Convergence Rate of Regularized Regression Associated with Zonal Translation Networks

1. Introduction

It is known that convolutional neural network provides various models and algorithms to process data model in many fields such as computer vision(see e.g. [1]), natural lagrange processing (see e.g. [2]), and sequence analysis in bio-informatics(see e.g. [3]). The regularized neural network learning has thus become an attractive research topic (see e.g.[4,5,6,7,8,9]). In the present paper, we shall give theory analysis for the convergence rate of regularized regression associated with zonal translation network on the unit sphere.

Let X be a compact subset in the d-dimensional Euclidean space $R^d$ with the usual norm $\parallel x\parallel =\sqrt{{\displaystyle \sum _{k=1}^d}{x}_k^2}$ for $x=(x_1,x_2,\cdots ,x_d)\in R^d$ and Y be a nonempty closed subset contained in $[-M,M]$ for a given $M>0$. The aim of the regression learning problem is to learn the target function which describes the relationship between the input $x\in X$ and the output $y\in Y$ from a hypothesis function space. In most of the cases, the target function is offered as a set of observations $z={\left\{z_i\right\}}_{i=1}^m={\left\{(x_i,y_i)\right\}}_{i=1}^m\in Z^m$ which has been drawn independently and identically distributed (i.i.d.) according to a probability joint distribution (measure) $\rho (x,y)={\rho }_X\left(x\right)\rho \left(y\right|x)$ on $Z=X\times Y$, where $\rho \left(y\right|x\left)\right(x\in X)$ is the conditional probability of y for a given x and ${\rho }_X\left(x\right)$ is the marginal probability about x, i.e.,for every integrable function $\phi :X\times Y\to R$ there holds

$$\begin{array}{c}\hfill {\int }_{X\times Y}\phi (x,y)d\rho ={\int }_X\left({\int }_Y\phi (x,y)d\rho (y\mid x)\right)d{\rho }_X.\end{array}$$

For a given normed space $(B,\parallel ·{\parallel }_B)$ consisting of real functions on X we define the regularized learning framework with B as

$$\begin{array}{c}\hfill f_{z,\lambda }:=arg{\displaystyle \underset{f\in B}{min}}\left({\mathcal{E}}_z\left(f\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f\parallel }_B^2\right),\end{array}$$

(1)

where $\lambda >0$ is the regularization parameter,${\mathcal{E}}_z\left(f\right)$ is the empirical mean

$${\mathcal{E}}_z\left(f\right)={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}{(y_i-f\left(x_i\right))}^2.$$

The optimal target function is the regression function

$$f_{\rho }\left(x\right)={\int }_{Y}yd\rho \left(y\right|x)$$

satisfying

$$\begin{array}{c}\hfill f_{\rho }={\displaystyle \underset{f}{inf}}{\mathcal{E}}_{\rho }\left(f\right),\end{array}$$

where ${\mathcal{E}}_{\rho }\left(f\right)={\int }_Z{(y-f\left(x\right))}^{2}d\rho$ and the inf is taken over all the ${\rho }_X$-measurable functions f. Moreover, there holds the famous equality (see e.g.[10])

$$\begin{array}{c}\hfill \parallel f-f_{\rho }{\parallel }_{L^2\left({\rho }_X\right)}^2={\mathcal{E}}_{\rho }\left(f\right)-{\mathcal{E}}_{\rho }\left(f_{\rho }\right),f\in L^2\left({\rho }_X\right).\end{array}$$

(2)

The choices for the hypothesis space B in (1) are riches. For example, C.P.An et al choose the algebraic polynomial class as B (see [11,12,13],). In [14], C. De Mol et al chose the dictionary as B. Recently some papers chose the Sobolev space as the hypothesis space B (see [15,16]). By kernel method we mean traditionally replacing B with a reproducing kernel Hilbert space (RKHS) $(H_K,{\langle ·,·\rangle }_K)$ which is a Hilbert space consisting of real functions defined on X and there is a Mercer kernel $K_x\left(y\right)=K(x,y)$ on $X\times X$ (i.e., $K_x\left(y\right)$ is a continuous and symmetric function on $X\times X$ and for any $n\ge 1$ and any $\{x_1,x_2,\cdots ,x_n\}\subset X$ the Mercer matrices ${\left(K(x_i,x_j)\right)}_{i,j=1,2,\cdots ,n}$ are positive semi-definite) such that

$$\begin{array}{c}\hfill f\left(x\right)={\langle f,K_x\rangle }_K,\forall x\in X,\forall f\in H_K.\end{array}$$

(3)

and there hold the embeded inequality

$$\begin{array}{c}\hfill \left|f\left(x\right)\right|\le {c\parallel f\parallel }_K,\forall x\in X,\forall f\in H_K,\end{array}$$

(4)

where c is a constant independent of f and x. There are two results about the optimal solution $f_{z,\lambda }$. The reproducing property (3) yields the representation

$$\begin{array}{c}\hfill f_{z,\lambda }\left(x\right)={\displaystyle \sum _{k=1}^m}{c}_k{K}_{x_k}\left(x\right),\forall x\in X.\end{array}$$

(5)

The embeded inequality (4) yields the inequality

$$\begin{array}{c}\hfill |f_{z,\lambda }\left(x\right)|\le \sqrt{{\displaystyle \frac{2M}{\lambda }}}\forall x\in X.\end{array}$$

(6)

Representation (5) is the theory basis for kernel regularized regression (see e.g.[17,18]). Inequality (6) is the key inequality for bounding the learning rate with covering number method (see e.g.[19,20,21]). For other skills of the kernel method one can cite [10,22,23,24] et al.

It is particularly important to mention here that translation networks have recently been used for the hypothesis space of regularized learning (see e.g. [25,26]). From the view of approximation theory, a simple single layer translation network with m neurons is a function space produced by translating a given function $\varphi$ which can be written as

$$\begin{array}{c}\hfill {\Delta }_{\varphi ,\overline{X}}^X=\{{\displaystyle \sum _{i=1}^m}{c}_i{T}_{x_i}(\varphi ,·):c_i\in R,x_i\in X,i=1,2,\cdots ,m\},\end{array}$$

where $\overline{X}={\left\{x_i\right\}}_{i=1}^m\subset X$ is a given node set, and for a given $x\in X$$T_x(\varphi ,y)$ is a translation operator corresponding to X. For example, when $X=R^d$ and $X={[-\pi ,\pi ]}^d$, we choose $T_x(\varphi ,y)$ as the usual convolution translation operator $\varphi (x-y)$ for the $\varphi$ defined on $R^d$ or an $2\pi$-periodic function $\varphi$ (see [27,28]. When $X=S^{d-1}=\{x\in R^d:\parallel x\parallel =1\}$ is the unit sphere in $R^d$, one can choose $T_x(\varphi ,y)$ as the zonal translation operator $\varphi \left(xy\right)$ for a given $\varphi$ defined on the interval $[-1,1]$ (see [29]). In [30], we defined a kind of translation operator $T_x(\varphi ,y)$ for $X=[-1,1]$.

It is easy to see that the approximation ability and the construction of a translation network depend upon the nodes set $\overline{X}={\left\{x_i\right\}}_{i=1}^m$ (see e.g. [31,32,33]). On the other hand, according to the view of [34], the quadrature rule and the Marcinkiewicz-Zygmund (M-Z) inequality associated with $\overline{X}$ also influence the construction of the translation network ${\Delta }_{\varphi ,\overline{X}}$. For a bounded closed set $\Omega \subset R^d$ with measure $d\omega$ satisfying ${\int }_{\Omega }d\omega =V<+\infty$. We denote by $P_n\subset L^2(\Omega )$ the linear space of polynomials on $\Omega$ of degree at most n, equipped with the $L^2$-inner product $\langle v,z\rangle ={\int }_{\Omega }vzd\omega .$ The m-point quadrature rule (QR) is

$$\begin{array}{c}\hfill {\int }_{\Omega }gd\omega \approx {\displaystyle \sum _{j=1}^m}{w}_{j}g\left(x_j\right),\end{array}$$

(7)

where $\overline{X}={\left\{x_j\right\}}_{j=1}^m\subset \Omega$ and weights $w_j$ are all positive for $j=1,2,\cdots ,m.$ We say the QR (7) has polynomial exactness n if

$$\begin{array}{c}\hfill {\int }_{\Omega }gd\omega ={\displaystyle \sum _{j=1}^m}{w}_{j}g\left(x_j\right),\forall g\in P_n.\end{array}$$

(8)

The Marcinkiewicz-Zygmund (MZ) inequality based on a set $\overline{X}={\left\{x_j\right\}}_{j=1}^m\subset \Omega$ is

$$\begin{array}{c}\hfill {\left({\displaystyle \sum _{j=1}^m}{w}_j{\left|g\left(x_j\right)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}\sim {\left({\int }_{\Omega }{\left|g\left(\omega \right)\right|}^{2}d\omega \right)}^{{\displaystyle \frac{1}{2}}},\forall g\in P_n,\end{array}$$

(9)

where the weights $w_j$ in (9) may be not the same as the $w_j$ in (7) and (8).

Accord to the view of [34], the quadrature rule (QR) follows automatically from the Marcinkiewicz-Zygmund (MZ) inequality. H.N.Mhaskar et al gave a method of transition from MZ inequality to polynomial exact GR in [35]. So the MZ inequality (9) is an important features for describing the nodes set $\overline{X}$. Since this reason the node set $\overline{X}={\left\{x_j\right\}}_{j=1}^m\subset \Omega$ which yields an MZ inequality is given a specially terminology called Marcinkiewicz-Zygmynd Family (MZF) (see [34,36,37,38]). However, from these literatures we know that the MZFs do not totally coincide with the Lagrange interpolation nodes in the case of $d>1$. The hyperinterpolations are then developed with the help of exact QR (see [39,40,41,42,43]) and are applied to approximation theory and regularized learning (see e.g.[11,12,13,44]). On the other hand, we find the problem of polynomial exact QR is investigated individually (see e,g, [45,46]). The concept of spherical t-design was first defined in [47] and is given investigations by many papers subsequently, one can see the classical references [48,49]. We say $T_t={\left\{x_i\right\}}_{i=1}^{|T_t|}\subset S^{d-1}$ is a spherical t-design if

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{|T_t|}}{\displaystyle \sum _{i=1}^{|T_t|}}\pi \left(x_i\right)={\displaystyle \frac{1}{{\omega }_{d-1}}}{\int }_{S^{d-1}}\pi \left(x\right)d\omega \left(x\right),\end{array}$$

(10)

where ${\omega }_{d-1}$ is the volume of $S^{d-1}$ and $\pi \left(x\right)$ is a spherical polynomial with degree t. Moreover, in many applications the polynomial exact QR and the MZFS has been used as assumptions. For example, C.P.An et al gave approximation order for the hyperinterpolation approximation under the assumptions that (8), (10) and the MZ inequality (9) hold (see [50,51]). Also, in [25], Lin et al gave investigations on regularized regression associated with zonal translation network by assuming the nodes set $\overline{X}={\left\{x_j\right\}}_{j=1}^m\subset S^{d-1}$ is a type of spherical t-design.

Polynomial exact QR is also a good tool in approximation theory. For example, we had ever used it to bound the norm for some Mercer matrices (see [52,53,54,55]). In particular, H.N.Mhaskar et al use polynomial exact QR to construct the first periodic translation operators (see [27]) and the zonal translation network operators (see [29]). Along this line, the translation operators defined on the unit ball, on the Euclidean space $R^d$ and on the interval $[-1,1]$ are constructed (see [28,30,56]).

Above investigations encourage us to use (8), (10) and (9) as hypothetical conditions to describe the approximation ability of zonal translation networks ${\Delta }_{\varphi ,\overline{X}}^X$. To ensure the single layer translation network ${\Delta }_{\varphi ,\overline{X}}^X$ can approximation the constant function, ${\Delta }_{\varphi ,\overline{X}}^X$ is modified as

$$\begin{array}{c}\hfill N_{\varphi ,\overline{X}}^X=\{{\displaystyle \sum _{i=1}^m}{c}_i{T}_{x_i}(\varphi ,·)+c_0:c_0,c_i\in R,x_i\in X,i=1,2,\cdots ,m\}.\end{array}$$

(11)

In the case of $T_y(\varphi ,y)=\sigma (w^{\top }x+b)$ and $w,x\in R^d,b\in R$, R.D.Nowak et al used (11) to design regularized learning frameworks (see [57]). An algorithm is provided by S.B.Lin et al [26] for designing such kind of network and is applied to construct regularized learning algorithms. In [5]$N_{\varphi ,\overline{X}}^X$ is used to construct deep neural network learning frameworks. The same type of investigations are given in [58,59] and [60].

In the present paper, we shall design the translation networks $N_{\varphi ,\overline{X}}^X$ by taking $X=S^{d-1}$, assuming $\overline{X}={\Omega }^{\left(m\right)}={\left\{x_j\right\}}_{j=1}^m\subset S^{d-1}$ satisfies equalities (8) and (9), and choosing the zonal translation $T_x(\varphi ,y)=\varphi \left(xy\right)$ with $\varphi$ being a given integrable function $\varphi$ on $[-1,1]$. Under these assumptions we shall provide a learning framework with $N_{\varphi ,{\Omega }^{\left(m\right)}}^{S^{d-1}}$ as the hypothesis space and give error analysis.

The contributions of the present paper are two folds. First, after absorbing the ideas of [34,36,37,38] and the successful experience of [11,25,27,29,50,51,61,62], we propose the concept of Marcinkiewicz-Zygmund Inequality Setting (MZIS) for the scattered nodes on the sphere unit, based on this as an assumption we show the reproducing property for the convolutional zonal translation network associated with the scattered nodes ${\Omega }^{\left(m\right)}$. Second, we give investigation on the kernel regularized neural network learning by combining classical the kernel approach and the convex analysis method, according to this method, the convergence rate given can be dimensional independent and capacity independent. Since the translation networks are produced by the zonal translations of convolutional kernels, we call them the convolutional zonal translation networks.

The paper is organized as follows. In Section 2 we first show the density for the zonal translations class and then show the reproducing property for the translation network ${\Delta }_{\varphi ,{\Omega }^{\left(m\right)}}^{S^{d-1}}$. In Section 3, we shall provide some results in the present paper, for example, a new regression learning framework and a learning setting, the error decomposition for the error analysis,and an estimate for the convergence rate. In Section 4, we shall give some lemmas which are used to prove the main results. The proofs for all the theorems and propositions are given in Section 5.

Throughout the paper, we write $A=O\left(B\right)$ if there is a positive constant C independent of A and B such that $A\le CB$.In particular, by $A=O\left(1\right)$ we show A is a bounded quantity. We write $A\sim B$ if both $A=O\left(B\right)$ and $B=O\left(A\right)$.

2. Some Properties of the Translation Network on the Unit Sphere

Let $\varphi \in L_{W_{\eta }}^1={\{\varphi :\parallel \varphi \parallel }_{1,W_{\eta }}={\int }_{-1}^1\left|\varphi \left(x\right)\right|W_{\eta }\left(x\right)dx<+\infty \},W_{\eta }\left(x\right)={(1-x^2)}^{\eta -{\displaystyle \frac{1}{2}}},\eta >-{\displaystyle \frac{1}{2}}$. Then H.N.Mhaskar et al constructed in [29] a sequence of approximation operators to show that the zonal translation class

$$\begin{array}{ccc}\hfill {\Delta }_{\varphi }^{S^{d-1}}& =& cl\{\varphi \left(xy\right):y\in S^{d-1}\}\cup \left\{1\right\}\hfill \\ & =& cl\{{\displaystyle \sum _{i=1}^m}{c}_i\varphi (x_i·)+c_0:c_0,c_i\in R,x_i\in S^{d-1},i=1,2,\cdots ,m;m=1,2,\cdots \}\hfill \end{array}$$

is density in $L^p\left(S^{d-1}\right)(1\le p\le +\infty )$ if $\widehat{a_l^{\eta }\left(\varphi \right)}\ne 0$ for all $l=0,1,2,\cdots ,$ where

$$\widehat{a_l^{\eta }\left(\varphi \right)}=c_{\eta }{\int }_{-1}^1\varphi \left(x\right){\displaystyle \frac{C_l^{\eta }\left(x\right)}{C_l^{\eta }\left(1\right)}}{W}_{\eta }\left(x\right)dx,\eta ={\displaystyle \frac{d-2}{2}}$$

and $C_n^{\eta }\left(x\right)=p_n^{(\eta -{\displaystyle \frac{1}{2}},\eta -{\displaystyle \frac{1}{2}})}\left(x\right)$ is the n-th Legendre polynomial satisfies the orthogonal relation

$$\begin{array}{c}\hfill c_{\eta }{\int }_{-1}^1{C}_n^{\eta }\left(x\right)C_m^{\eta }\left(x\right)W_{\eta }\left(x\right)dx=h_n^{\eta }{\delta }_{n,m}.\end{array}$$

with $c_{\eta }={\displaystyle \frac{\Gamma (\eta +1)}{\Gamma \left(\frac{1}{2}\right)\Gamma (\eta +\frac{1}{2})}}$, $h_n^{\eta }={\displaystyle \frac{\eta }{n+\eta }}{C}_n^{\eta }\left(1\right)$ and it is known that (see it from (B.2.1),(B.2.2)and (B.5.1) of [63])$C_n^{\eta }\left(x\right)\le C_n^{\eta }\left(1\right)=n^{2\eta -1},x\in [-1,1].$ It follows that

$$\begin{array}{c}\hfill \varphi \left(t\right)={\displaystyle \sum _{l=0}^{+\infty }}\widehat{a_l^{\eta }\left(\varphi \right)}{\displaystyle \frac{n+\eta }{\eta }}{C}_l^{\eta }\left(t\right)={\displaystyle \sum _{l=0}^{+\infty }}\widehat{a_l^{\eta }\left(\varphi \right)}{Z}_l^{\eta }\left(t\right),\end{array}$$

(12)

where $Z_l^{\eta }\left(t\right)={\displaystyle \frac{l+\eta }{\eta }}{C}_l^{\eta }\left(t\right),\eta ={\displaystyle \frac{d-2}{2}}$.

For a given real number $p\ge 1$,we denote by $L^p\left({\rho }_X\right)$ the class of ${\rho }_X$-measurable functions f satisfying ${\parallel f\parallel }_{L^p\left({\rho }_X\right)}=({\int }_X{\left|f\left(x\right)\right|}^{p}d{\rho }_X)<+\infty$.

Let $P_n^d$ denote the space of all homogeneous polynomials of degree n in d variables. We denote by $L^p\left(S^{d-1}\right)$ the class of all measurable functions defined on $S^{d-1}$ with the finite norm

$$\begin{array}{ccc}& & {\parallel f\parallel }_{p,S^{d-1}}=\left\{\begin{array}{cc}{\left({\int }_{S^{d-1}}{\left|f\left(x\right)\right|}^{p}d\sigma \left(x\right)\right)}^{{\displaystyle \frac{1}{p}}},\hfill & 1\le p<+\infty ,\hfill \\ \\ {\displaystyle \underset{x\in S^{d-1}}{max}}\left|f\left(x\right)\right|,\hfill & p=+\infty ,\hfill \end{array}\right.\hfill \end{array}$$

and for $p=+\infty$ we assume that $L^{+\infty }\left(S^{d-1}\right)$ is the space $C\left(S^{d-1}\right)$ of continuous functions on $S^{d-1}$ with the uniform norm.

For a given integer $n\ge 0$, the restriction to $S^{d-1}$ of a homogeneous harmonic polynomial of degree n is called a spherical harmonics of degree n. If $Y\in P_n^d$, then $Y\left(x\right)={\parallel x\parallel }^{n}Y\left(x^{\prime }\right),x^{\prime }={\displaystyle \frac{x}{\parallel x\parallel }}$, so that Y is determined by its restriction on the unit sphere. Let $H_n\left(S^{d-1}\right)$ denote the space of spherical harmonics of degree n. Then

$$dim{H}_n\left(S^{d-1}\right)=(\begin{array}{c}n+d-2\\ n\end{array})+(\begin{array}{c}n+d-3\\ n-1\end{array}),n=1,2,3,\cdots ,$$

Spherical harmonics of different degrees are orthogonal on the unit sphere. For further properties about harmonics one can refer to [64].

For $n=0,1,2,\cdots ,$ let $\{Y_l^n:1\le l\le dim{H}_n\left(S^{d-1}\right)\}$ be an orthonormal basis of $H_n\left(S^{d-1}\right)$. Then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\omega }_{d-1}}}{\int }_{S^{d-1}}{Y}_l^n\left(\xi \right)Y_{l^{\prime }}^m\left(\xi \right)d\sigma \left(\xi \right)={\delta }_{l,l^{\prime }}{\delta }_{m,n},\end{array}$$

where ${\omega }_{d-1}$ denotes the surface area of $S^{d-1}$ and ${\omega }_{d-1}={\displaystyle \frac{2{\pi }^{\frac{d}{2}}}{\Gamma \left(\frac{d}{2}\right)}}$. Furthermore, by (1.2.8) in [63] we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{j=1}^{dim{H}_n\left(S^{d-1}\right)}}{Y}_l^n\left(x\right)Y_l^n\left(y\right)={\displaystyle \frac{n+\eta }{\eta }}{C}_n^{\eta }\left(xy\right),x,y\in S^{d-1},\end{array}$$

(13)

where $C_n^{\eta }\left(t\right)$ is the n-th generalized Legendre polynomial the same as in (12). Combining (12) and (13) we have

$$\begin{array}{c}\hfill \varphi \left(xy\right)={\displaystyle \sum _{l=0}^{+\infty }}\widehat{a_l^{\eta }\left(\varphi \right)}{\displaystyle \frac{l+\eta }{\eta }}{C}_l^{\eta }(x·y)={\displaystyle \sum _{l=0}^{+\infty }}\widehat{a_l^{\eta }\left(\varphi \right)}{\displaystyle \sum _{k=1}^{dim{H}_l\left(S^{d-1}\right)}}{Y}_k^l\left(x\right)Y_k^l\left(y\right),x,y\in S^{d-1}.\end{array}$$

(14)

2.1. Density

We first give a general discrimination method for density.

Proposition 2.1. (see Lemma 1 in Chapter 18 of [65]) For a subset V in a normed linear space E, the following two properties are equivalent:

(a) V is fundamental in E (that is, its linear span is dense in E).

(b) $V^{\perp }=\left\{0\right\}$ ( that is, 0 is the only element of $E^{*}$ that annihilates V).

Based on this proposition, we can show the density of ${\Delta }_{\varphi }^{S^{d-1}}$.

Theorem 2.1. Let $\varphi \in L_{W_{\eta }}^2$ satisfy $\widehat{a_l^{\eta }\left(\varphi \right)}>0$ for all $l=0,1,2,\cdots .$ Then ${\Delta }_{\varphi }^{S^{d-1}}$ is dense in $L^2\left(S^{d-1}\right)$.

Proof. See the proof in Section 5.

2.2. Reproducing Property

We first restate a proposition.

Proposition 2.2. For any given $n\ge 1$ there exist a finite subset ${\Omega }^{\left(n\right)}\subset S^{d-1}$ and corresponding positive numbers $\{{\mu }_{\omega }:\omega \in {\Omega }^{\left(n\right)}\}$ such that

$$\begin{array}{c}\hfill {\int }_{S^{d-1}}f\left(x\right)d\sigma \left(x\right)={\displaystyle \sum _{\omega \in {\Omega }^{\left(n\right)}}}{\mu }_{\omega }^{\left(n\right)}f\left(\omega \right),f\in H_{3n}\left(S^{d-1}\right),\end{array}$$

(15)

and for some $1\le p<+\infty$

$$\begin{array}{c}\hfill {\int }_{S^{d-1}}{\left|f\left(x\right)\right|}^{p}d\sigma \left(x\right)\sim {\displaystyle \sum _{\omega \in {\Omega }^{\left(n\right)}}}{\mu }_{\omega }^{\left(n\right)}{\left|f\left(\omega \right)\right|}^p,f\in H_n\left(S^{d-1}\right).\end{array}$$

(16)

Moreover, for any $m\ge n$ and $p\ge 1$ there exists a constant $c_{p,d}>0$ such that

$$\begin{array}{c}\hfill {\displaystyle \sum _{\omega \in {\Omega }^{\left(n\right)}}}{\mu }_{\omega }^{\left(n\right)}{\left|f\left(\omega \right)\right|}^p\le c_{p,d}{\left({\displaystyle \frac{m}{n}}\right)}^{d-1}{\int }_{S^{d-1}}{\left|f\left(x\right)\right|}^{p}d\sigma \left(x\right),f\in H_m\left(S^{d-1}\right).\end{array}$$

(17)

Proof. (15)-(16) were proved by H.N.Mhaskar et al in [61] and now have been extended to other domains (see e.g.[66,67]). Inequality (17) is proved by [68].

(15)-(16) show the existence of ${\Omega }^{\left(n\right)}\subset S^{d-1}$ which has the polynomial exact QR (8) and satisfies MZ inequality (9). Inequality (17) often goes with (15) and (16) and are needed in discussing approximation order(see e.g. [62]).

Based above analysis we propose the following definition.

Definition 2.1(Marcinkiewicz-Zygmund inequalities setting (MZIS)). We say a given finite node set ${\Omega }^{\left(n\right)}\subset S^{d-1}$ forms a Marcinkiewicz-Zygmund inequality setting if (15)-(16) and (17) simultaneously hold.

Let $\varphi \in L_{W_{\eta }}^2$. For a given finite set ${\Omega }^{\left(n\right)}\subset S^{d-1}$ satisfy Definition 2.1 and the positive numbers $\{{\mu }_{\omega }:\omega \in {\Omega }^{\left(n\right)}\}$ are defined as in Proposition 2.2. We define a zonal translation network by

$$\begin{array}{c}\hfill H_{{\Omega }^{\left(n\right)}}:=cl\{f\left(x\right)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right)+c_0:c_k\in R,k=0,1,2,\cdots ,|{\Omega }^{\left(n\right)}\left|\right\},\end{array}$$

where $T_x\left(\varphi \right)\left(y\right)=\varphi \left(xy\right)$. Then it is easy to see that

$$\begin{array}{c}\hfill H_{{\Omega }^{\left(n\right)}}=H_{\varphi }^{\left(n\right)}⨁R,\end{array}$$

where for $A,B\subset R$ and $A\bigcap B=\left\{0\right\}$ we define $A⨁B=\{a+b:a\in A,b\in B\}$ and

$$\begin{array}{c}\hfill H_{\varphi }^{\left(n\right)}:=cl\{f\left(x\right)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right):c_k\in R,k=0,1,2,\cdots ,|{\Omega }^{\left(n\right)}\left|\right\}.\end{array}$$

For $f\left(x\right)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right),g\left(x\right)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{d}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right)$ we define a bivariate operation as

$$\begin{array}{c}\hfill {\langle f,g\rangle }_{\varphi }={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{d}_k{\mu }_k^{\left(n\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\parallel f\parallel }_{\varphi }={\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{\left|c_k\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

Since (14), we have by Theorem 4 in Chapter 17 of [65] that the matrix ${\left(T_{x_i}(\varphi ,x_j)\right)}_{i,j=1,2,\cdots ,|{\Omega }^{\left(n\right)}|}$ is positive definite for a given n. It follows that the vector $c={\left\{{\mu }_k^{\left(n\right)}{c}_k\right\}}_{x_k\in {\Omega }^{\left(n\right)}}$ is uniqueness. Then $(H_{\varphi }^{\left(n\right)},\parallel ·{\parallel }_{\varphi })$ is a Hilbert space which is isometric isomorphism with $l^2\left({\Omega }^{\left(n\right)}\right)$,where

$$\begin{array}{c}\hfill l^2\left({\Omega }^{\left(n\right)}\right)=\{c={\left\{{\mu }_k^{\left(n\right)}{c}_k\right\}}_{x_k\in {\Omega }^{\left(n\right)}}{:\parallel c\parallel }_{l^2\left({\Omega }^{\left(n\right)}\right)}={\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{\left|c_k\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}<+\infty \}.\end{array}$$

Let $C\left(\right[-1,1\left]\right)$ be the set of all continuous functions defined on $[-1,1]$ and

$${\parallel \varphi \parallel }_{C\left(\right[-1,1\left]\right)}={\displaystyle \underset{x\in [-1,1]}{sup}}\left|\varphi \left(x\right)\right|.$$

We then have the following proposition.

Proposition 2.3. Let $\varphi \in C[-1,1]$ satisfy $\widehat{a_n^{\eta }\left(\varphi \right)}>0$ for all $n\ge 1$ and ${\Omega }^{\left(n\right)}\subset S^{d-1}$ be an MZIS. Then $(H_{\varphi }^{\left(n\right)},\parallel ·{\parallel }_{\varphi })$ is a reproducing kernel Hilbert space associated with kernel

$$\begin{array}{c}\hfill K_x^{*}\left(\varphi \right)\left(y\right)=K^{*}(\varphi ,x,y)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{T}_{x_k}(\varphi ,x)T_{x_k}(\varphi ,y),\end{array}$$

(18)

i.e.,

$$\begin{array}{c}\hfill f\left(x\right)={〈f,K_x^{*}\left(\varphi \right)(·)〉}_{\varphi }\forall x\in S^{d-1},\forall f\in H_{\varphi }^{\left(n\right)}\end{array}$$

(19)

and there is a constant $k^{*}>0$ such that

$$\begin{array}{c}\hfill \left|f\left(x\right)\right|\le k^{*}{\parallel f\parallel }_{\varphi }\forall f\in H_{\varphi }^{\left(n\right)},\forall x\in S^{d-1}.\end{array}$$

(20)

Proof. See the proof in Section 5.

Corollary 2.1. Under the assumption of Proposition 2.3, $H_{{\Omega }^{\left(n\right)}}$ is a reproducing kernel Hilbert space associated with the inner product defined by

$$\begin{array}{ccc}\hfill & & {\langle f,g\rangle }_{H_{{\Omega }^{\left(n\right)}}}={\langle f_1,g_1\rangle }_{\varphi }+c_0{d}_0,\hfill \\ \hfill where\\ & & f\left(x\right)=f_1\left(x\right)+c_0,g\left(x\right)=g_1\left(x\right)+d_0,\hfill \\ & & f_1\left(x\right)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right),g_1\left(x\right)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{d}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right)\}.\hfill \end{array}$$

and the corresponding reproducing kernel $K_x\left(\varphi \right)\left(y\right)$ is

$$\begin{array}{c}\hfill K_x\left(\varphi \right)\left(y\right)=K_x^{*}\left(\varphi \right)\left(y\right)+1,x,y\in S^{d-1}.\end{array}$$

Furthermore, there is a constant $\kappa >0$ such that

$$\begin{array}{c}\hfill \left|f\left(x\right)\right|\le {\kappa \parallel f\parallel }_{H_{{\Omega }^{\left(n\right)}}},\forall f{\in }_{H_{{\Omega }^{\left(n\right)}}},\forall x\in S^{d-1}.\end{array}$$

(21)

Proof. The results can be obtained from Proposition 2.3, Lemma 4.2 and the fact that the real set R is a reproducing kernel Hilbert space whose reproducing kernel is 1 and the inner product is the usual product for two real numbers.

3. Convergence Rate of the Kernel Regularized Regression

We now make a combination of the zonal translation network with the kernel regularized regression.

3.1. Learning Framework

For a set of observations $z={\left\{(x_i,y_i)\right\}}_{i=1}^m$ drawn i.i.d. according to a joint distribution $\rho (x,y)$ on $Z=S^{d-1}\times Y,Y=[-M,M],M>0isagivenrealnumber$, satisfying $\rho (x,y)=\rho \left(y\right|x){\rho }_{S^{d-1}}\left(x\right)$ we define a regularized framework as

$$\begin{array}{c}\hfill f_{z,\lambda }^{{\Omega }^{\left(m\right)}}:=arg{\displaystyle \underset{f\in H_{{\Omega }^{\left(m\right)}}}{min}}\left({\mathcal{E}}_z\left(f\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f\parallel }_{H_{{\Omega }^{\left(m\right)}}}^2\right),\end{array}$$

(22)

where $\lambda =\lambda \left(m\right)$ is the regularization parameter, and

$${\mathcal{E}}_z\left(f\right)={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}{(y_i-f\left(x_i\right))}^2.$$

The general integral model of (22) is

$$\begin{array}{c}\hfill f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}:=arg{\displaystyle \underset{f\in H_{{\Omega }^{\left(m\right)}}}{min}}\left({\mathcal{E}}_{\rho }\left(f\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f\parallel }_{H_{{\Omega }^{\left(m\right)}}}^2\right),\end{array}$$

(23)

where

$${\mathcal{E}}_{\rho }\left(f\right)={\int }_Z{(y-f\left(x\right))}^{2}d\rho .$$

To show the convergence analysis for (22) we need to bound the error

$$\begin{array}{c}\hfill \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho }{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)},\end{array}$$

which is an approximation problem, whose convergence rate depends upon the approximation ability of ${\mathcal{H}}_{{\Omega }^{\left(m\right)}}$. An error decomposition will be given in Section 3.2.

3.2. Error Decomposition

By (2) and the definition of $f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}$ we have the following decompositions

$$\begin{array}{ccc}\hfill & & \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho }{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}\hfill \\ & \le & \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}+{\parallel f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho }\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}\hfill \\ & =& \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}+\sqrt{{\mathcal{E}}_{\rho }\left(f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\right)-{\mathcal{E}}_{\rho }\left(f_{\rho }\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\parallel }_{H_{{\Omega }^{\left(m\right)}}}^2}\hfill \\ & \le & \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}+D^{{\Omega }^{\left(m\right)}}{(f_{\rho },\lambda )}_{L^2\left({\rho }_{S^{d-1}}\right)},\hfill \end{array}$$

(24)

where

$$\begin{array}{c}\hfill D^{{\Omega }^{\left(m\right)}}(f_{\rho },\lambda )={\displaystyle \underset{g\in H_{{\Omega }^{\left(m\right)}}}{inf}}\left(\parallel g-f_{\rho }{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}+\sqrt{{\displaystyle \frac{\lambda }{2}}}{\parallel g\parallel }_{H_{{\Omega }^{\left(m\right)}}}\right).\end{array}$$

We have used the fact that for $a>0,b>0,c>0$ and $0<p\le 1$ there holds

$$\begin{array}{c}\hfill {(a+b+c)}^p\le a^p+b^p+c^p.\end{array}$$

The K-functional $D^{{\Omega }^{\left(m\right)}}(f_{\rho },\lambda )$ shows the approximation error whose decay have been described in [69]. So the main estimate we need to deal with is the sample error

$$\begin{array}{c}\hfill \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}.\end{array}$$

3.3. The Convergence Rate

To give a capacity independent generalization error for algorithm (22), we need some concepts of convex analysis.

$G\widehat{a}teaux$ differentiable.Let $(H,\parallel ·{\parallel }_H)$ be a Hilbert space, $F\left(f\right):H\to R\cup \{\mp \infty \}$ be a real function. We say F is $G\widehat{a}teaux$ differentiable at $f_0\in H$ if there is an $\xi \in H$ such that for any $g\in H$ there holds

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to 0}{lim}}{\displaystyle \frac{F(f_0+tg)-F\left(f_0\right)}{t}}={\langle g,\xi \rangle }_H\end{array}$$

and write $F_G^{\prime }\left(f_0\right)=\xi$ or ${\nabla }_{f}F\left(f\right)=\xi$. It known that for a differentiable convex function $F\left(f\right)$ on H $f_0=arg{\displaystyle \underset{f\in H}{min}}F\left(f\right)$ if and only if $\left({\nabla }_{f}F\left(f\right)\right){|}_{f=f_0}=0$. (see Proposition 17.4 in [70]).

In learning theory, to form an explicit learning rate, one often assume the K-functional has a power convergence rate, i.e., we assume that there exists a $0<\beta \le 1$ such that

$$\begin{array}{c}\hfill D^{{\Omega }^{\left(m\right)}}(f_{\rho },\lambda )=O\left({\lambda }^{\beta }\right),\lambda \to 0^{+}(m\to +\infty ).\end{array}$$

(25)

Since $H_{{\Omega }^{\left(m\right)}}$ is a reproducing kernel Hilbert space, the decay of $D^{{\Omega }^{\left(m\right)}}(f_{\rho },\lambda )$ can be discussed with the method of [69]. In particular, it is shown in Theorem 2.3 of [71] that one can choose an MZIS ${\Omega }^{\left(m\right)}\subset S^{d-1},\varphi \in L_{\eta }^{+\infty }$ and suitable real numbers $\alpha ,\beta$ in such a way that

$$\begin{array}{c}\hfill D^{{\Omega }^{\left(m\right)}}\left(f_{\rho },{\displaystyle \frac{1}{2^{\alpha m}}}\right)=O\left({\displaystyle \frac{1}{2^{m\beta }}}\right)\end{array}$$

if ${\partial }_{\beta }{f}_{\rho }\in L^2\left(S^{d-1}\right)$, where we say ${\partial }_{\beta }{f}_{\rho }\in L^2\left(S^{d-1}\right)$ if there is a function $\phi \in L^2\left(S^{d-1}\right)$ such that

$$\phi \left(x\right)\sim {\displaystyle \sum _{l=0}^{+\infty }}{l}^{\beta }{Y}_l(f,x),Y_l(f,x)={\displaystyle \sum _{k=1}^{dim{H}_l\left(S^{d-1}\right)}}{\widehat{f}}_{k,l}{Y}_k^l\left(x\right),{\widehat{f}}_{k,l}={\int }_{S^{d-1}}f\left(u\right)Y_k^l\left(u\right)d\sigma \left(u\right),$$

which shows that (25) is reasonable.

Theorem 3.1.Let $\varphi \in C[-1,1]$ satisfy $\widehat{a_l^{\eta }\left(\varphi \right)}>0$ for all $l\ge 0$ and ${\Omega }^{\left(m\right)}\subset S^{d-1}$ be an MZIS. Then for any $\delta \in (0,1)$, with confidence $1-\delta$, hold

$$\begin{array}{ccc}\hfill & & \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho }{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}\hfill \\ & \le & 4\kappa \left({\displaystyle \frac{M}{\lambda \sqrt{m}}}+{\displaystyle \frac{D^{{\Omega }^{\left(m\right)}}{(f_{\rho },\lambda )}_{L^2\left({\rho }_{S^{d-1}}\right)}}{{\lambda }^{\frac{3}{2}}\sqrt{m}}}\right)log{\displaystyle \frac{4}{\delta }}+D^{{\Omega }^{\left(m\right)}}{(f_{\rho },\lambda )}_{L^2\left({\rho }_{S^{d-1}}\right)}.\hfill \end{array}$$

(26)

Proof. See the proof in Section 5.

3.4. Conclusion and Discussion

We propose the concept of MZIS for the scattered node set on the unit sphere, with which show that the related convolutional zonal translation network is a reproducing kernel Hilbert space, and show the convergence rate for the kernel regularized least square regression model. We give further analysis to show the advantages of the present paper.

(1)The zonal translation networks which we have chosen is a finite dimensional reproducing kernel Hilbert space, our discussions belong to the scope of kernel method, which is a combination and application of (neural) translation networks to learning theory.

(2)Compared with the existing convergence rate estimate of neural network learning, our upper estimates are dimensional independent (see Theorem in [25], Theorem 3.1 in [72], Theorem 7 in [73], Theorem 1 in [26]).

(3) It is hopeful that the node set in (15)-(16) and (17) may be replaced by a set of uniform distribution observations or a set of random samples (see [72,73]).

(4)The density derivation for the zonal translation network is qualitative, the density is described with the Fourier orthogonal coefficients. We think that this method can be extended to other domains such as the unit ball, the Euclidean space $R^d$ and $R_{+}=[0,+\infty )$ et al.

(5)It is hopeful that with the help of the concept MZIS one may show reproducing property for the deep translation networks and thus give investigations for the performance of deep convolutional translation learning with the kernel method (see e.g.[6,7,59]).

4. Some Lemmas

To prove the main results, we need some lemmas.

Lemma 4.1.Let $(H,\parallel ·\parallel )$ be a Hilbert space and ξ be a random variable on $(Z,\rho )$ with values in H and ${\left\{z_i\right\}}_{i=1}^m$ be independent samples drawers of ρ.Assume ${\parallel \xi \parallel }_H\le \tilde{M}<+\infty$ almost surely. Then, for any $0<\delta <1$, with confidence $1-\delta$,it holds

$$\begin{array}{c}\hfill \parallel {\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}\xi \left(z_i\right)-E\left(\xi \right){\parallel }_H\le {\displaystyle \frac{2\tilde{M}log\left(\frac{2}{\delta }\right)}{\sqrt{m}}}\end{array}$$

(27)

Proof. Find it from [74].

Lemma 4.2.Let $(H,\parallel ·{\parallel }_H)$ be a Hilbert space over X with respect to kernel K. If $(E,\parallel ·{\parallel }_E)$ and $(F,\parallel ·{\parallel }_F)$ are closed subspaces of H such that $E\perp F$ and $E⨁F=H$, then $K=L+M$, where L and M are the reproducing kernels of E and F respectively. Moreover, for $h=e+f,e\in E,f\in F$, we have

$$\begin{array}{c}\hfill {\parallel h\parallel }_H={\left({\parallel e\parallel }_E^2+{\parallel f\parallel }_F^2\right)}^{{\displaystyle \frac{1}{2}}}.\end{array}$$

Proof. See Corollary 1 in Chapter 31 of [65] or the Theorem in Section 6 in part I of [75].

Lemma 4.3. There hold the following equalities:

$$\begin{array}{c}\hfill {\nabla }_f{\mathcal{E}}_z\left(f\right)(·)=-{\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}(y_i-f\left(x_i\right))K_{x_i}\left(\varphi \right)(·),f\in {\mathcal{H}}_{{\Omega }^{\left(m\right)}}.\end{array}$$

(28)

and

$$\begin{array}{c}\hfill {\nabla }_f{\mathcal{E}}_{\rho }\left(f\right)(·)=-2{\int }_Z(y-f\left(x\right))K_x\left(\varphi \right)(·)d\rho ,f\in {\mathcal{H}}_{{\Omega }^{\left(m\right)}}.\end{array}$$

(29)

Proof of (28). By the equality

$$\begin{array}{c}\hfill a^2-b^2=2(a-b)b+{(a-b)}^2,a\in R,b\in R\end{array}$$

(30)

we have

$$\begin{array}{ccc}& & {\displaystyle \underset{t\to 0}{lim}}{\displaystyle \frac{{\mathcal{E}}_z(f+tg)-{\mathcal{E}}_z\left(f\right)}{t}}\hfill \\ & =& {\displaystyle \underset{t\to 0}{lim}}{\displaystyle \frac{\frac{1}{m}{\sum }_{i=1}^m(-2t{y}_i-f\left(x_i\right))g\left(x_i\right)+t^{2}g{\left(x_i\right)}^2}{t}}\hfill \\ & =& -{\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}(y_i-f\left(x_i\right))g\left(x_i\right)\hfill \end{array}$$

Since $g\left(x\right)={\langle g,K_x(\varphi ;·)\rangle }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}$ and the definition of $G\widehat{a}teaux$ derivative, we have by above equality that

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to 0}{lim}}{\displaystyle \frac{{\mathcal{E}}_z(f+tg)-{\mathcal{E}}_z\left(f\right)}{t}}={〈g,-{\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}(y_i-f\left(x_i\right))g\left(x_i\right)K_{x_i}\left(\varphi \right)(·)〉}_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}.\end{array}$$

We then have (28). By the same way we can have (29).

Lemma 4.4. Let $(H,{\langle ·\rangle }_H)$ be a Hilbert space consisting of real functions on X. Then

$$\begin{array}{c}\hfill {\parallel f\parallel }_H^2-{\parallel g\parallel }_H^2=2{\langle f-g,g\rangle }_H+{\parallel f-g\parallel }_H^2,\forall f,g\in H\end{array}$$

(31)

and

$$\begin{array}{c}\hfill {\nabla }_f{(\parallel f\parallel }_H^2)(·)=2f(·),\forall f\in H.\end{array}$$

(32)

Proof. (31) is the deformation of the parallelogram formula. (32) can be shown with (31).

Lemma 4.5. (22) has a unique solution $f_{z,\lambda }^{{\Omega }^{\left(m\right)}}$ and (23) has a unique solution $f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}$. Moreover,

(i) There hold the bounds

$$\begin{array}{c}\hfill \parallel f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{C\left(S^{d-1}\right)}\le {\displaystyle \frac{2\kappa D^{{\Omega }^{\left(m\right)}}\left(f_{\rho }\right)}{\sqrt{\lambda }}},\end{array}$$

(33)

where $\kappa$ is defined as in (21).

(ii) There holds the equality

$$\begin{array}{c}\hfill \lambda f_{z,\lambda }^{{\Omega }^{\left(m\right)}}(·)={\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}\left(y_i-f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)K_{x_i}\left(\varphi \right)(·)\end{array}$$

(34)

and the equality

$$\begin{array}{c}\hfill \lambda f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}(·)=2{\int }_Z\left(y-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)\right)K_x\left(\varphi \right)(·)d\rho \end{array}$$

(35)

Proof.Proof of (i). Since ${\mathcal{E}}_{\rho }\left(f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\right)\ge {\mathcal{E}}_{\rho }\left(f_{\rho }\right)$, we have by (2) that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\lambda }{2}}{\parallel f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2& \le & {\mathcal{E}}_{\rho }\left(f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\right)-{\mathcal{E}}_{\rho }\left(f_{\rho }\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\parallel }_{H_{{\Omega }^{\left(m\right)}}}^2\hfill \\ & =& {\displaystyle \underset{g\in H_{{\Omega }^{\left(m\right)}}}{inf}}\left({\mathcal{E}}_{\rho }\left(g\right)-{\mathcal{E}}_{\rho }\left(f_{\rho }\right)+{\displaystyle \frac{\lambda }{2}}{\parallel g\parallel }_{H_{{\Omega }^{\left(m\right)}}}^2\right)\hfill \\ & =& {\displaystyle \underset{g\in H_{{\Omega }^{\left(m\right)}}}{inf}}\left(\parallel g-f_{\rho }{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}^2+{\displaystyle \frac{\lambda }{2}}{\parallel g\parallel }_{H_{{\Omega }^{\left(m\right)}}}^2\right).\hfill \end{array}$$

(36)

By (36) we have (33).

Proof of (ii). By the definition of $f_{z,\lambda }^{{\Omega }^{\left(m\right)}}$ and (32) we have

$$\begin{array}{c}\hfill 0={\nabla }_f\left({\mathcal{E}}_z\left(f\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2\right){|}_{f=f_{z,\lambda }^{{\Omega }^{\left(m\right)}}},\end{array}$$

i.e.,

$$\begin{array}{ccc}\hfill 0& =& {\nabla }_f{\mathcal{E}}_z{\left(f\right)|}_{f=f_{z,\lambda }^{{\Omega }^{\left(m\right)}}}+\lambda {\nabla }_f({\displaystyle \frac{1}{2}}{\parallel f\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2{\left)\right|}_{f=f_{z,\lambda }^{{\Omega }^{\left(m\right)}}}\hfill \\ & =& -{\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}(y_i-f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right))K_{x_i}\left(\varphi \right)(·)+\lambda f_{z,\lambda }^{{\Omega }^{\left(m\right)}}(·).\hfill \end{array}$$

Hence, (34) holds. We can prove (35) in the same way.

Lemma 4.6. The solutions $f_{z,\lambda }^{{\Omega }^{\left(m\right)}}$ and $f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}$ satisfy the inequality

$$\begin{array}{c}\hfill \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\le {\displaystyle \frac{2A\left(z\right)}{\lambda }},\end{array}$$

(37)

where

$$A\left(z\right)=\parallel {\int }_Z\left(y-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)\right)K_x\left(\varphi \right)(·)d\rho -{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}\left(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)K_{x_i}\left(\varphi \right)(·){\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}.$$

Proof. By (30) we have

$$\begin{array}{c}\hfill a^2-b^2\ge 2(a-b)b,a\in R,b\in R\end{array}$$

It follows that

$$\begin{array}{ccc}\hfill & & {\mathcal{E}}_z\left(f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\right)-{\mathcal{E}}_z\left(f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\right)\hfill \\ & =& {\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}\left[{(y_i-f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right))}^2-{(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right))}^2\right]\hfill \\ & \ge & -{\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right))\times \left(f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)\hfill \\ & =& {〈f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}},-{\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}\left(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)K_{x_i}\left(\varphi \right)(·)〉}_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}},\hfill \end{array}$$

(38)

where we have used the reproducing property

$$f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)={〈f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}},K_{x_i}\left(\varphi \right)(·)〉}_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}.$$

By the definition of $f_{z,\lambda }^{{\Omega }^{\left(m\right)}}$ we have

$$\begin{array}{ccc}\hfill 0& \ge & \left({\mathcal{E}}_z\left(f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2\right)-\left({\mathcal{E}}_z\left(f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\right)+{\displaystyle \frac{\lambda }{2}}{\parallel f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2\right).\hfill \end{array}$$

On the other hand, by above inequality, (38) and (31) we have

$$\begin{array}{ccc}\hfill 0& \ge & \left({\mathcal{E}}_z\left(f_{z,\lambda }^{{\Omega }^{\left(m\right)}}\right)-{\mathcal{E}}_z\left(f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\right)\right)+{\displaystyle \frac{\lambda }{2}}\left(\parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2-{\parallel f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2\right)\hfill \\ & \ge & {〈f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}},-{\displaystyle \frac{2}{m}}{\displaystyle \sum _{i=1}^m}\left(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)K_{x_i}\left(\varphi \right)(·)〉}_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\hfill \\ & +& {〈f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}},\lambda j_q\left(f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\right)〉}_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}+\lambda \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2.\hfill \end{array}$$

Since (35), we have

$$\begin{array}{ccc}\hfill 0& \ge & 2\langle f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}},{\int }_Z\left(y-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)\right)K_x\left(\varphi \right)(·)d\rho \hfill \\ & & -{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}\left(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)K_{x_i}\left(\varphi \right)(·){\rangle }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}+\lambda \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2.\hfill \end{array}$$

By the Cauchy inequality we have

$$\begin{array}{ccc}& & \lambda \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}^2\hfill \\ & \le & 2\langle f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}-f_{z,\lambda }^{{\Omega }^{\left(m\right)}},{\int }_Z\left(y-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)\right)K_x\left(\varphi \right)(·)d\rho \hfill \\ & & -{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}\left(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)K_{x_i}\left(\varphi \right)(·){\rangle }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\hfill \\ & \le & \parallel {\int }_Z\left(y-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)\right)K_x\left(\varphi \right)(·)d\rho \hfill \\ & & -{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}\left(y_i-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)\right)K_{x_i}\left(\varphi \right)(·){\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\times 2\parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}.\hfill \end{array}$$

We then have (37).

Lemma 4.7. For the solutions $f_{z,\lambda }^{{\Omega }^{\left(m\right)}}$ and $f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}$ we have

$$\begin{array}{c}\hfill \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\le 4\kappa \left({\displaystyle \frac{M}{\lambda \sqrt{m}}}+{\displaystyle \frac{D^{{\Omega }^{\left(m\right)}}{(f_{\rho },\lambda )}_{L^2\left({\rho }_{S^{d-1}}\right)}}{{\lambda }^{\frac{3}{2}}\sqrt{m}}}\right)log{\displaystyle \frac{4}{\delta }}.\end{array}$$

(39)

Proof. It is easy to see that

$$\begin{array}{ccc}\hfill A\left(z\right)& \le & \parallel {\int }_{Z}y{K}_x\left(\varphi \right)(·)d\rho -{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}{y}_i{K}_{x_i}\left(\varphi \right)(·){\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\hfill \\ & & +\parallel {\int }_Z{f}_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)K_x\left(\varphi \right)(·)d{\rho }_{S^{d-1}}-{\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^m}{f}_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x_i\right)K_{x_i}\left(\varphi \right)(·){\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\hfill \\ & =& B\left(z\right)+C\left(z\right).\hfill \end{array}$$

(40)

Take $\xi (x,y)(·)=y{K}_x\left(\varphi \right)(·)$. Then,

$$\begin{array}{c}\hfill {\parallel \xi (x,y)(·)\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}=\left|y\right||K_x\left(\varphi \right)\left(x\right)|\le M\kappa .\end{array}$$

(41)

Combining (41) with (27) we have, with confidence $1-\delta$, that

$$\begin{array}{c}\hfill B\left(z\right)\le {\displaystyle \frac{2M\kappa }{\sqrt{m}}}log{\displaystyle \frac{2}{\delta }}.\end{array}$$

(42)

Also, take $\eta \left(x\right)(·)=f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)K_x\left(\varphi \right)(·)$. Then

$$\begin{array}{ccc}\hfill {\parallel \eta \left(x\right)\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}& =& |f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}\left(x\right)\left|\right|K_x\left(\varphi \right)\left(x\right)|\hfill \\ & & \le {\displaystyle \frac{4\kappa D^{{\Omega }^{\left(m\right)}}{(f_{\rho },\lambda )}_{L^2\left({\rho }_{S^{d-1}}\right)}}{\sqrt{\lambda }}}\hfill \end{array}$$

(43)

Combing (43) with (27) we have, with confidence $1-\delta$, that

$$\begin{array}{c}\hfill C\left(z\right)\le {\displaystyle \frac{4\kappa D^{{\Omega }^{\left(m\right)}}{(f_{\rho },\lambda )}_{L^2\left({\rho }_{S^{d-1}}\right)}}{\sqrt{\lambda m}}}log{\displaystyle \frac{2}{\delta }}.\end{array}$$

(44)

Collecting (44), (42),(40) and (37), we arrive (39).

5. Proof for Theorems and Propositions

Proof of Theorem $2.1.$ If ${\Delta }_{\varphi }^{S^{d-1}}$ is not dense in $L^2\left(S^{d-1}\right)$, i.e.,

$$clspan\left({\Delta }_{\varphi }^{S^{d-1}}\right)\ne L^2\left(S^{d-1}\right).$$

Then by Proposition 2.1 we know ${\left(clspan\left({\Delta }_{\varphi }^{S^{d-1}}\right)\right)}^{\perp }\ne \left\{0\right\}$ and there is a nonzero functional $F\in L^2\left(S^{d-1}\right)$ such that

$$F\left(f\right)=0,f\in clspan\left({\Delta }_{\varphi }^{S^{d-1}}\right).$$

It follows that $F\left(\varphi \right(·y\left)\right)=0$ for all $y\in S^{d-1}$. By the Riesz representation Theorem, F corresponds to a nonzero $h\in L^2\left(S^{d-1}\right)$ in such a way that

$$\begin{array}{c}\hfill F\left(f\right)={\int }_{S^{d-1}}f\left(x\right)h\left(x\right)d\sigma \left(x\right),\forall f\in L^2\left(S^{d-1}\right).\end{array}$$

Consequently, ${\int }_{S^{d-1}}\varphi \left(xy\right)h\left(y\right)d\sigma \left(y\right)=0,\forall x\in S^{d-1},$ which gives

$$\begin{array}{c}\hfill {\int }_{S^{d-1}}\left({\int }_{S^{d-1}}\varphi \left(xy\right)h\left(y\right)d\sigma \left(y\right)\right)h\left(x\right)d\sigma \left(x\right)=0.\end{array}$$

(45)

Combining (45) with (14) we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{l=0}^{+\infty }}\widehat{a_l^{\eta }\left(\varphi \right)}{\displaystyle \sum _{k=1}^{dim{H}_l\left(S^{d-1}\right)}}{\left({\int }_{S^{d-1}}h\left(y\right)Y_k^l\left(y\right)d\sigma \left(y\right)\right)}^2=0.\end{array}$$

It follows that ${\int }_{S^{d-1}}h\left(y\right)Y_k^l\left(y\right)d\sigma \left(y\right)=0$ for all $l\ge 0.$ Therefore, $h=0.$ We have obtained a contradiction.

Proof of Proposition 2.3. By the the definition of ${\langle ·,·\rangle }_{\varphi }$ and the definition kernel $K_x^{*}(\varphi ,y)$ in (18) we have for any $f\left(x\right)={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right)$ that

$$\begin{array}{c}\hfill {\langle f,K_x^{*}(\varphi ,·)\rangle }_{\varphi }={\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right)=f\left(x\right).\end{array}$$

(19) then holds. We now show (20). In fact, by the Cauchy’s inequality we have

$$\begin{array}{ccc}\hfill \left|f\right(x\left)\right|& =& |{\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{c}_k{\mu }_k^{\left(n\right)}{T}_x\left(\varphi \right)\left(x_k\right)|\hfill \\ & \le & {\parallel c\parallel }_{L^2\left({\Omega }^{\left(n\right)}\right)}{\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{\left|T_x\left(\varphi \right)\left(x_k\right)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& {\parallel f\parallel }_{\varphi }{\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{\left|\varphi (x_k·x)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

On the other hand, by the localized kernel theory about the Jacobi polynomials in [76] (see Lemma 2.5 in [76]), for a given n, we have an operator $V_n:L_{W_{\eta }}^1\to {\Pi }_n[-1,1]$ (where ${\Pi }_n[-1,1]$ is the set of all the algebraic polynomials of order $\le n$ defined on $[-1,1]$ and $\eta ={\displaystyle \frac{d-2}{2}}$) such that $V_n\left(\varphi \right)\in {\Pi }_{2n}[-1,1],V_n\left(g\right)=g$ for any $g\in {\Pi }_n[-1,1]$ and there is constant $c>0$ such that

$$\begin{array}{c}\hfill \parallel V_n{\left(\varphi \right)\parallel }_{C\left(\right[-1,1\left]\right)}\le {c\parallel \varphi \parallel }_{C\left(\right[-1,1\left]\right)},{\parallel V_n\left(\varphi \right)-\varphi \parallel }_{C\left(\right[-1,1\left]\right)}\le c{E}_n{\left(\varphi \right)}_{C\left(\right[-1,1\left]\right)},\end{array}$$

where

$$E_n{\left(\varphi \right)}_{C\left(\right[-1,1\left]\right)}={\displaystyle \underset{p\in {\Pi }_n[-1,1]}{inf}}{\parallel \varphi -p\parallel }_{C\left(\right[-1,1\left]\right)}.$$

It follows by the Minkowski inequality and inequality (17) that

$$\begin{array}{ccc}& & {\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{\left|\varphi (x_k·x)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \le & {\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{\left|(\varphi -V_n\left(\varphi \right))(x_k·x)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}+{\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}{\left|V_n\left(\varphi \right)(x_k·x)\right|}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \le & c{E}_n{\left(\varphi \right)}_{C\left(\right[-1,1\left]\right)}{\left({\displaystyle \sum _{x_k\in {\Omega }^{\left(n\right)}}}{\mu }_k^{\left(n\right)}\right)}^{{\displaystyle \frac{1}{2}}}+O\left(1\right){\left({\int }_{S^{d-1}}{\left|V_n\left(\varphi \right)(x·y)\right|}^{2}d\sigma \left(x\right)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& O\left(1\right)E_n{\left(\varphi \right)}_{C\left(\right[-1,1\left]\right)}+O\left(1\right){\left({\int }_{-1}^1{\left|V_n\left(\varphi \right)\left(u\right)\right|}^2{W}_{\eta }\left(u\right)du\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & =& O\left(1\right).\hfill \end{array}$$

(20) thus holds, where we have used the Funk-Hecke formula (see (1.2.11) in [63])

$$\begin{array}{c}\hfill {\int }_{S^{d-1}}f(x·y)Y_n\left(y\right)d\sigma \left(y\right)={\lambda }_n\left(f\right)Y_n\left(x\right),Y_n\in H_n\left(S^{d-1}\right)\end{array}$$

and

$$\begin{array}{c}\hfill {\lambda }_n\left(f\right)={\omega }_{d-1}{\int }_{-1}^{1}f\left(t\right){\displaystyle \frac{C^{\eta }\left(t\right)}{C^{\eta }\left(1\right)}}dt,\eta ={\displaystyle \frac{d-1}{2}}.\end{array}$$

Proof of Theorem 3.2. Since (39) we have by (21) that

$$\begin{array}{ccc}\hfill & & \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{L^2\left({\rho }_{S^{d-1}}\right)}\hfill \\ & \le & \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{C\left(S^{S^{d-1}}\right)}\hfill \\ & \le & \kappa \parallel f_{z,\lambda }^{{\Omega }^{\left(m\right)}}-f_{\rho ,\lambda }^{{\Omega }^{\left(m\right)}}{\parallel }_{{\mathcal{H}}_{{\Omega }^{\left(m\right)}}}\hfill \\ & \le & 4\kappa \left({\displaystyle \frac{M}{\lambda \sqrt{m}}}+{\displaystyle \frac{D^{{\Omega }^{\left(m\right)}}{(f_{\rho },\lambda )}_{L^2\left({\rho }_{S^{d-1}}\right)}}{{\lambda }^{\frac{3}{2}}\sqrt{m}}}\right)log{\displaystyle \frac{4}{\delta }}.\hfill \end{array}$$

(46)

Taking (46) into (24) we have (26).