polynomials on ${{S}}^{d}$ of degree
$\leq n$ consists of the restrictions to ${{S}}^{d}$ of all polynomials on
${R}^{d+1}$ of total degree $\leq n$. The dimension of $\Pi_{n}^{d}$ is given
dy

$d_{n}:=\dim(\Pi_{n}^{d})=\frac{(2n+d)\Gamma(n+d)}{\Gamma(d+1)\Gamma(n+1)}.$

Let $S_{n}$ be the orthogonal projection from $L_{2}({{S}}^{d})$ onto
$\Pi_{n}^{d}$, i. e., for $f\in L_{2}({{S}}^{d})$,

(1.2) $S_{n}f({x})=\int_{{{S}}^{d}}f(y)E_{n}(x,y)d\sigma(y),$

where $E_{n}(x,y)$ is the reproducing kernel for $\Pi_{n}^{d}$ with respect to
the inner product (1.1). We note that $E_{n}(x,y)$ satisfies the following
properties:

(1) For any $x,y\in{{S}}^{d}$, $E_{n}(x,y)=E_{n}(y,x)$;

(2) For any fixed $y\in{{S}}^{d}$, $E_{n}(\cdot,y)\in\Pi_{n}^{d}$;

(3) For any $p\in\Pi_{n}^{d}$ and $x\in{{S}}^{d}$, $p(x)=S_{n}p(x)=\langle
p,E_{n}(\cdot,x)\rangle$.

This paper is concerned with constructive polynomial approximation on
${{S}}^{d}$ that uses function values (the samples) at selected well-
distributed points (sometimes called standard information). Here the “well-
distributed” points indicate that those points constitute a Marcinkiewicz-
Zygmund (MZ) family on ${{S}}^{d}$ defined as follows.

######  Definition 1.1.

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

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

The ratio $\kappa=B/A$ is the global condition number of the Marcinkiewicz-
Zygmund family $\mathcal{X}$, and
$\mathcal{X}_{n}=\\{x_{n,k}:k=1,\dots,l_{n}\\}$ is the $n$th layer of
$\mathcal{X}$.

The MZ inequality (1.3) means that the $L_{2}$-norm of a spherical polynomial
of degree at most $n$ is comparable to the discrete version given by the
weighted $\ell_{2}$-norm of its restriction to $\mathcal{X}_{n}$. It follows
from [19, 2, 7] that such MZ families exist if the families are dense enough.
We remark that Fekete points of degree $\lfloor n(1+\varepsilon)\rfloor\
(\varepsilon>0)$ on the sphere are MZ families with the equal weights
$\tau_{n,k}=\frac{1}{d_{n}}$, see [17], where $\lfloor a\rfloor$ is the
largest integer not exceeding $a\in{R}$. Also, sufficient conditions and
necessary density conditions for MZ families with the equal weights
$\tau_{n,k}=\frac{1}{d_{n}}$ on the sphere are obtained in [18] and [16],
respectively.

Given a MZ family on a usual compact space $\mathcal{M}$, Gröchenig in [11]
introduced the weighted least squares polynomials and the least squares
quadrature from pointwise samples of a function, and obtained approximation
theorems and quadrature errors. However, the obtained error estimates in [11]
are not optimal due to the generality of $\mathcal{M}$. In this paper we
confine the study to the sphere ${{S}}^{d}$.

Let $\mathcal{X}$ be a MZ family with associated weight $\tau$. Given the
samples $\\{f(x_{n,k})\\}$ of a continuous function $f$ on ${{S}}^{d}$ on the
$n$th layer $\mathcal{X}_{n}$, we want to approximate $f$ using only these
samples. For this we solve a sequence of weighted least squares problems with
samples taken from the samples $\mathcal{X}_{n}$:

(1.4) $L_{n}f=\arg\min_{p\in\Pi_{n}^{d}}\
\sum_{k=1}^{l_{n}}|f(x_{n,k})-p(x_{n,k})|^{2}\tau_{n,k}.$

This procedure yields a sequence $\\{L_{n}f\\}$ of the best weighted
$\ell_{2}$-approximation of the data $\\{f(x_{n,k})\\}$ by a spherical
polynomial in $\Pi_{n}^{d}$ for every $f\in C({{S}}^{d})$. We call $L_{n}f$
the weighted least squares polynomial, and $L_{n}$ the weighted least squares
operator. Clearly, $L_{n}$ is the projection onto $\Pi_{n}^{d}$, i.e., $L_{n}$
is a bounded linear operator on $C({{S}}^{d})$ satisfying that
$L_{n}^{2}=L_{n}$ and the range of $L_{n}$ is $\Pi_{n}^{d}$. It follows from
(1.3) that for $p\in\Pi_{n}^{d}$, $p=0$ whenever $p(x_{n,k})=0$. This means
that usually $\mathcal{X}_{n}$ contains more than $d_{n}=\dim\Pi_{n}^{d}$
points, so that it is not an interpolating set for $\Pi_{n}^{d}$. Therefore,
$L_{n}f$ is usually a quasi-interpolant.

Let $E_{n}(x,y)$ be the reproducing kernel of $\Pi_{n}^{d}$ with respect to
the inner product (1.1). Then the Marcinkiewicz-Zygmund inequalities (1.3) say
that every set $\\{\tilde{e}_{n,k}:k=1,\dots,l_{n}\\},\
\tilde{e}_{n,k}=\tau^{1/2}_{n,k}E_{n}(\cdot,x_{n,k})$ is a frame for
$\Pi_{n}^{d}$ with uniform frame bounds $A,B>0$, i.e., for all
$p\in\Pi_{n}^{d}$, we have

$A\|p\|_{2}^{2}\leq\sum_{k=1}^{l_{n}}|\langle
p,\tilde{e}_{n,k}\rangle|^{2}\leq B\|p\|_{2}^{2}.$

It follows that the associated frame operator

$T_{n}p(\cdot)=\sum_{k=1}^{l_{n}}\langle
p,\tilde{e}_{n,k}\rangle\tilde{e}_{n,k}(\cdot)=\sum_{k=1}^{l_{n}}\tau_{n,k}p(x_{n,k})E_{n}(\cdot,x_{n,k})$

is invertible on $\Pi_{n}^{d}$ for every $n\in{N}$. We obtain the dual frame
$\\{e_{n,k}=T_{n}^{-1}(\tilde{e}_{n,k}),\ k=1,\dots,l_{n}\\}$ for
$\Pi_{n}^{d}$ with uniform frame bounds $B^{-1}$ and $A^{-1}$, and every
polynomial $p\in\Pi_{n}^{d}$ can be reconstructed from the samples on
$\mathcal{X}_{n}$ by

$p=T_{n}^{-1}T_{n}p=\sum_{k=1}^{l_{n}}\langle p,\tilde{e}_{n,k}\rangle
T_{n}^{-1}\tilde{e}_{n,k}=\sum_{k=1}^{l_{n}}\tau_{n,k}^{1/2}p(x_{n,k})e_{n,k}.$

The weights for the quadrature rules are defined by

$w_{n,k}=\langle\tau^{1/2}_{n,k}e_{n,k},1\rangle=\tau^{1/2}_{n,k}\int_{{{S}}^{d}}e_{n,k}(x)d\sigma(x)$

and the corresponding quadrature rule is defined by

$I_{n}(f)=\sum_{k=1}^{l_{n}}w_{n,k}f(x_{n,k}).$

Such quadrature rules are usually named least squares quadrature.

Let $\mathcal{X}$ be a MZ family with associated weight $\tau$. For a function
$f$ in the Sobolev space $H^{\sigma}(\mathbb{S}^{d}),\ \sigma>d/2$ (see
Section 2 for definition of $H^{\sigma}({{S}}^{d})$), Gröchenig obtained in
[11] that

$\|f-L_{n}f\|_{2}\leq c(1+\kappa^{2})^{1/2}n^{-\sigma+d/2}\|f\|_{H^{\sigma}},$

and

$\big{|}\int_{{{S}}^{d}}f(x)d\sigma(x)-I_{n}(f)\big{|}\leq
c(1+\kappa^{2})^{1/2}n^{-\sigma+d/2}\|f\|_{H^{\sigma}},$

where $c$ depends on $d,\ s$, but not on $f$ or $\kappa$ or the MZ family
$\mathcal{X}$.

We remark that the condition $\sigma>d/2$ is the well known necessary and
sufficient condition for $H^{\sigma}(\mathbb{S}^{d})$ to be continuously
embedded in $C(\mathbb{S}^{d})$, and is unavoidable in any approximation
scheme that employs function values.

In this paper we improve the above results and obtain the optimal estimates.
One of our main results can be formulated as follows.

######  Theorem 1.2.

Let $\mathcal{X}$ be a MZ family on ${{S}}^{d}$ with associated weight $\tau$
and global condition number $\kappa=B/A$, $L_{n}$ be the weighted least
squares operators defined by (1.4), and let $\\{I_{n}\,|\,n\in{N}\\}$ be the
sequence of the least squares quadrature. If $f\in H^{\sigma}({{S}}^{d}),\
\sigma>d/2$, then we have

(1.5) $\|f-L_{n}f\|_{2}\leq
c(1+\kappa^{2})^{1/2}n^{-\sigma}\|f\|_{H^{\sigma}},$

and

(1.6) $\Big{|}\int_{{{S}}^{d}}f(x)d\sigma(x)-I_{n}(f)\Big{|}\leq
c(1+\kappa^{2})^{1/2}n^{-\sigma}\|f\|_{H^{\sigma}},$

where $c$ depends on $d,\ s$, but not on $f$ or $\kappa$ or the MZ family
$\mathcal{X}$.

Remarkably, the convergence rates for the weighted least squares approximation
and for the least squares quadrature errors are optimal (as explained in
Remark 1.3 below) in a variety of Sobolev space settings. When the global
condition number $\kappa=1$, the weighted least squares operators are just the
hyperinterpolation operators on the sphere (see Subsection 2.5). In this case,
the inequality (1.5) was obtained in [28].

######  Remark 1.3.

For $N\in{N}$, the sampling numbers (or the optimal recovery) of the Sobolev
classes $BH^{\sigma}({{S}}^{d})$ (the unit ball of the space
$H^{\sigma}({{S}}^{d})$) in $L_{2}({{S}}^{d})$ are defined by

$g_{N}(BH^{\sigma}({{S}}^{d}),L_{2}({{S}}^{d})):=\inf_{\begin{subarray}{c}\xi_{1},\dots,\xi_{N}\in{{S}}^{d}\\\
\varphi:\,{R}^{N}\to L_{2}({{S}}^{d})\end{subarray}}\sup_{f\in
BH^{\sigma}({{S}}^{d})}\|f-\varphi(f(\xi_{1}),\dots,f(\xi_{N}))\|_{2},$

where the infimum is taken over all $N$ points $\xi_{1},\dots,\xi_{N}$ in
${{S}}^{d}$ and all mappings $\varphi$ from ${R}^{N}$ to $L_{2}({{S}}^{d})$.
The optimal quadrature errors of the Sobolev classes $BH^{\sigma}({{S}}^{d})$
are defined by

$e_{N}(\text{Int};BH^{\sigma}({{S}}^{d})):=\inf_{\begin{subarray}{c}{\lambda}_{1},\dots,{\lambda}_{N}\in\mathbb{R}\\\
\xi_{1},\dots,\xi_{N}\in\mathbb{S}^{d}\end{subarray}}\sup_{f\in
BH^{\sigma}({{S}}^{d})}\Bigl{|}\int_{\mathbb{S}^{d}}f(x)\,d\sigma(x)-\sum_{j=1}^{N}{\lambda}_{j}f(\xi_{j})\Bigr{|}.$

It follows from [31], [1] and [13] that

(1.7) $g_{N}(BH^{\sigma}({{S}}^{d}),L_{2}({{S}}^{d}))\asymp N^{-\sigma/d}\ \ \
{\rm and}\ \ \ e_{N}(\text{Int};BH^{\sigma}({{S}}^{d}))\asymp N^{-\sigma/d},$

where the notation $A(N)\asymp B(N)$ means that $A(N)\ll B(N)$ and $A(N)\gg
B(N)$, $A(N)\ll B(N)$ means that there exists a positive constant $c$
independent of $N$ such that $A(N)\leq cB(N)$, and $A(N)\gg B(N)$ means
$B(N)\ll A(N)$.

It follows from [16, 17, 18] that there exist MZ families with $l_{n}\asymp
d_{n}\asymp N\asymp n^{d}$. For such MZ family it follows from (1.7) that

$\sup_{f\in BH^{\sigma}({{S}}^{d})}\|f-L_{n}(f)\|_{2}\asymp N^{-s/d}\asymp
g_{N}(BH^{\sigma}({{S}}^{d}),L_{2}({{S}}^{d})),$

which implies that the weighted least squares operators $L_{n}$ are
asymptotically optimal algorithms in the sense of optimal recovery. Also, we
have

$\sup_{f\in
BH^{\sigma}({{S}}^{d})}\Bigl{|}\int_{\mathbb{S}^{d}}f(x)\,d\sigma(x)-I_{n}(f)\Bigr{|}\asymp
N^{-s/d}\asymp e_{N}(\text{Int};BH^{\sigma}({{S}}^{d})),$

which means that the least squares quadrature rules are the asymptotically
optimal quadrature formulas.

For a linear operator $L$ on $C({{S}}^{d})$, the operator norm $\|L\|$ of $L$
is given by

$\|L\|:=\sup\\{\|Lf\|\ \big{|}\ f\in C({{S}}^{d}),\ \|f\|\leq 1\\}.$

The quantity $\|L\|$ of a projection $L$ is also called the Lebesgue constant
of $L$ and the estimation of $\|L\|$ is extremely important in numerical
computation. We use the Christoffel functions to get the following estimation.

######  Theorem 1.4.

Let $\mathcal{X}$ be a MZ family on ${{S}}^{d}$ with associated weight $\tau$
and global condition number $\kappa=B/A$, and let $L_{n}$ be the weighted
least squares operators defined by (1.4). Then we have

$n^{(d-1)/2}\ll\|L_{n}\|\ll\kappa^{1/2}n^{d/2}.$

######  Remark 1.5.

When $\kappa=1$, the weighted least squares operators are just the
hyperinterpolation operators on the sphere (see Subsection 2.5). In this case,
we have (see [23, 10, 20])

$\|L_{n}\|\asymp n^{(d-1)/2}.$

Hence, we conjecture that the upper bound of the operator norm of $L_{n}$ is
$n^{(d-1)/2}$ multiplied by a positive constant independent of $n$. However,
we have not been able to prove it.

######  Remark 1.6.

Since the weighted least squares operators $L_{n}$ is the projection onto
$\Pi_{n}^{d}$, by the Lebesgue theorem the hyperinterpolation error in the
uniform norm can be estimated as

$\|f-L_{n}(f)\|\leq(1+\|L_{n}\|)E_{n}(f)\ll\kappa^{1/2}n^{d/2}E_{n}(f),$

where

$E_{n}(f):=\inf_{p\in\Pi_{n}^{d}}\|f-p\|$

is the best approximation of $f$ by $\Pi_{n}^{d}$.

##  2\. Preliminary

This section is devoted to giving some basic facts about spherical harmonics
(see for example, [7]) and some preliminary knowledge.

###  2.1. Harmonic analysis on the sphere

Let ${{S}}^{d}=\\{x\in{R}^{d+1}\ |\ |x|=1\\}$ denote the unit sphere in
${R}^{d+1}$, where $(x,y)=x\cdot y$ is the usual inner product and
$|x|=(x,x)^{1/2}$ is the Euclidean norm. We denote by $\mathcal{H}_{\ell}^{d}$
the space of all spherical harmonics of degree $l$ on ${{S}}^{d}$, i.e., the
space of the restrictions to ${{S}}^{d}$ of all harmonic homogeneous
polynomials of exact degree $\ell$ on ${R}^{d+1}$. It is well known that the
dimension of $\mathcal{H}_{\ell}^{d}$ is

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

The spaces $\mathcal{H}_{k}^{d},\ k=0,1,2,\dots$ are just the eigenspaces
corresponding to the eigenvalues $-k(k+d-1)$ of the Laplace-Beltrami operator
$\triangle$ on the sphere ${{S}}^{d}$ and are mutually orthogonal with respect
to the inner product

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

Let

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

be a fixed orthonormal basis for $\mathcal{H}_{\ell}^{d}$. We have the
addition theorem for the spherical harmonics of degree $\ell$:

$\sum_{k=1}^{N(d,\ell)}Y_{\ell,k}(x)Y_{\ell,k}(y)=\frac{\ell+\lambda}{\lambda}C_{\ell}^{\lambda}(x\cdot
y),\ \ x,y\in{{S}}^{d},\ \lambda=\frac{d-1}{2},\ \ d\geq 2,$

where $C_{\ell}^{\lambda}(t)$ is the usual ultraspherical (Gegenbeuer)
polynomial of order ${\lambda}$ normalized by
$C_{\ell}^{\lambda}(1)=\frac{\Gamma(\ell+2\lambda)}{\Gamma(2\lambda)\Gamma(\ell+1)}$
and generated by

$\frac{1}{(1-2zt+z^{2})^{\lambda}}=\sum_{k=0}^{\infty}C_{k}^{\lambda}(t)z^{k}\
\ (0\leq z<1).$

(See ([26, p. 81]).

Since the space $\Pi_{n}^{d}$ can be expressed as a direct sum of

$\Pi_{n}^{d}=\mathcal{H}_{0}^{d}\bigoplus\mathcal{H}_{1}^{d}\bigoplus\cdots\bigoplus\mathcal{H}_{n}^{d},$

we obtain that

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

forms an $L_{2}$-orthonormal basis of $\Pi_{n}^{d}$. Hence, the
$L_{2}$-reproducing kernel $E_{n}(x,y)$ of $\Pi_{n}^{d}$ has the explicit
expression:

$E_{n}(x,y)=\sum_{\ell=0}^{n}\sum_{k=1}^{N(d,\ell)}Y_{\ell,k}(x)Y_{\ell,k}(y)=\sum_{\ell=0}^{n}\frac{\ell+\lambda}{\lambda}C_{\ell}^{\lambda}(x\cdot
y),\ \lambda=\frac{d-1}{2},\ x,y\in{{S}}^{d}.$

Specifically, we have

(2.1)
$E_{n}(x,x)=\sum_{\ell=0}^{n}\frac{\ell+\lambda}{\lambda}\frac{\Gamma(\ell+2\lambda)}{\Gamma(2\lambda)\Gamma(\ell+1)}\asymp
n^{d},\ \ \lambda=\frac{d-1}{2},\ \ x\in{{S}}^{d}.$

We remark that

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

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:

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

where

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

are the Fourier coefficients of $f$. The Sobolev space
$H^{\sigma}({{S}}^{d})$, where $\sigma>0$, is defined as the space of all
functions in $L_{2}({{S}}^{d})$ for which the norm

$\|f\|_{H^{\sigma}}:=\bigg{(}\sum_{\ell=0}^{\infty}\Big{(}1+\ell(\ell+d-1)\Big{)}^{\sigma}\sum_{k=1}^{N(d,\ell)}\big{|}\langle
f,Y_{\ell,k}\rangle\big{|}^{2}\bigg{)}^{1/2}$

is finite. The space $H^{\sigma}({{S}}^{d})$ is a Hilbert space with the inner
product

$\langle
f,g\rangle_{{}_{H^{\sigma}}}:=\sum_{\ell=0}^{\infty}\Big{(}1+\ell(\ell+d-1)\Big{)}^{\sigma}\sum_{k=1}^{N(d,\ell)}\langle
f,Y_{\ell,k}\rangle\,\langle g,Y_{\ell,k}\rangle,$

which induces the norm $\|\cdot\|_{H^{\sigma}}$. It is easily seen that for
$f\in H^{\sigma}({{S}}^{d})$,

(2.2) $\|f-S_{n}f\|_{2}\leq cn^{-\sigma}\|f\|_{H^{\sigma}},$

where $S_{n}f$ is given by (1.2).

If $\sigma>d/2$, then the space $H^{\sigma}(\mathbb{S}^{d})$ is continuously
embedded in $C(\mathbb{S}^{d})$ and is a reproducing kernel Hilbert space with
the reproducing kernel

$K_{\sigma}(x,y)=\sum_{\ell=0}^{\infty}\Big{(}1+\ell(\ell+d-1)\Big{)}^{-\sigma}\sum_{k=1}^{N(d,\ell)}Y_{\ell,k}(x)Y_{\ell,k}(y).$

###  2.2. Two examples of MZ families

We shall give two examples of MZ families with equal weights. Let $X$ be a
finite subset of ${{S}}^{d}$ with cardinality $d_{n}:=\dim\Pi_{n}^{d}$. The
points $\xi_{1},\dots,\xi_{d_{n}}$ in $X$ maximizing a Vandermonde-type
determinant

$\big{|}\Delta(\xi_{1},\dots,\xi_{d_{n}})\big{|}=\big{|}\det\big{(}Q_{i}(\xi_{j})\big{)}_{i,j=1}^{d_{n}}\big{|}$

are called Fekete points of degree $n$ for ${{S}}^{d}$ (these points are
sometimes called extremal fundamental systems of points, as in [24]), where
$Q_{i},\ i=1,\dots,d_{n}$ are a basis of $\Pi_{n}^{d}$. It is known that
Fekete points are independent of the choice of the polynomial basis. For any
fixed $\varepsilon>0$, if $\mathcal{X}=\\{\mathcal{X}_{n}\\}$ and
$\mathcal{X}_{n}$ are Fekete points of degree $\lfloor
n(1+\varepsilon)\rfloor$ for ${{S}}^{d}$, then $\mathcal{X}$ is a MZ family
with equal weights $\tau_{n,k}=\frac{1}{d_{n}}$ (see [17]).

Let $X$ be a finite subset of ${{S}}^{d}$. The mesh norm of $X$ is

$\rho(X)=\sup_{u\in{{S}}^{d}}\inf_{z\in X}d(u,z),$

where $d(x,y)$ denotes the standard geodesic distance $\arccos x\cdot y$
between two points $x$ and $y$ on ${{S}}^{d}$. Let
$\mathcal{X}=\\{\mathcal{X}_{n}\\}=\\{x_{n,k}:n=1,2,\dots,\,k=1,\dots,l_{n}\\}$
be a doubly-indexed set of points in ${{S}}^{d}$. We say that $\mathcal{X}$ is
uniformly separated if there is a positive number $\varepsilon>0$ such that

$d(x_{n,k},x_{n,l})\geq\frac{\varepsilon}{n+1}\ \ {\rm if}\ \ k\neq l$

for all $n\in{N}$. Let $\mathcal{X}=\\{\mathcal{X}_{n}\\}$ be a uniformly
separated array in ${{S}}^{d}$ such that for all $n\geq 1$,

$\rho(\mathcal{X}_{n})\leq\frac{\eta}{n},$

where $\eta<\pi/2$. Then $\mathcal{X}$ is an MZ family with the equal weights
$\tau_{n,k}=\frac{1}{d_{n}}$ (see [18]).

However, in [17] and [18], the authors do not give estimates of the global
condition numbers of two MZ families on the sphere.

###  2.3.  The weighted least squares polynomials $L_{n}f$

Now let $\mathcal{X}$ be a MZ family with associated weight $\tau$. We use the
weighted discretized inner product

(2.3) $\langle
f,g\rangle_{(n)}:=\sum_{k=1}^{l_{n}}f(x_{n,k})g(x_{n,k})\tau_{n,k}$

and the discretized norm

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

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

$L_{n}f=\arg\min_{p\in\Pi_{n}^{d}}\|f-p\|_{(n)}^{2}=\arg\min_{p\in\Pi_{n}^{d}}\
\sum_{k=1}^{l_{n}}|f(x_{n,k})-p(x_{n,k})|^{2}\tau_{n,k}.$

We shall give a formal expression for $L_{n}f$. Let $\varphi_{i},\
i=1,\dots,d_{n}$ be an orthonormal basis of $\Pi_{n}^{d}$ with respect to the
weighted discrete scalar product (2.3), i.e.,

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

Such $\varphi_{i},\ i=1,\dots,d_{n}$ can be obtained by applying Gram-Schmidt
orthogonalization process to the basis $\\{Y_{\ell,k}\ |\
k=1,\dots,N(d,\ell),\ \ell=0,1,\dots,n\\}$ of $\Pi_{n}^{d}$. We set

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

Clearly, the weighted least squares polynomial $L_{n}f$ is just the orthogonal
projection of $f$ onto $\Pi_{n}^{d}$ with respect to the weighted discrete
scalar product (2.3), i. e.,

(2.4) $L_{n}f({x})=\langle
f,D_{n}(\cdot,x)\rangle_{(n)}=\sum_{k=1}^{l_{n}}f(x_{n,k})D(x_{n,k},x)\tau_{n,k},$

and $D_{n}(x,y)$ is the reproducing kernel for $\Pi_{n}^{d}$ with respect to
the weighted discrete scalar product (2.3) satisfying the following
properties:

(1) For any $x,y\in{{S}}^{d}$, $D_{n}(x,y)=D_{n}(y,x)$;

(2) For any fixed $y\in{{S}}^{d}$, $D_{n}(\cdot,y)\in\Pi_{n}^{d}$;

(3) For any $p\in\Pi_{n}^{d}$ and $x\in{{S}}^{d}$,

(2.5) $p(x)=L_{n}p(x)=\langle
p,D_{n}(\cdot,x)\rangle_{(n)}=\sum_{k=1}^{l_{n}}p(x_{n,k})D(x_{n,k},x)\tau_{n,k}.$

According to the definition of the operator norm and (2.4), by the standard
argument we have

(2.6)
$\|L_{n}\|=\max_{x\in{{S}}^{d}}\sum_{k=1}^{l_{n}}|D(x_{n,k},x)|\tau_{n,k}.$

###  2.4. The least squares quadrature

Let $\mathcal{X}$ be a MZ family with a weight $\tau$ and let $E_{n}(x,y)$ be
the reproducing kernel of $\Pi_{n}^{d}$. Then the frame operator

$T_{n}p(\cdot)=\sum_{k=1}^{l_{n}}\langle
p,\tilde{e}_{n,k}\rangle\tilde{e}_{n,k}(\cdot)=\sum_{k=1}^{l_{n}}\tau_{n,k}p(x_{n,k})E_{n}(\cdot,x_{n,k})$

is invertible on $\Pi_{n}^{d}$, where
$\tilde{e}_{n,k}=\tau^{1/2}_{n,k}E_{n}(\cdot,x_{n,k})$. We set

$e_{n,k}=T_{n}^{-1}(\tilde{e}_{n,k})\ \ {\rm and}\ \
w_{n,k}=\langle\tau^{1/2}_{n,k}e_{n,k},1\rangle=\tau^{1/2}_{n,k}\int_{{{S}}^{d}}e_{n,k}(x)d\sigma(x).$

Then the least squares quadrature is defined by

$I_{n}(f)=\sum_{k=1}^{l_{n}}w_{n,k}f(x_{n,k}).$

In this subsection we shall show that

(2.7)
$I_{n}(f)=\int_{{{S}}^{d}}L_{n}f(x)d\sigma(x)=\sum_{k=1}^{l_{n}}f(x_{n,k})\tau_{n,k}\int_{{{S}}^{d}}D(x_{n,k},x)d\sigma(x).$

In order to prove (2.7) it suffices to show that

$\tau^{1/2}_{n,k}e_{n,k}=\tau_{n,k}D(x_{n,k},\cdot).$

Indeed, by (2.4) and (2.5) we have for $x,y\in{{S}}^{d}$

$\displaystyle T_{n}(D_{n}(\cdot,y))(x)$
$\displaystyle=\sum_{k=1}^{l_{n}}\tau_{n,k}D(x_{n,k},y)E_{n}(x,x_{n,k})$
$\displaystyle=L_{n}(E(\cdot,x))(y)=E_{n}(x,y).$

It follows that

$\tau^{1/2}_{n,k}e_{n,k}=\tau_{n,k}T_{n}^{-1}(E_{n}(\cdot,x_{n,k}))=\tau_{n,k}D(x_{n,k},\cdot).$

Hence, (2.7) holds. By (2.7) and the Hölder inequality we obtain that

(2.8)
$\big{|}\int_{{{S}}^{d}}f(x)d\sigma(x)-I_{n}(f)\big{|}\leq\int_{{{S}}^{d}}|f(x)-L_{n}f(x)|d\sigma(x)\leq\|f-L_{n}f\|_{2}.$

###  2.5. Hyperinterpolation and the weighted least squares polynomials

Hyperinterpolation was originally introduced by I. H. Sloan (see [22]). 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 cubature formula. Hence, hyperinterpolation is a
discretized orthogonal projection on polynomial subspaces and provides a
polynomial approximation which relies only on a discrete set of data. In
recent years, hyperinterpolation has attracted much interest, and a great
number of interesting results have been obtained (see [3, 4, 5, 6, 10, 12, 14,
15, 20, 21, 22, 23, 25, 27, 28, 29, 30, 32]).

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

$\int_{{{S}}^{d}}f(x)d\sigma(x)=Q_{n}(f)=\sum_{k=1}^{l_{n}}\tau_{n,k}f(x_{n,k}).$

For any $p\in\Pi_{n}^{d}$, using $f=p^{2}$ in the above equality we obtain
that the family $\mathcal{X}$ is a MZ family with the constants $A=B=1$ and
the global condition number $\kappa$ is equal to $1$.

The hyperinterpolation operator $H_{n}$ on ${{S}}^{d}$ is defined by

$H_{n}f(x)=\sum\limits_{k=1}^{l_{n}}\tau_{n,k}f(x_{n,k})E_{n}(x,x_{n,k}),\ \ \
f\in C({{S}}^{d}).$

We use the discretized inner product

$\langle f,g\rangle_{(n)}:=\sum_{k=1}^{l_{n}}f(x_{n,k})g(x_{n,k})\tau_{n,k}$

and the discretized norm $\|f\|_{(n)}^{2}=\langle f,f\rangle_{(n)}$. Since the
quadrature formula $Q_{n}$ is exact for $\Pi_{2n}^{d}$, we get for all
$p,q\in\Pi_{n}^{d}$,

$\langle p,q\rangle=\int_{{{S}}^{d}}p(x)q(x)d\sigma(x)=Q_{n}(p\,q)=\langle
p,q\rangle_{(n)}.$

Hence,

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

forms an orthonormal basis of $\Pi_{n}^{d}$ with respect to
$\langle\cdot,\cdot\rangle_{(n)}$. It follows that

$H_{n}f(x)=\langle
f,E_{n}(\cdot,x)\rangle_{(n)}=\sum_{\ell=0}^{n}\sum_{k=1}^{N(d,\ell)}\langle
f,Y_{\ell,k}\rangle_{(n)}Y_{\ell,k}(x),$

which means that the hyperinterpolation operator $H_{n}$ is just the
discretized orthogonal projection on the polynomial subspace $\Pi_{n}^{d}$
with respect to $\langle\cdot,\cdot\rangle_{(n)}$, and satisfies

$\|f-H_{n}f\|_{(n)}^{2}=\min_{p\in\Pi_{n}^{d}}\|f-p\|_{(n)}^{2}=\min_{p\in\Pi_{n}^{d}}\sum_{k=1}^{l_{n}}|f(x_{n,k})-p(x_{n,k})|^{2}\tau_{n,k}.$

Hence, the hyperinterpolation $H_{n}f$ is just the weighted least squares
polynomial for a sequence of positive quadrature formulas on ${{S}}^{d}$ which
are exact for $\Pi_{2n}^{d}$.

On the other hand, if the global condition number $\kappa$ of a MZ family
$\mathcal{X}$ with the weight $\tau$ is equal to $1$, then for all
$p\in\Pi_{n}^{d}$, we have

$A\|p\|_{2}^{2}=\sum_{k=1}^{l_{n}}|p(x_{n,k})|^{2}\tau_{n,k}.$

It follows that for $f,g\in\Pi_{n}^{d}$,

$A\langle f,g\rangle=\sum_{k=1}^{l_{n}}f(x_{n,k})g(x_{n,k})\tau_{n,k}.$

Since any function in $\Pi_{2n}^{d}$ can be expressed as a linear combination
of the product of the functions $x^{\alpha}$ and $x^{\beta}$, we obtain that
$\mathcal{X}$ determines a sequence of positive quadrature formulas
$Q_{n}(f)=\frac{1}{A}\sum\limits_{k=1}^{l_{n}}\tau_{n,k}f(x_{n,k})$ on
${{S}}^{d}$ which are exact for $\Pi_{2n}^{d}$, where, $x\in{{S}}^{d},\
\alpha,\beta\in{N}_{0}^{d+1},|\alpha|\leq n,\ |\beta|\leq n,\
x^{\alpha}:=x_{1}^{\alpha_{1}}\dots x_{d+1}^{\alpha_{d+1}},\
|\alpha|=\alpha_{1}+\dots+\alpha_{d+1}$. This means that the weighted least
squares operators $L_{n}$ are just the hyperinterpolation operators $H_{n}$.

Hence, the hyperinterpolation is just the weighted least squares polynomial
for a MZ family with the global condition number $\kappa=1$, and the weighted
least squares polynomial may be viewed as a generalization of the
hyperinterpolation.

##  3\. Proofs of Theorems 1.2 and 1.4

In order to prove Theorem 1.2, we shall use the following lemma.

######  Lemma 3.1.

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

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

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

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

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

Proof of Theorem 1.2.

We rewrite the orthonormal basis $\\{Y_{\ell,k}\ |\ k=1,\dots,N(d,\ell),\
\ell=0,1,\dots,n\\}$ of $\Pi_{n}^{d}$ as $\\{\phi_{k}\ |\ k=1,\dots,d_{n}\\}$.
Let

${\bf
y}_{n}=(\tau_{n,1}^{1/2}f(x_{n,1}),\dots,\tau_{n,l_{n}}^{1/2}f(x_{n,l_{n}}))^{*}$

be the given sampling data column vector, $U_{n}$ be the $l_{n}\times d_{n}$
matrix with entries

$(U_{n})_{kl}=\tau_{n,k}^{1/2}\phi_{l}(x_{n,k}),\ \ \ k=1,\dots,l_{n},\
l=1,\dots,d_{n},$

and let $R_{n}=U_{n}^{*}U_{n}$, where $U_{n}^{*}$ is the conjugate transpose
of the matrix $U_{n}$. Suppose that the solution to the weighted least squares
problem (1.4) is the polynomial

$L_{n}f=\sum_{k=1}^{d_{n}}a_{n,k}\phi_{k}\in\Pi_{n}^{d}$

with coefficient column vector ${\bf a}_{n}$. According to the standard
formula for the solution of a least squares problem by means of the Moore-
Penrose inverse
$U_{n}^{+}=(U_{n}^{*}U_{n})^{-1}U_{n}^{*}=R_{n}^{-1}U_{n}^{*}$, we obtain that

${\bf a}_{n}=U_{n}^{+}{\bf y}_{n}=R_{n}^{-1}U^{*}_{n}{\bf y}_{n}.$

For any $c\in{R}^{d_{n}}$, $p=\sum_{k=1}^{d_{n}}c_{k}\phi_{k}$, we have

$\|p\|_{2}^{2}=|c|^{2}\ \ {\rm and}\ \
(R_{n}c,c)=(U_{n}c,U_{n}c)=\sum_{k=1}^{n}|p(x_{n,k}|^{2}\tau_{n,k}.$

It follows from (1.3) that

$A|c|^{2}\leq(R_{n}c,c)\leq B|c|^{2},$

which means that the spectrum of every $R_{n}$ is contained in the interval
$[A,B]$. We obtain that the operator norm of $R_{n}^{-1}$ is bounded by
$A^{-1}$ and the operator norm of $U_{n}^{*}$ is bounded by

$\|U_{n}^{*}\|=\|U_{n}\|=\|U_{n}^{*}U_{n}\|^{1/2}=\|R_{n}\|^{1/2}\leq
B^{1/2},$

where the operator norm of a matrix $A$ is defined by
$\|A\|=\sup_{|x|=1}|Ax|.$

We use the orthogonal decomposition

$\|f-L_{n}f\|_{2}^{2}=\|f-S_{n}f\|_{2}^{2}+\|S_{n}f-L_{n}f\|_{2}^{2}.$

We recall that

$S_{n}f=\sum_{k=1}^{d_{n}}f_{n,k}\phi_{k}$

with coefficient column vector ${\bf f}_{n}$, where $f_{n,k}=\langle
f,\phi_{k}\rangle$. By the Parseval equality we obtain that

$\displaystyle\|S_{n}f-L_{n}f\|_{2}^{2}$ $\displaystyle=|{\bf f}_{n}-{\bf
a}_{n}|^{2}=|{\bf f}_{n}-R_{n}^{-1}U_{n}^{*}{\bf y}_{n}|^{2}$
$\displaystyle=|R_{n}^{-1}U_{n}^{*}(U_{n}{\bf f}_{n}-{\bf y}_{n})|^{2}$
$\displaystyle\leq A^{-2}B|U_{n}{\bf f}_{n}-{\bf y}_{n}|^{2}.$

Finally, we have

$(U_{n}{\bf f}_{n})_{k}=\tau_{n,k}^{1/2}S_{n}f(x_{n,k}).$

Hence,

$|U_{n}{\bf f}_{n}-{\bf
y}_{n}|^{2}=\sum_{k=1}^{l_{n}}|f(x_{n,k})-S_{n}f(x_{n,k})|^{2}\tau_{n,k}=\|f-S_{n}f\|_{(n)}^{2}.$

It follows that

(3.1) $\|S_{n}f-L_{n}f\|_{2}^{2}\leq A^{-2}B\|f-S_{n}f\|_{(n)}^{2}$

For $f\in H^{\sigma}({{S}}^{d}),\ \sigma>d/2$, we define

$A_{0}f=S_{n}f,\ \ A_{k}f=S_{2^{k}n}f-S_{2^{k-1}n}f\ \ {\rm for}\ \ k\geq 1.$

Then for $f\in H^{\sigma}({{S}}^{d}),\ s>d/2$, the series
$\sum\limits_{k=0}^{\infty}A_{k}f(x)$ converges to $f(x)$ uniformly in
${{S}}^{d}$. By (2.2) we have

(3.2) $\|A_{k}f\|_{2}\leq\|f-S_{2^{k}n}f\|_{2}+\|f-S_{2^{k-1}n}f\|_{2}\ll
2^{-k\sigma}n^{-\sigma}\|f\|_{H^{\sigma}}.$

Note that

$f(x)-S_{n}f(x)=\sum_{k=1}^{\infty}A_{k}f(x),$

and the right series converges uniformly on ${{S}}^{d}$. Hence, by (3.1) and
the trigonometric inequality we have

$\|S_{n}f-L_{n}f\|_{2}\leq A^{-1}B^{1/2}\|f-S_{n}f\|_{(n)}\leq
A^{-1}B^{1/2}\sum_{k=1}^{\infty}\|A_{k}f\|_{(n)}.$

Note that $A_{k}f\in\Pi_{2^{k}n}^{d}$. Using Lemma 3.1 with $p_{0}=q=2$ and
(3.2), we obtain that

$\displaystyle\|A_{k}f\|_{(n)}^{2}$
$\displaystyle=\sum_{k=1}^{l_{n}}|A_{k}f(x_{n,k})|^{2}\tau_{n,k}$
$\displaystyle\leq
cB\Big{(}\frac{2^{k}n}{n}\Big{)}^{d}\int_{{{S}}^{d}}|A_{k}f(x)|^{2}\,d\sigma(x)$
$\displaystyle\ll B2^{kd}2^{-2k\sigma}n^{-2\sigma}\|f\|_{H^{\sigma}}^{2}.$

We have

$\displaystyle\|S_{n}f-L_{n}f\|_{2}$ $\displaystyle\leq
A^{-1}B^{1/2}\sum_{k=1}^{\infty}\|A_{k}f\|_{(n)}$ $\displaystyle\ll
A^{-1}B\sum_{k=1}^{\infty}2^{kd/2}2^{-k\sigma}n^{-\sigma}\|f\|_{H^{\sigma}}$
$\displaystyle\ll A^{-1}Bn^{-\sigma}\|f\|_{H^{\sigma}}.$

It follows that

$\displaystyle\|f-L_{n}f\|_{2}$
$\displaystyle=\big{(}\|f-S_{n}f\|_{2}^{2}+\|S_{n}f-L_{n}f\|_{2}^{2}\big{)}^{1/2}$
$\displaystyle\ll(1+\kappa^{2})^{1/2}n^{-\sigma}\|f\|_{H^{\sigma}}.$

This completes the proof of (1.5). Inequality (1.6) follows from (1.5) and
(2.8) directly.

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

######  Remark 3.2.

The inequality (3.1) was proved in [11]. For the convenience of the reader, we
provide details of the proof.

Proof of Theorem 1.4.

In order to give the lower estimate of the operator norm $\|L_{n}\|$, we use
the Daugavet theorem to obtain that

$\|L_{n}\|\geq\|S_{n}\|\asymp n^{(d-1)/2}.$

So it suffices to get the upper estimate of $\|L_{n}\|$.

Let the functions

$\Phi_{n}(x)=\inf_{p\in\Pi_{n}^{d},\ |p(x)|=1}\|f\|_{2}^{2},\ \
x\in{{S}}^{d},$

and

$\Psi_{n}(x)=\inf_{p\in\Pi_{n}^{d},\ |p(x)|=1}\|f\|_{(n)}^{2},\ \
x\in{{S}}^{d}$

be the Christoffel functions with respect to the inner
$\langle\cdot,\cdot\rangle$ and $\langle\cdot,\cdot\rangle_{(n)}$,
respectively. We remark that estimates for Christoffel functions are useful in
comparing different norms of functions in $\Pi_{n}^{d}$, and they are also
basic tools in the theory of orthogonal polynomials.

It follows from (1.3) that

(3.3) $A\Phi_{n}(x)\leq\Psi_{n}(x)\leq B\Phi_{n}(x),\ \ x\in{{S}}^{d}.$

Now let $E_{n}(x,y)$ and $D_{n}(x,y)$ be the reproducing kernels of
$\Pi_{n}^{d}$ with respect to the inner $\langle\cdot,\cdot\rangle$ and
$\langle\cdot,\cdot\rangle_{(n)}$, respectively. According to [9, Theorem
3.5.6], we have

$E_{n}(x,x)^{-1}=\Phi_{n}(x)\ \ {\rm and}\ \ D_{n}(x,x)^{-1}=\Psi_{n}(x).$

It follows from (3.3) that

(3.4) $B^{-1}E_{n}(x,x)\leq D_{n}(x,x)\leq A^{-1}E_{n}(x,x),\ \
x\in{{S}}^{d}.$

Using $p=1$ in (1.3) we get that

(3.5) $A\leq\sum_{k=1}^{l_{n}}\tau_{n,k}\leq B.$

Using (2.6), the Cauchy-Schwartz inequality, (2.5), (3.5), (3.4), and (2.1),
we have

$\displaystyle\|L_{n}\|$
$\displaystyle=\max_{x\in{{S}}^{d}}\sum_{k=1}^{l_{n}}\tau_{n,k}|D_{n}(x,x_{n,k})|$
$\displaystyle\leq\max_{x\in{{S}}^{d}}\Big{(}\sum_{k=1}^{l_{n}}\tau_{n,k}(D_{n}(x,x_{n,k}))^{2}\Big{)}^{1/2}\Big{(}\sum_{k=1}^{l_{n}}\tau_{n,k}\Big{)}^{1/2}$
$\displaystyle\leq B^{1/2}\max_{x\in{{S}}^{d}}D_{n}(x,x)^{1/2}$
$\displaystyle\ll\kappa^{1/2}\max_{x\in{{S}}^{d}}E_{n}(x,x)^{1/2}$
$\displaystyle\ll\kappa^{1/2}n^{d/2}.$

This completes the proof of Theorem 1.4. $\hfill\Box$

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