Exponential Convergence-(t,s)-Weak Tractability of Approximation in Weighted Hilbert Spaces

Abstract

We study $L_2$-approximation problems in the weighted Hilbert spaces in the worst case setting. Three interesting weighted Hilbert spaces appear in this paper, whose weights are equipped with two positive parameters ${\gamma }_j$ and ${\alpha }_j$ for $j\in \mathbb{N}$. We consider algorithms using the class of arbitrary linear functionals. We discuss the exponential convergence-$(t,s)$-weak tractability of these $L_2$-approximation problems. In particular, we obtain the sufficient and necessary conditions on the weights for exponential convergence-weak tractability and exponential convergence-$(t,1)$-weak tractability with $t<1$.

1. Introduction

We study multivariate approximation problems $\mathrm{APP}={\left\{{\mathrm{APP}}_d\right\}}_{d\in \mathbb{N}}$ of functions defined over Hilbert spaces with large or huge d in the worst case setting (approximation error by the worst case error). Such problems appear in quantum physics (see [1]), computational chemistry (see [2]), and economics (see [3]). We consider algorithms using the class of arbitrary linear functionals. The information complexity $n(\epsilon ,{\mathrm{APP}}_d)$ is the minimal number n of linear functionals for which the approximation error of some algorithm is at most $\epsilon$. Tractability describes the dependence of the information complexity $n(\epsilon ,{\mathrm{APP}}_d)$ on the threshold $\epsilon$ and the dimension d. We consider the classical tractability which is polynomially convergent, and the exponential convergence-tractability (EC-tractability) which is exponentially convergent. Recently many authors discuss classical tractability and EC-tractability in weighted Hilbert spaces (see [4] by linear information, ref. [5] by standard information for functionals, and [6] by standard information for operators), especially in analytic Korobov spaces, such as exponential convergence and uniform exponential convergence (see [7]), classical tractability (see [8]) and EC-tractability for $L_2$-approximation (see [9] for exponential convergence-$(t,s)$-weak tractability and [10] for other EC-tractability results by algorithms using continuous linear functionals, and see [11] for EC-tractability by algorithms using function values), and EC-tractability for $L_p$-approximation with $1\le p\le \infty$ by algorithms using continuous linear functionals (see [12]). Some authors consider tractability in weighted Hilbert spaces, such as classical tractability in weighted Korobov spaces (see [13] for strong polynomial tractability and polynomial tractability, [14] for other classical tractability results by algorithms using continuous linear functionals, and [15] by algorithms using function values), EC-tractability in weighted Korobov spaces (see [16]), and classical tractability in weighted Gaussian ANOVA spaces (see [17,18] with different weights, respectively).

In this paper, we investigate EC-tractability of $L_2$-approximation problems from the weighted Hilbert spaces with some weights. Let $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ be a Hilbert space with weight $R_{d,\mathit{\alpha },\mathit{\gamma }}$, where $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ are two positive sequences satisfying $1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0$ and $1<{\alpha }_1\le {\alpha }_2\le \cdots .$ In the worst case setting, we consider the $L_2$-approximation problem

$${\mathrm{APP}}_d:H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})\to L_2({[0,1]}^d)\mathrm{with}{\mathrm{APP}}_d(f)=f.$$

The classical tractability for $L_2$-approximation problem $\mathrm{APP}=\left\{{\mathrm{APP}}_d\right\}$ in weighted Korobov spaces $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ such as strong polynomial tractability and polynomial tractability were discussed in [13,15,17]; quasi-polynomial tractability, uniform weak tractability, weak tractability and $(t,s)$-weak tractability were investigated in [14,17]. Additionally, ref. [17] also discussed classical tractability in several weighted Hilbert spaces, including weighted Korobov spaces and weighted Gaussian ANOVA spaces. The EC-tractability of the problem $\mathrm{APP}=\left\{{\mathrm{APP}}_d\right\}$ in weighted Korobov spaces such as EC-$(t,1)$-weak tractability for $0<t\le 1$ were studied in [16]. However, the above weighted Hilbert spaces $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with weights $R_{d,\mathit{\alpha },\mathit{\gamma }}$ satisfy $1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0$ and $1<{\alpha }_1={\alpha }_2=\cdots .$

In this paper we present three cases of weighted Hilbert spaces $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with weights $R_{d,\mathit{\alpha },\mathit{\gamma }}$ for $1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0$ and $1<{\alpha }_1\le {\alpha }_2\le \cdots$ that appear in the reference [18]. These weighted Hilbert spaces are similar but also different. The authors in [18] studied the polynomial tractability, strong polynomial tractability, weak tractability, and $(t,s)$-weak tractability for $t>1$ and $s>0$ of the problems $\mathrm{APP}=\left\{{\mathrm{APP}}_d\right\}$ in these three weighted Hilbert spaces. However, there are no results about EC-tractability of the approximation problems $\mathrm{APP}=\left\{{\mathrm{APP}}_d\right\}$ in the above three weighted Hilbert spaces. We will study exponential convergence-$(t,s)$-weak tractability (EC-$(t,s)$-WT) for some $t>0$, $s>0$ and obtain the complete sufficient and necessary conditions for $t=s=1$ and $t<1$, $s=1$, respectively.

The paper is structured in the following ways. We present three cases of weighted Hilbert spaces in Section 2. Section 3 gives preliminaries about the $L_2$-approximation problem in the weighted Hilbert space. Section 4.1 is devoted to recall some notions about the tractability, such as classical tractability and exponential convergence-tractability and state the main results. In Section 4.2 we give the proof of Theorem 1. In Section 5 we present a summary.

2. Weighted Reproducing Kernel Hilbert Spaces

In this section we consider weighted reproducing kernel Hilbert spaces with different weights.

Let $H(K_d)$ be a Hilbert space defined in ${[0,1]}^d$. The function $K_d(\mathit{x},\mathit{y})$ of $\mathit{x},\mathit{y}\in {[0,1]}^d$ is called a reproducing kernel of $H(K_d)$ if for every $\mathit{y}\in {[0,1]}^d$ and every $f\in H(K_d)$,

$$f(\mathit{y})={⟨f(\mathit{x}),K_d(\mathit{x},\mathit{y})⟩}_{H(K_d)}.$$

The Hilbert space is a so-called reproducing kernel Hilbert space. We can study more details on reproducing kernel Hilbert spaces in the reference [19].

In this paper, let $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ be two positive sequences of the Hilbert space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with $R_{d,\mathit{\alpha },\mathit{\gamma }}$ satisfying

$$1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0,\mathrm{and}1<{\alpha }_1\le {\alpha }_2\le \cdots .$$

(1)

Assume that the function $K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}$ of the space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with $K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}:{[0,1]}^d\times {[0,1]}^d\to \mathbb{C}$ is of product form

$$K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y}):={\displaystyle \prod _{k=1}^d}{K}_{R_{1,{\alpha }_k,{\gamma }_k}}(x_k,y_k),$$

where $K_{R_{1,\alpha ,\gamma }}:[0,1]\times [0,1]\to \mathbb{C}$ is a universal weighted function

$$K_{R_{1,\alpha ,\gamma }}(x,y):=\sum _{h\in {\mathbb{N}}_0}{R}_{\alpha ,\gamma }(h)exp(2\pi ih(x-y)),x,y\in [0,1].$$

Here, let weight $R_{\alpha ,\gamma }:{\mathbb{N}}_0\to {\mathbb{R}}^{+}$ be a summable function, i.e., ${\displaystyle \sum _{k\in {\mathbb{N}}_0}}{R}_{\alpha ,\gamma }(k)<\infty$. Then we have

$$K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})=\sum _{\mathit{h}\in {\mathbb{N}}_0^d}{R}_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})exp(2\pi i\mathit{h}·(\mathit{x}-\mathit{y})),\mathit{x},\mathit{y}\in {[0,1]}^d,$$

(2)

the inner product

$${⟨f,g⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}=\sum _{\mathit{h}\in \in {\mathbb{N}}_0^d}\frac{1}{R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})}\widehat{f}(\mathit{h})\overline{\widehat{g}}(\mathit{h}),$$

(3)

and

$${\left|\right|f\left|\right|}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}=\sqrt{{⟨f,f⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}},$$

where

$$R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h}):={\displaystyle \prod _{j=1}^d}{R}_{{\alpha }_j,{\gamma }_j}(h_j),\mathit{h}=(h_1,h_2,\dots ,h_d)\in {\mathbb{N}}_0^d,$$

$$\mathit{x}·\mathit{y}:={\displaystyle \sum _{h=1}^d}{x}_h·y_h,\mathit{x}=(x_1,x_2,\dots ,x_d),\mathit{y}=(y_1,y_2,\dots ,y_d)\in {[0,1]}^d,$$

and

$$\widehat{f}(\mathit{h})=\underset{{[0,1]}^d}{\int }f(\mathit{x})exp(-2\pi i\mathit{h}·\mathit{x})d\mathit{x}.$$

We can ascertain that $K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})$ is well defined for $1<{\alpha }_1\le {\alpha }_2\le \cdots$ and for all $\mathit{x},\mathit{y}\in {[0,1]}^d$, since

$$|K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})|\le \sum _{\mathit{h}\in {\mathbb{N}}_0^d}{R}_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})={\displaystyle \prod _{j=1}^d}(\sum _{h\in {\mathbb{N}}_0}{R}_{{\alpha }_j,{\gamma }_j}(h))<\infty .$$

Note that the Hilbert space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ is a reproducing kernel Hilbert space with the reproducing kernel $K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}$. Indeed, for every $f\in H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ we have

$$f(\mathit{y})={⟨f(\mathit{x}),K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}.$$

The kernel $K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}$ with weight $R_{d,\mathit{\alpha },\mathit{\gamma }}$ is called a weighted reproducing kernel and the space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ is called a weighted reproducing kernel Hilbert space. If ${\gamma }_1={\gamma }_2=\cdots =1$ and $1<{\alpha }_1={\alpha }_2=\cdots$, then the space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ is called unweighted space. Here, ${\mathbb{N}}_0=\{0,1,\dots \}$ and $\mathbb{N}=\{1,2,\dots \}$.

There are many ways to introduce weighted reproducing kernel Hilbert spaces with weights $R_{d,\mathit{\alpha },\mathit{\gamma }}$. In this paper we consider three weights like the cases in the reference [18].

2.1. A Weighted Korobov Space

Let $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ be two sequences satisfying (1). We consider a weighted Korobov space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with weight

$$R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})=r_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h}):={\displaystyle \prod _{j=1}^d}{r}_{{\alpha }_j,{\gamma }_j}(h_j),$$

where

$$r_{\alpha ,\gamma }(h):=\left\{\begin{array}{cc}& 1,\mathrm{for}h=0,\\ & \frac{\gamma }{h^{\lceil \alpha \rceil }},\mathrm{for}h\ge 1\end{array}\right.$$

for $\alpha >1$ and $\gamma \in (0,1]$. We can see the case in the references [18,20]. Then we have the kernel function (2) with

$$K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})=K_{r_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})=\sum _{\mathit{h}\in {\mathbb{N}}_0^d}{r}_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})exp(2\pi i\mathit{h}·(\mathit{x}-\mathit{y}))$$

for $\mathit{x},\mathit{y}\in {[0,1]}^d$, and the inner product (3) with

$${⟨f,g⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}={⟨f,g⟩}_{H(K_{r_{d,\mathit{\alpha },\mathit{\gamma }}})}=\sum _{\mathit{h}\in \in {\mathbb{N}}_0^d}\frac{1}{r_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})}\widehat{f}(\mathit{h})\overline{\widehat{g}}(\mathit{h}).$$

Remark 1.

Obviously, the kernel $K_{r_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})$ is well defined for $\mathit{\alpha }$ and $\mathit{\gamma }$ satisfying (1), due to

$$|K_{r_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})|\le \sum _{\mathit{k}\in {\mathbb{N}}_0^d}{r}_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{k})={\displaystyle \prod _{j=1}^d}(1+\zeta (\lceil {\alpha }_j\rceil ){\gamma }_j)<\infty ,$$

where $\zeta (·)$ is the Riemann zeta function.

2.2. A First Variant of the Weighted Korobov Space

Let $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ be two sequences satisfying (1). We discuss a first variant of the weighted Korobov space with weight

$$R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})={\psi }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h}):={\displaystyle \prod _{j=1}^d}{\psi }_{{\alpha }_j,{\gamma }_j}(h_j),$$

where 1.5

$${\psi }_{\alpha ,\gamma }(h):=\left\{\begin{array}{cc}1,& \mathrm{for}h=0,\\ \frac{\gamma }{h!},& \mathrm{for}1\le h<\lceil \alpha \rceil ,\\ \frac{\gamma (h-\lceil \alpha \rceil )!}{h!},& \mathrm{for}h\ge \lceil \alpha \rceil \end{array}\right.$$

for $\alpha >1$ and $\gamma \in (0,1]$.

Then we have the kernel function (2) with

$$K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})=K_{{\psi }_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})=\sum _{\mathit{h}\in {\mathbb{N}}_0^d}{\psi }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})exp(2\pi i\mathit{h}·(\mathit{x}-\mathit{y}))$$

for $\mathit{x},\mathit{y}\in {[0,1]}^d$ and the inner product (3) with

$${⟨f,g⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}={⟨f,g⟩}_{H(K_{{\psi }_{d,\mathit{\alpha },\mathit{\gamma }}})}=\sum _{\mathit{h}\in \in {\mathbb{N}}_0^d}\frac{1}{{\psi }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})}\widehat{f}(\mathit{h})\overline{\widehat{g}}(\mathit{h}).$$

Lemma 1

([18] Lemma 2). For all $j,k\in \mathbb{N}$ we have

$${\psi }_{{\alpha }_j,{\gamma }_j}(k)\le {\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(k).$$

Remark 2.

From Lemma 1 and $1<{\alpha }_1\le {\alpha }_2\le \cdots$ we get

$$\begin{array}{cc}\hfill |K_{{\psi }_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})|& \le \sum _{\mathit{k}\in {\mathbb{N}}_0^d}{\psi }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{k})={\displaystyle \prod _{j=1}^d}(1+\sum _{k\in \mathbb{N}}{\psi }_{{\alpha }_j,{\gamma }_j}(k))\hfill \\ \hfill & \le {\displaystyle \prod _{j=1}^d}(1+\sum _{k\in \mathbb{N}}{\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(k))\hfill \\ \hfill & ={\displaystyle \prod _{j=1}^d}(1+{\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }\zeta (\lceil {\alpha }_j\rceil ){\gamma }_j)\hfill \\ \hfill & <\infty .\hfill \end{array}$$

Hence, the kernel $K_{{\psi }_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})$ is well defined.

2.3. A Second Variant of the Weighted Korobov Space

Let $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ be two sequences satisfying (1). We study a second variant of the weighted Korobov space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ (see the references [18,21]) with weight

$$R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})={\omega }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h}):={\displaystyle \prod _{j=1}^d}{\omega }_{{\alpha }_j,{\gamma }_j}(h_j),$$

where

$${\omega }_{\alpha ,\gamma }(h):={\left(1+\frac{1}{\gamma }{\displaystyle \sum _{l=1}^{\lceil \alpha \rceil }}{\theta }_l(h)\right)}^{-1}$$

for $\alpha >1$ and $\gamma \in (0,1]$ and 1.5

$${\theta }_l(h):=\left\{\begin{array}{cc}\frac{h!}{(h-l)!},& \mathrm{for}h\ge l,\\ 0,& \mathrm{for}0\le h<l.\end{array}\right.$$

Then we have the kernel function (2) with

$$K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})=K_{{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})=\sum _{\mathit{h}\in {\mathbb{N}}_0^d}{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})exp(2\pi i\mathit{h}·(\mathit{x}-\mathit{y}))$$

for $\mathit{x},\mathit{y}\in {[0,1]}^d$, and the inner product (3) with

$${⟨f,g⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}={⟨f,g⟩}_{H(K_{{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}})}=\sum _{\mathit{h}\in \in {\mathbb{N}}_0^d}\frac{1}{{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})}\widehat{f}(\mathit{h})\overline{\widehat{g}}(\mathit{h}).$$

Lemma 2

([18] Lemma 3). For all $j,k\in \mathbb{N}$ we have

$${\omega }_{{\alpha }_j,{\gamma }_j}(k)\le {\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(k).$$

Remark 3.

We note that the kernel $K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})$ is also well defined. Indeed, it follows from Lemma 2 and $1<{\alpha }_1\le {\alpha }_2\le \cdots$ that

$$\begin{array}{cc}\hfill |K_{{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}}(\mathit{x},\mathit{y})|& \le \sum _{\mathit{k}\in {\mathbb{N}}_0^d}{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{k})={\displaystyle \prod _{j=1}^d}(1+\sum _{k\in \mathbb{N}}{\omega }_{{\alpha }_j,{\gamma }_j}(k))\hfill \\ \hfill & \le {\displaystyle \prod _{j=1}^d}(1+\sum _{k\in \mathbb{N}}{\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(k))\hfill \\ \hfill & ={\displaystyle \prod _{j=1}^d}(1+{\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }\zeta (\lceil {\alpha }_j\rceil ){\gamma }_j)\hfill \\ \hfill & <\infty .\hfill \end{array}$$

Lemma 3.

Let $R_{{\alpha }_j,{\gamma }_j}\in \left\{r_{{\alpha }_j,{\gamma }_j},{\psi }_{{\alpha }_j,{\gamma }_j},{\omega }_{{\alpha }_j,{\gamma }_j}\right\}$ for all $j\in \mathbb{N}$. Then we have for all $j\in \mathbb{N}$, $k\in {\mathbb{N}}_0$

$$R_{{\alpha }_j,{\gamma }_j}(k)\le {\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(k).$$

Particularly, we have for all $j\in \mathbb{N}$, $k\in {\mathbb{N}}_0$

$$R_{{\alpha }_j,{\gamma }_j}(k)\le {\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{r}_{{\alpha }_1,{\gamma }_j}(k).$$

Proof. 

On the one hand, it is obvious from Lemmas 1 and 2 that

$$R_{{\alpha }_j,{\gamma }_j}(k)\le {\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(k)$$

(4)

for all $j,k\in \mathbb{N}$. Since for all $j\in \mathbb{N}$

$$r_{{\alpha }_j,{\gamma }_j}(0)={\psi }_{{\alpha }_j,{\gamma }_j}(0)={\omega }_{{\alpha }_j,{\gamma }_j}(0)=1,$$

we have

$$R_{{\alpha }_j,{\gamma }_j}(0)=1\le {\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }={\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(0).$$

Thus we have for all $j\in \mathbb{N}$, $k\in {\mathbb{N}}_0$ that

$$R_{{\alpha }_j,{\gamma }_j}(k)\le {\lceil {\alpha }_j\rceil }^{\lceil {\alpha }_j\rceil }{r}_{{\alpha }_j,{\gamma }_j}(k).$$

On the other hand, noting for all $j,k\in \mathbb{N}$

$$r_{{\alpha }_j,{\gamma }_j}(k)\le r_{{\alpha }_1,{\gamma }_j}(k),{\psi }_{{\alpha }_j,{\gamma }_j}(k)\le {\psi }_{{\alpha }_1,{\gamma }_j}(k),{\omega }_{{\alpha }_j,{\gamma }_j}(k)\le {\omega }_{{\alpha }_1,{\gamma }_j}(k),$$

and for all $j\in \mathbb{N}$

$$r_{{\alpha }_j,{\gamma }_j}(0)=r_{{\alpha }_1,{\gamma }_j}(0)=1,{\psi }_{{\alpha }_j,{\gamma }_j}(0)={\psi }_{{\alpha }_1,{\gamma }_j}(0)=1,{\omega }_{{\alpha }_j,{\gamma }_j}(0)={\omega }_{{\alpha }_1,{\gamma }_j}(0)=1,$$

we have for all $j\in \mathbb{N}$, $k\in {\mathbb{N}}_0$ that

$$R_{{\alpha }_j,{\gamma }_j}(k)\le R_{{\alpha }_1,{\gamma }_j}(k).$$

Hence, by (4) we further get for all $j\in \mathbb{N}$, $k\in {\mathbb{N}}_0$ that

$$R_{{\alpha }_j,{\gamma }_j}(k)\le R_{{\alpha }_1,{\gamma }_j}(k)\le {\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{r}_{{\alpha }_1,{\gamma }_j}(k).$$

□

Remark 4.

Let $R_{{\alpha }_j,{\gamma }_j}\in \left\{r_{{\alpha }_j,{\gamma }_j},{\psi }_{{\alpha }_j,{\gamma }_j},{\omega }_{{\alpha }_j,{\gamma }_j}\right\}$ for all $j\in \mathbb{N}$. Then we obtain

$$R_{{\alpha }_j,{\gamma }_j}(0)=1\mathit{and}{R}_{{\alpha }_j,{\gamma }_j}(1)\ge \frac{{\gamma }_j}{2}$$

(5)

for all $j\in \mathbb{N}$. Indeed, for all $j\in \mathbb{N}$ we have

$${\psi }_{{\alpha }_j,{\gamma }_j}(0)=r_{{\alpha }_j,{\gamma }_j}(0)={\omega }_{{\alpha }_j,{\gamma }_j}(0)=1,$$

which means $R_{{\alpha }_j,{\gamma }_j}(0)=1$. As a result of all $j\in \mathbb{N}$, we get

$${\psi }_{{\alpha }_j,{\gamma }_j}(1)=r_{{\alpha }_j,{\gamma }_j}(1)={\gamma }_j\mathit{and}{\omega }_{{\alpha }_j,{\gamma }_j}(1)={\left(1+\frac{1}{{\gamma }_j}\right)}^{-1}\ge \frac{{\gamma }_j}{2},$$

which yields $R_{{\alpha }_j,{\gamma }_j}(1)\ge \frac{{\gamma }_j}{2}.$

3. ${\mathit{L}}_{\mathbf{2}}$-Approximation in the Weighted Hilbert Spaces

In this paper we investigate the $L_2$-approximation ${\mathrm{APP}}_d:H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})\to L_2({[0,1]}^d)$ given by

$${\mathrm{APP}}_d(f)=f\mathrm{for}\mathrm{all}f\in H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$$

in weighted Hilbert space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with weight $R_{d,\mathit{\alpha },\mathit{\gamma }}\in \{r_{d,\mathit{\alpha },\mathit{\gamma }},{\psi }_{d,\mathit{\alpha },\mathit{\gamma }},{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}\}$. We note from Remarks 1–3, and [15] that this $L_2$-approximation is compact for $1<{\alpha }_1\le {\alpha }_2\le \cdots$.

We approximate ${\mathrm{APP}}_d$ by using the algorithm $A_{n,d}$ of the form

$$A_{n,d}(f)={\displaystyle \sum _{i=1}^n}{T}_i(f)g_i,\mathrm{for}f\in H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}),$$

(6)

where $g_1,g_2,\dots ,g_n$ belong to $L_2({[0,1]}^d)$ and $T_1,T_2,\dots ,T_n$ are continuous linear functionals on $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$.

We consider the worst case setting in which the error of the algorithm $A_{n,d}$ of the form (6) is defined as

$$e(A_{n,d}):=\underset{{\left|\right|f\left|\right|}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}\le 1}{sup}\left|\right|{\mathrm{APP}}_d(f)-A_{n,d}(f){\left|\right|}_{L_2}.$$

The error $e(A_{n,d})$ is also called the worst case error. The nth minimal worst case error is defined as

$$e(n,{\mathrm{APP}}_d):=\underset{A_{n,d}}{inf}e(A_{n,d})\mathrm{for}n\ge 1,$$

which is the infimum error among all algorithms (6). For $n=0$, we set $A_{0,d}=0$. We call

$$e(0,{\mathrm{APP}}_d)=\underset{{\left|\right|f\left|\right|}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}\le 1}{sup}\left|\right|{\mathrm{APP}}_d(f){\left|\right|}_{L_2}$$

the initial error of the problem ${\mathrm{APP}}_d$.

We are interested in how the worst case error for the algorithm $A_{n,d}$ depends on the numbers n and d. We define the information complexity as

$$n(\epsilon ,{\mathrm{APP}}_d):=min\{n\in {\mathbb{N}}_0:e(n,{\mathrm{APP}}_d)\le \epsilon \},$$

where $\epsilon \in (0,1)$ and $d\in \mathbb{N}$. In this paper, we set ${\mathbb{N}}_0=\{0,1,\dots \}$ and $\mathbb{N}=\{1,2,\dots \}$.

By the references [2,4] we know that the nth minimal worst case errors $e(n,{\mathrm{APP}}_d)$ and the information complexity $n(\epsilon ,{\mathrm{APP}}_d)$ are related to the eigenvalues of the continuously linear operator $W_d={\mathrm{APP}}_d^{*}{\mathrm{APP}}_d:H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})\to H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$, where ${\mathrm{APP}}_d^{*}$ is the operator dual to ${\mathrm{APP}}_d$. The eigenvalues of $W_d$ are denoted by ${\left\{{\lambda }_{d,j}\right\}}_{j\in \mathbb{N}}$ satisfying

$${\lambda }_{d,1}\ge {\lambda }_{d,2}\ge \cdots \ge 0$$

and the corresponding orthogonal eigenvectors of ${\left\{{\lambda }_{d,j}\right\}}_{j\in \mathbb{N}}$ by ${\left\{{\eta }_{d,j}\right\}}_{j\in \mathbb{N}}$ satisfying

$${⟨{\eta }_{d,i},{\eta }_{d,j}⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}={\delta }_{i,j},\mathrm{for}\mathrm{all}i,j\in \mathbb{N},$$

where

$$W_d{\eta }_{d,j}={\lambda }_{d,j}{\eta }_{d,j},\mathrm{for}\mathrm{all}j\in \mathbb{N}.$$

Here ${\delta }_{i,j}=1$ for $i=j$ and ${\delta }_{i,j}=0$ for $i\ne j$. Then the nth minimal worst case error is attained for the algorithm

$$A_{n,d}^{\diamond }f={\displaystyle \sum _{i=1}^n}{⟨f,{\eta }_{d,i}⟩}_{H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})}{\eta }_{d,i},\mathrm{for}\mathrm{all}n\in \mathbb{N}$$

and

$$e(n,{\mathrm{APP}}_d)=e(A_{n,d}^{\diamond })=\sqrt{{\lambda }_{d,n+1}},\mathrm{for}\mathrm{all}n\in \mathbb{N}.$$

The initial error $e(0,{\mathrm{APP}}_d)=\sqrt{{\lambda }_{d,1}}$. Hence, we have $e(n,{\mathrm{APP}}_d)=\sqrt{{\lambda }_{d,n+1}}$ for all $n\in {\mathbb{N}}_0$. This deduces that the information complexity is equal to

$$\begin{array}{cc}\hfill n(\epsilon ,{\mathrm{APP}}_d)& =min\left\{n\in {\mathbb{N}}_0:\sqrt{{\lambda }_{d,n+1}}\le \epsilon \right\}=min\left\{n\in {\mathbb{N}}_0:{\lambda }_{d,n+1}\le {\epsilon }^2\right\}.\hfill \end{array}$$

(7)

Since the eigenvalues ${\lambda }_{d,j}$ with $j\in \mathbb{N}$ of the operator $W_d$ are $R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{k})$ with $\mathit{k}\in {\mathbb{N}}_0^d$ (see [4] p. 215), by (7) the information complexity of ${\mathrm{APP}}_d$ from the space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ is equal to

$$\begin{array}{cc}\hfill n(\epsilon ,{\mathrm{APP}}_d)& =min\left\{n\in {\mathbb{N}}_0:{\lambda }_{d,n+1}\le {\epsilon }^2\right\}=\left|\left\{n\in \mathbb{N}:{\lambda }_{d,n}>{\epsilon }^2\right\}\right|\hfill \\ \hfill & =\left|\left\{\mathit{h}\in {\mathbb{N}}_0^d:R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})>{\epsilon }^2\right\}\right|=\left|\left\{\mathit{h}\in {\mathbb{N}}_0^d:{\displaystyle \prod _{j=1}^d}{R}_{{\alpha }_j,{\gamma }_j}(h_j)>{\epsilon }^2\right\}\right|,\hfill \end{array}$$

(8)

with $\epsilon \in (0,1)$ and $d\in \mathbb{N}$, where $\left|A\right|$ denotes the cardinality of set A.

Note that for the $L_2$-approximation ${\mathrm{APP}}_d$ from the space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ the absolute error criterion and the normalized error criterion are the same, since the initial error $e(0,{\mathrm{APP}}_d)=\sqrt{{\lambda }_{d,1}}=1$.

4. Tractability in Weighted Hilbert Spaces and Main Results

In this paper we will study the classical tractability and the exponential convergence-tractability (EC-tractability) for the problem $\mathrm{APP}={\left\{{\mathrm{APP}}_d\right\}}_{d\in \mathbb{N}}$ in the weighted Hilbert space $H_{d,\mathit{\alpha },\mathit{\gamma }}$.

4.1. Tractability and Main Results

We focus on the behaviours of the information complexity $n(\epsilon ,{\mathrm{APP}}_d)$ depending on the dimension d and the error threshold $\epsilon$. Hence, we will study several notions about the classical tractability and the exponential convergence-tractability (EC-tractability) notions (see [4,5,6,7,8,9,11,12,16,22]).

Definition 1.

Let $APP={\left\{{APP}_d\right\}}_{d\in N}$. We say the following:

• Strong polynomial tractability (SPT) if there are positive numbers C and p such that

  $$n(\epsilon ,{APP}_d)\le C{({\epsilon }^{-1})}^p\mathit{for}\mathit{all}d\in \mathbb{N},\epsilon \in (0,1).$$
  In this case we define the exponent $p^{\mathit{str}}$ of SPT as

  $$p^{\mathit{str}}:=inf\{p:\exists C>0\mathit{such}\mathit{that}n(\epsilon ,{APP}_d)\le C{({\epsilon }^{-1})}^p,\forall d\in \mathbb{N},\epsilon \in (0,1)\}.$$
• Polynomial tractability (PT) if there are positive numbers C, p, and q such that

  $$n(\epsilon ,{APP}_d)\le C{d}^q{({\epsilon }^{-1})}^p\mathit{for}\mathit{all}d\in \mathbb{N},\epsilon \in (0,1).$$
• Quasi-polynomial tractability (QPT) if there are positive numbers C and t such that

  $$n(\epsilon ,{APP}_d)\le Cexp\left(t\left(1+lnd)(1+ln{\epsilon }^{-1}\right)\right)\mathit{for}\mathit{all}d\in \mathbb{N},\epsilon \in (0,1).$$
• Uniform weak tractability (UWT) if for all $t,s>0$,

  $$\underset{{\epsilon }^{-1}+d\to \infty }{lim}\frac{lnn(\epsilon ,{APP}_d)}{d^t+{({\epsilon }^{-1})}^s}=0.$$
• Weak tractability (WT) if

  $$\underset{{\epsilon }^{-1}+d\to \infty }{lim}\frac{lnn(\epsilon ,{APP}_d)}{d+{\epsilon }^{-1}}=0.$$
• $(t,s)$-weak tractability ($(t,s)$-WT) for fixed positive t and s if

  $$\underset{{\epsilon }^{-1}+d\to \infty }{lim}\frac{lnn(\epsilon ,{APP}_d)}{d^t+{({\epsilon }^{-1})}^s}=0.$$

We find that (1,1)-WT is the same as WT and

$$\mathrm{SPT}⟹\mathrm{PT}⟹\mathrm{QPT}⟹\mathrm{UWT}⟹\mathrm{WT}.$$

In the above definitions regarding classical tractability, replacing ${\epsilon }^{-1}$ with $(1+ln({\epsilon }^{-1}))$, we will have the following definitions about exponential convergence-tractability (EC-tractability).

Definition 2.

Let $\mathit{APP}={\left\{{\mathit{APP}}_d\right\}}_{d\in N}$. We say we have the following:

• Exponential convergence-strong polynomial tractability (EC-SPT) if there are positive numbers C and p such that

  $$n(\epsilon ,{\mathit{APP}}_d)\le C{\left(1+ln({\epsilon }^{-1})\right)}^p\mathit{for}\mathit{all}d\in \mathbb{N},\epsilon \in (0,1).$$
  The exponent of EC-SPT is defined as

  $$inf\{p:\exists C>0\mathit{such}\mathit{that}n(\epsilon ,{\mathit{APP}}_d)\le C{\left(1+ln({\epsilon }^{-1})\right)}^p,\forall d\in \mathbb{N},\epsilon \in (0,1)\}.$$
• Exponential convergence-polynomial tractability (EC-PT) if there are positive numbers C, p, and q such that

  $$n(\epsilon ,{\mathit{APP}}_d)\le C{d}^q{\left(1+ln({\epsilon }^{-1})\right)}^p\mathit{for}\mathit{all}d\in \mathbb{N},\epsilon \in (0,1).$$
• Exponential convergence-uniform weak tractability (EC-UWT) if for all $t,s>0$

  $$\underset{{\epsilon }^{-1}+d\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d^t+{\left(1+ln({\epsilon }^{-1})\right)}^s}=0.$$
• Exponential convergence-weak tractability (EC-WT) if

  $$\underset{{\epsilon }^{-1}+d\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d+ln({\epsilon }^{-1})}=0.$$
• Exponential convergence-$(t,s)$-weak tractability (EC-$(t,s)$-WT) for fixed positive t and s if

  $$\underset{{\epsilon }^{-1}+d\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d^t+{\left(1+ln({\epsilon }^{-1})\right)}^s}=0.$$

We note that EC-$(1,1)$-WT is the same as EC-WT, and

$$\mathrm{EC}-\mathrm{SPT}⟹\mathrm{EC}-\mathrm{PT}⟹\mathrm{EC}-\mathrm{QPT}⟹\mathrm{EC}-\mathrm{UWT}⟹\mathrm{EC}-\mathrm{WT}.$$

Obviously, if the problem $\mathrm{APP}$ has exponential convergence-tractability, then it has classical tractability and

$$\mathrm{EC}-(t,s)-\mathrm{WT}⟹(t,s)-\mathrm{WT},\mathrm{EC}-\mathrm{UWT}⟹\mathrm{UWT},\mathrm{EC}-\mathrm{WT}⟹\mathrm{WT}.$$

In the worst case setting the classical tractability and EC-tractability of the problem $\mathrm{APP}={\left\{{\mathrm{APP}}_d\right\}}_{d\in N}$ in the weighted Hilbert space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ satisfying

$$1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0,\mathrm{and}1<{\alpha }^{*}={\alpha }_1={\alpha }_2=\cdots$$

have been solved by [13,14,16,18] as follows:

• For $R_{d,{\alpha }^{*},\mathit{\gamma }}\in \{r_{d,{\alpha }^{*},\mathit{\gamma }},{\psi }_{d,{\alpha }^{*},\mathit{\gamma }},{\omega }_{d,{\alpha }^{*},\mathit{\gamma }}\}$, SPT holds iff PT holds iff

  $$s_{\mathit{\gamma }}:=inf\left\{\kappa >0:{\displaystyle \sum _{j=1}^{\infty }}{\gamma }_j^{\kappa }<\infty \right\}<\infty$$

  and the exponent of SPT is

  $$p^{\mathrm{str}}=2max\left(s_{\mathit{\gamma }},\frac{1}{\alpha }\right).$$
• For $R_{d,{\alpha }^{*},\mathit{\gamma }}=r_{d,{\alpha }^{*},\mathit{\gamma }}$, QPT, UWT, and WT are equivalent and hold iff

  $${\mathit{\gamma }}_I:=\underset{j\in \mathbb{N}}{inf}{\gamma }_j<1.$$
  For $R_{d,{\alpha }^{*},\mathit{\gamma }}\in \{{\psi }_{d,{\alpha }^{*},\mathit{\gamma }},{\omega }_{d,{\alpha }^{*},\mathit{\gamma }}\}$,

  $${\mathit{\gamma }}_I<\infty$$

  implies QPT.
• For $R_{d,{\alpha }^{*},\mathit{\gamma }}\in \{r_{d,{\alpha }^{*},\mathit{\gamma }},{\psi }_{d,{\alpha }^{*},\mathit{\gamma }},{\omega }_{d,{\alpha }^{*},\mathit{\gamma }}\}$ and $t>1$, $(t,s)$-WT holds for all $1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0$.
• For $R_{d,{\alpha }^{*},\mathit{\gamma }}=r_{d,{\alpha }^{*},\mathit{\gamma }}$, EC-WT holds iff

  $$\underset{j\to \infty }{lim}{\gamma }_j=0.$$
• For $R_{d,{\alpha }^{*},\mathit{\gamma }}=r_{d,{\alpha }^{*},\mathit{\gamma }}$ and $t<1$, EC-$(t,1)$-WT holds iff

  $$\underset{j\to \infty }{lim}\frac{lnj}{ln({\gamma }_j^{-1})}=0.$$

In the worst case setting the classical tractability such as SPT, PT, and WT of the problem $\mathrm{APP}={\left\{{\mathrm{APP}}_d\right\}}_{d\in N}$ in the weighted Hilbert space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ satisfying (1), i.e.,

$$1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0,\mathrm{and}1<{\alpha }_1\le {\alpha }_2\le \cdots$$

has been solved by [18] as follows:

• For $R_{d,\mathit{\alpha },\mathit{\gamma }}\in \{r_{d,\mathit{\alpha },\mathit{\gamma }},{\psi }_{d,\mathit{\alpha },\mathit{\gamma }},{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}\}$, SPT holds iff PT holds iff

  $$\delta :=\underset{j\to \infty }{lim\; inf}\frac{ln{\gamma }_j^{-1}}{lnj}>0.$$
  The exponent of SPT is

  $$p^{\mathrm{str}}=2max\left\{\frac{1}{\delta },\frac{1}{\lceil {\alpha }_1\rceil }\right\}.$$
• For $R_{d,\mathit{\alpha },\mathit{\gamma }}=r_{d,\mathit{\alpha },\mathit{\gamma }}$, WT holds iff

  $$\underset{j\to \infty }{lim}{\gamma }_j<1.$$
• For $R_{d,\mathit{\alpha },\mathit{\gamma }}\in \{r_{d,\mathit{\alpha },\mathit{\gamma }},{\psi }_{d,\mathit{\alpha },\mathit{\gamma }},{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}\}$ and $t>1$, $(t,s)$-WT holds.

In this paper, we investigate the EC-tractability of the problem $\mathrm{APP}={\left\{{\mathrm{APP}}_d\right\}}_{d\in N}$ in the weighted Hilbert space $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ satisfying (1). We obtain sufficient and necessary conditions for EC-$(t,1)$-WT with $0<t<1$ and $t=1$.

Theorem 1.

Let $\mathit{\gamma }={\left\{{\gamma }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ satisfy (1). Then the problem $\mathit{APP}={\left\{{\mathit{APP}}_d\right\}}_{d\in N}$ in the weighted Hilbert spaces $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with $R_{d,\mathit{\alpha },\mathit{\gamma }}\in \{r_{d,\mathit{\alpha },\mathit{\gamma }},{\psi }_{d,\mathit{\alpha },\mathit{\gamma }},{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}\}$

(1) 

is EC-WT, if and only if

$$\underset{j\to \infty }{lim}{\gamma }_j=0.$$

(2) 

is EC-$(t,1)$-WT with $t<1$, if and only if

$$\underset{j\to \infty }{lim}\frac{lnj}{ln({\gamma }_j^{-1})}=0.$$

4.2. The Proof

In order to prove Theorem 1 we need the following Lemmas.

Lemma 4.

Let $\eta >0$, $\epsilon \in (0,1)$. We have for any $d\in N$

$$n(\epsilon ,{\mathrm{APP}}_d)\le {\epsilon }^{-2\eta }{\displaystyle \prod _{j=1}^d}\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }\right).$$

Proof. 

By Lemma 3 we have

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=1}^{\infty }}{\lambda }_{d,k}^{\eta }& =\sum _{\mathit{k}\in N_0^d}{\left(R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{k})\right)}^{\eta }={\displaystyle \prod _{j=1}^d}\left(1+{\displaystyle \sum _{k=1}^{\infty }}{\left(R_{{\alpha }_j,{\gamma }_j}(k)\right)}^{\eta }\right)\hfill \\ \hfill & \le {\displaystyle \prod _{j=1}^d}\left(1+{\displaystyle \sum _{k=1}^{\infty }}{\left({\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{r}_{{\alpha }_1,{\gamma }_j}(k)\right)}^{\eta }\right)\hfill \\ \hfill & ={\displaystyle \prod _{j=1}^d}\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }{\displaystyle \sum _{k=1}^{\infty }}{\left(r_{{\alpha }_1,{\gamma }_j}(k)\right)}^{\eta }\right)\hfill \\ \hfill & ={\displaystyle \prod _{j=1}^d}\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }{\displaystyle \sum _{k=1}^{\infty }}{\left(\frac{{\gamma }_j}{k^{\lceil {\alpha }_1\rceil }}\right)}^{\eta }\right)\hfill \\ \hfill & ={\displaystyle \prod _{j=1}^d}\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta \left(\lceil {\alpha }_1\rceil \eta \right){\gamma }_j^{\eta }\right).\hfill \end{array}$$

This yields

$$n{\lambda }_{d,n}^{\eta }\le {\displaystyle \sum _{k=1}^n}{\lambda }_{d,k}^{\eta }\le {\displaystyle \sum _{k=1}^{\infty }}{\lambda }_{d,k}^{\eta }\le {\displaystyle \prod _{j=1}^d}\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta \left(\lceil {\alpha }_1\rceil \eta \right){\gamma }_j^{\eta }\right),$$

which means

$${\lambda }_{d,n}\le \frac{{\displaystyle \prod _{j=1}^d}{\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }\right)}^{1/\eta }}{n^{1/\eta }}.$$

It follows from the above inequality and (7)

$$n(\epsilon ,{\mathrm{APP}}_d)=min\left\{n\in {\mathbb{N}}_0:{\lambda }_{d,n+1}\le {\epsilon }^2\right\},$$

that

$$n(\epsilon ,{\mathrm{APP}}_d)\le {\epsilon }^{-2\eta }{\displaystyle \prod _{j=1}^d}\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }\right).$$

This proof is complete. □

Lemma 5.

Let $\epsilon \in (0,1)$. We have for any $d\ge 2$

$$n(\epsilon ,{\mathit{APP}}_d)\ge ⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉.$$

Proof. 

Set

$$H=H(\epsilon ,d,\alpha ):=\left\{h\in N_0:h\le ⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉-1\right\}.$$

If $h>⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉-1$ and $d\ge 2$, by Lemma 3 we have

$${\displaystyle \prod _{j=1}^{d-1}}{R}_{{\alpha }_j,{\gamma }_j}(h_j)R_{{\alpha }_d,{\gamma }_d}(h)\le R_{{\alpha }_d,{\gamma }_d}(h)\le {\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{r}_{{\alpha }_1,{\gamma }_d}(h)={\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }\frac{{\gamma }_d}{h^{\lceil {\alpha }_1\rceil }}\le {\epsilon }^2$$

for any $\{h_1,\cdots ,h_{d-1}\}\in N_0^{d-1}$, which means

$$\left\{\mathit{h}\in N_0^{d-1}:{\displaystyle \prod _{j=1}^{d-1}}{R}_{{\alpha }_j,{\gamma }_j}(h_j)R_{{\alpha }_d,{\gamma }_d}(h)>{\epsilon }^2\right\}=⌀$$

(9)

for all $h>⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉-1$. It follows from (8) and (9) that

$$\begin{array}{cc}\hfill n(\epsilon ,{\mathrm{APP}}_d)& =\left|\left\{\mathit{h}\in N_0^d:{\displaystyle \prod _{j=1}^d}{R}_{{\alpha }_j,{\gamma }_j}(h_j)>{\epsilon }^2\right\}\right|\hfill \\ \hfill & =\left|\left\{\mathit{h}\in N_0^d:{\displaystyle \prod _{j=1}^{d-1}}{R}_{{\alpha }_j,{\gamma }_j}(h_j)R_{{\alpha }_d,{\gamma }_d}(h_d)>{\epsilon }^2\right\}\right|\hfill \\ \hfill & =\sum _{h\in N_0}\left|\left\{\mathit{h}\in N_0^{d-1}:{\displaystyle \prod _{j=1}^{d-1}}{R}_{{\alpha }_j,{\gamma }_j}(h_j)R_{{\alpha }_d,{\gamma }_d}(h)>{\epsilon }^2\right\}\right|\hfill \\ \hfill & =\sum _{h\in H}\left|\left\{\mathit{h}\in N_0^{d-1}:{\displaystyle \prod _{j=1}^{d-1}}{R}_{{\alpha }_j,{\gamma }_j}(h_j)R_{{\alpha }_d,{\gamma }_d}(h)>{\epsilon }^2\right\}\right|\hfill \\ \hfill & =\sum _{h\in (H\setminus \{0\})}\left|\left\{\mathit{h}\in N_0^{d-1}:{\displaystyle \prod _{j=1}^{d-1}}{R}_{{\alpha }_j,{\gamma }_j}(h_j)>{\epsilon }^2{R}_{{\alpha }_d,{\gamma }_d}^{-1}(h)\right\}\right|\hfill \\ \hfill & +\left|\left\{\mathit{h}\in N_0^{d-1}:{\displaystyle \prod _{j=1}^{d-1}}{R}_{{\alpha }_j,{\gamma }_j}(h_j)>{\epsilon }^2\right\}\right|\hfill \\ \hfill & =\sum _{h\in (H\setminus \{0\})}n(\epsilon R_{{\alpha }_d,{\gamma }_d}^{-1/2}(h),{\mathrm{APP}}_{d-1})+n(\epsilon ,{\mathrm{APP}}_{d-1})\hfill \\ \hfill & ={\displaystyle \sum _{h=1}^{⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉-1}}n(\epsilon R_{{\alpha }_d,{\gamma }_d}^{-1/2}(h),{\mathrm{APP}}_{d-1})+n(\epsilon ,{\mathrm{APP}}_{d-1})\hfill \\ \hfill & \ge ⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉.\hfill \end{array}$$

This finishes the proof. □

Lemma 6.

For ${\displaystyle \prod _{j=1}^d}\left(\frac{{\gamma }_j}{2}\right)>{\epsilon }^2$ and $\epsilon \in (0,1)$ we have

$$n(\epsilon ,{\mathrm{APP}}_d)\ge 2^d.$$

Proof. 

Set

$$\mathcal{A}(\epsilon ,d)=\left\{\mathit{h}\in {\mathbb{N}}_0^d:{\displaystyle \prod _{j=1}^d}{R}_{{\alpha }_j,{\gamma }_j}(h_j)>{\epsilon }^2\right\}.$$

If $\mathit{h}=\{h_1,h_2,\dots ,h_d\}\in {\{0,1\}}^d$, we have from (5) that

$$R_{d,\mathit{\alpha },\mathit{\gamma }}(\mathit{h})={\displaystyle \prod _{j=1}^d}{R}_{{\alpha }_j,{\gamma }_j}(h_j)\ge {\displaystyle \prod _{j=1}^d}\left(\frac{{\gamma }_j}{2}\right).$$

Thus, we have ${\{0,1\}}^d\in \mathcal{A}(\epsilon ,d)$ for ${\displaystyle {\displaystyle \prod _{j=1}^d}}\left(\frac{{\gamma }_j}{2}\right)>{\epsilon }^2$. Hence, it follows from (8) that

$$\begin{array}{cc}\hfill n(\epsilon ,{\mathrm{APP}}_d)& =\left|\mathcal{A}(\epsilon ,d)\right|\ge \left|\left\{\mathit{h}\in {\{0,1\}}^d:{\displaystyle \prod _{j=1}^d}{R}_{{\alpha }_j,{\gamma }_j}(h_j)>{\epsilon }^2\right\}\right|=2^d\hfill \end{array}$$

for ${\displaystyle \prod _{j=1}^d}\left(\frac{{\gamma }_j}{2}\right)>{\epsilon }^2$. This proof is complete. □

Proof of Theorem 1.

If there are infinitely many ${\gamma }_j=0$ for $j\in N$, the results are obviously true. Without loss of generality we discuss only that the ${\gamma }_j$ are positive for $j\in N$.

(1)

Let $\delta >0$ and take $\epsilon ={\displaystyle \prod _{j=1}^d}{\left(\frac{{\gamma }_j}{2}\right)}^{\frac{1+\delta }{2}}$, then we have

$${\displaystyle \prod _{j=1}^d}\left(\frac{{\gamma }_j}{2}\right)>{\epsilon }^2.$$

It follows from Lemma 6 that

$$\begin{array}{cc}\hfill \frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d+ln({\epsilon }^{-1})}& \ge \frac{dln2}{d+\frac{1+\delta }{2}·ln\left({\displaystyle \prod _{j=1}^d}(2{\gamma }_j^{-1})\right)}\hfill \\ \hfill & \ge \frac{dln2}{d+\frac{1+\delta }{2}·d·ln(2{\gamma }_d^{-1})}\hfill \\ \hfill & =\frac{ln2}{1+\frac{1+\delta }{2}·\left(ln2+ln({\gamma }_d^{-1})\right)}.\hfill \end{array}$$

(10)

Assume that App is EC-WT, i.e., for the above fixed $\epsilon$

$$\underset{d\to \infty }{lim}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d+ln({\epsilon }^{-1})}=0.$$

Combing (10) and the above equality we have

$$0=\underset{d\to \infty }{lim}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d+ln({\epsilon }^{-1})}\ge \underset{d\to \infty }{lim}\frac{ln2}{1+\frac{1+\delta }{2}·\left(ln2+ln({\gamma }_d^{-1})\right)}.$$

This implies $\underset{d\to \infty }{lim}{\gamma }_d=0$.

On the other hand, assume that we have $\underset{d\to \infty }{lim}{\gamma }_d=0$. For $\eta >0$ we obtain from the upper bound in Lemma 4 that

$$\begin{array}{cc}\hfill \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d+ln({\epsilon }^{-1})}& \le \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}ln\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }\right)}{d+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }}{d+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le 2\eta +\underset{d\to \infty }{lim\; sup}\frac{{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\displaystyle \sum _{j=1}^d}{\gamma }_j^{\eta }}{d}\hfill \\ \hfill & =2\eta ,\hfill \end{array}$$

where we used $ln(1+x)\le x$ for all $x\ge 0$ and $\underset{d\to \infty }{lim\; sup}\frac{{\sum }_{j=1}^d{\gamma }_j^{\eta }}{d}=0$ if $\underset{d\to \infty }{lim}{\gamma }_d=0$. Setting $\eta \to 0$, we have

$$\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d+ln({\epsilon }^{-1})}=0,$$

which yields that ET-WT holds.

(2)

Assume that APP is EC-$(t,1)$-WT for $t<1$. First, we note that $\underset{d\to \infty }{lim}{\gamma }_d=0$. Indeed, if $\underset{d\to \infty }{lim}{\gamma }_d\ne 0$, we deduce from Theorem 1 (1) that EC-WT doesn’t hold, i.e.,

$$0<\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d+ln({\epsilon }^{-1})}\le \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d^t+ln({\epsilon }^{-1})}.$$

This deduces that EC-$(t,1)$-WT for $t<1$ does not hold.

Next, we will prove $\underset{j\to \infty }{lim}\frac{lnj}{ln({\gamma }_j^{-1})}=0$. Let $\epsilon ={\epsilon }_d\in (0,1)$ such that

$$ln{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}=d^t$$

for large $d\in N$. From the lower bound in Lemma 5 we obtain

$$\begin{array}{cc}\hfill \frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d^t+ln({\epsilon }^{-1})}& \ge \frac{ln⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉}{d^t+ln({\epsilon }^{-1})}\ge \frac{ln{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & =\frac{d^t}{d^t+ln({\epsilon }^{-1})}=\frac{d^t}{d^t+\lceil {\alpha }_1\rceil d^t/2+ln({\gamma }_d^{-1})/2-\lceil {\alpha }_1\rceil (ln\lceil {\alpha }_1\rceil )/2}\hfill \\ \hfill & =\frac{1}{1+\lceil {\alpha }_1\rceil /2+ln({\gamma }_d^{-1})/(2d^t)-\lceil {\alpha }_1\rceil (ln\lceil {\alpha }_1\rceil )/(2d^t)}.\hfill \end{array}$$

It follows from the assumption that

$$\begin{array}{cc}\hfill 0=\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d^t+ln({\epsilon }^{-1})}& \ge \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{1}{1+\lceil {\alpha }_1\rceil /2+ln({\gamma }_d^{-1})/(2d^t)-\lceil {\alpha }_1\rceil (ln\lceil {\alpha }_1\rceil )/(2d^t)}\hfill \\ \hfill & =\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{1}{1+\lceil {\alpha }_1\rceil /2+ln({\gamma }_d^{-1})/(2d^t)},\hfill \end{array}$$

which implies

$$\underset{d\to \infty }{lim}\frac{d^t}{ln({\gamma }_d^{-1})}=0.$$

Using the fact that $d^t\ge ln{d}^t=tlnd\ge 0$ for large $d\in N$, we have

$$0\le \underset{d\to \infty }{lim}\frac{lnd}{ln({\gamma }_d^{-1})}\le \underset{d\to \infty }{lim}\frac{d^t}{tln({\gamma }_d^{-1})}=0,$$

i.e.,

$$\underset{d\to \infty }{lim}\frac{lnd}{ln({\gamma }_d^{-1})}=0.$$

On the other hand, assume that $\underset{j\to \infty }{lim}\frac{lnj}{ln({\gamma }_j^{-1})}=0$. Then we obtain that for all $\delta >0$ there exists a positive number $N_{\delta }>0$ such that

$$\begin{array}{c}\hfill {\gamma }_j\le j^{-\delta }\mathrm{for}\mathrm{all}j\ge N_{\delta }.\end{array}$$

(11)

Let $\eta >0$. We get from Lemma 4 that

$$\begin{array}{cc}\hfill lnn(\epsilon ,{\mathrm{APP}}_d)& \le 2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}ln\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }\right)\hfill \\ \hfill & \le 2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta },\hfill \end{array}$$

(12)

where we used $ln(1+x)\le x$ for all $x>0$. Choose $\delta =\frac{2}{\eta }$. By (11) and (12) we get

$$\begin{array}{cc}\hfill \frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d^t+ln({\epsilon }^{-1})}& \le \frac{2\eta ln({\epsilon }^{-1})+{\displaystyle {\displaystyle \sum _{j=1}^{N_{2/\eta }-1}}}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }+{\displaystyle {\displaystyle \sum _{j=N_{2/\eta }}^{max\{d,N_{2/\eta }\}}}}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le 2\eta +\frac{{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_1^{\eta }(N_{2/\eta }-1)+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\displaystyle {\displaystyle \sum _{j=N_{2/\eta }}^{max\{d,N_{2/\eta }\}}}}{j}^{-2}}{d^t+ln({\epsilon }^{-1})}.\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d^t+ln({\epsilon }^{-1})}& \le 2\eta +{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta )\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{{\displaystyle {\displaystyle \sum _{j=N_{2/\eta }}^{max\{d,N_{2/\eta }\}}}}{j}^{-2}}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le 2\eta +{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta )\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{{\displaystyle {\displaystyle \sum _{j=1}^{\infty }}}{j}^{-2}}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & =2\eta .\hfill \end{array}$$

Setting $\eta \to 0$, we have

$$\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d^t+ln({\epsilon }^{-1})}=0.$$

Therefore, Theorem 1 is proved.

□

Example 1.

An example for EC-WT.

Assume that ${\gamma }_j=j^{-2}$ and ${\alpha }_j=j+1$ for all $j\in \mathbb{N}$. Next, we will study EC-WT for the weighted Hilbert spaces $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with weight $R_{d,\mathit{\alpha },\mathit{\gamma }}\in \{r_{d,\mathit{\alpha },\mathit{\gamma }},{\psi }_{d,\mathit{\alpha },\mathit{\gamma }},{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}\}$.

Obviously, we have $\underset{j\to \infty }{lim}{\gamma }_j=0$. By Lemma 4 we get

$$\begin{array}{cc}\hfill \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathrm{APP}}_d)}{d+ln({\epsilon }^{-1})}& \le \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}ln\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }\right)}{d+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }}{d+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le 2\eta +\underset{d\to \infty }{lim\; sup}\frac{{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\displaystyle \sum _{j=1}^d}{\gamma }_j^{\eta }}{d}\hfill \\ \hfill & =2\eta +\underset{d\to \infty }{lim\; sup}\frac{{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\displaystyle \sum _{j=1}^d}{j}^{-2\eta }}{d}\hfill \\ \hfill & =2\eta ,\hfill \end{array}$$

where in the second inequality we used $ln(1+x)\le x$ for all $x\ge 0$ and $\underset{d\to \infty }{lim\; sup}\frac{{\sum }_{j=1}^d{j}^{-2\eta }}{d}=0$. Setting $\eta \to 0$, we have

$$\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d+ln({\epsilon }^{-1})}=0.$$

Hence, APP is EC-WT.

Example 2.

An example for EC-$(t,1)$-WT for $t<1$.

Assume that ${\gamma }_j=2^{-j}$ and ${\alpha }_j=2j$ for all $j\in \mathbb{N}$. Next, we will study EC-$(t,1)$-WT for $t<1$ for the weighted Hilbert spaces $H(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}})$ with weight $R_{d,\mathit{\alpha },\mathit{\gamma }}\in \{r_{d,\mathit{\alpha },\mathit{\gamma }},{\psi }_{d,\mathit{\alpha },\mathit{\gamma }},{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}\}$.

Note that $\underset{j\to \infty }{lim}\frac{lnj}{ln({\gamma }_j^{-1})}=\underset{j\to \infty }{lim}\frac{lnj}{jln2}=0$. It follows from Lemma 4 that

$$\begin{array}{cc}\hfill lnn(\epsilon ,{\mathit{APP}}_d)& \le 2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}ln\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\gamma }_j^{\eta }\right)\hfill \\ \hfill & =2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}ln\left(1+{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta )2^{-\eta j}\right)\hfill \\ \hfill & \le 2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta )2^{-\eta j},\hfill \end{array}$$

where in the last inequality we used $ln(1+x)\le x$ for all $x>0$. It yields that

$$\begin{array}{cc}\hfill \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d^t+ln({\epsilon }^{-1})}& \le \underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{2\eta ln({\epsilon }^{-1})+{\displaystyle \sum _{j=1}^d}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta )2^{-\eta j}}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le 2\eta +\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\displaystyle \sum _{j=1}^d}{2}^{-\eta j}}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & \le 2\eta +\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil \eta }\zeta (\lceil {\alpha }_1\rceil \eta ){\displaystyle \sum _{j=1}^{\infty }}{2}^{-\eta j}}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & =2\eta .\hfill \end{array}$$

Setting $\eta \to 0$, we have

$$\underset{d+{\epsilon }^{-1}\to \infty }{lim\; sup}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d^t+ln({\epsilon }^{-1})}=0.$$

Hence, APP is EC-$(t,1)$-WT for $t<1$.

Remark 5.

We note that for Example 1 with ${\gamma }_j=j^{-2}$ and ${\alpha }_j=j+1$ for all $j\in \mathbb{N}$, APP is EC-WT, but not EC-$(t,1)$-WT for $t<1$. Indeed, let $\epsilon ={\epsilon }_d\in (0,1)$ such that

$$ln{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}=d,\mathrm{i}.\mathrm{e}.,{\epsilon }^{-1}=\frac{d{e}^{d\lceil {\alpha }_1\rceil /2}}{{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil /2}}$$

for large $d\in N$. From Lemma 5 we have

$$\begin{array}{cc}\hfill \frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d^t+ln({\epsilon }^{-1})}& \ge \frac{ln⌈{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}⌉}{d^t+ln({\epsilon }^{-1})}\ge \frac{ln{\left({\epsilon }^{-2}{\lceil {\alpha }_1\rceil }^{\lceil {\alpha }_1\rceil }{\gamma }_d\right)}^{\frac{1}{\lceil {\alpha }_1\rceil }}}{d^t+ln({\epsilon }^{-1})}\hfill \\ \hfill & =\frac{d}{d^t+ln({\epsilon }^{-1})}=\frac{d}{d^t+\lceil {\alpha }_1\rceil d/2+lnd-\lceil {\alpha }_1\rceil (ln\lceil {\alpha }_1\rceil )/2}\hfill \\ \hfill & =\frac{1}{d^{t-1}+\lceil {\alpha }_1\rceil /2+lnd/d-\lceil {\alpha }_1\rceil (ln\lceil {\alpha }_1\rceil )/(2d)}.\hfill \end{array}$$

For the above fixed ε and $t<1$ we obtain

$$\underset{d\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d+ln({\epsilon }^{-1})}\ge \underset{d\to \infty }{lim}\frac{1}{d^{t-1}+\lceil {\alpha }_1\rceil /2+lnd/d-\lceil {\alpha }_1\rceil (ln\lceil {\alpha }_1\rceil )/(2d)}=\frac{2}{\lceil {\alpha }_1\rceil }.$$

This means that APP is not EC-$(t,1)$-WT for $t<1$.

Remark 6.

Obviously, for Example 2 with ${\gamma }_j=2^{-j}$ and ${\alpha }_j=2j$ for all $j\in \mathbb{N}$, APP is also EC-WT. Indeed, if APP is EC-$(t,1)$-WT for $t<1$, then it is EC-WT. Assume that APP is EC-$(t,1)$-WT for $t<1$, then we have

$$\underset{d+{\epsilon }^{-1}\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d^t+ln({\epsilon }^{-1})}=0.$$

Since

$$0\le \underset{d+{\epsilon }^{-1}\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d+ln({\epsilon }^{-1})}\le \underset{d+{\epsilon }^{-1}\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d^t+ln({\epsilon }^{-1})},$$

we further get

$$\underset{d+{\epsilon }^{-1}\to \infty }{lim}\frac{lnn(\epsilon ,{\mathit{APP}}_d)}{d+ln({\epsilon }^{-1})}=0,$$

which means that APP is EC-WT.

5. Conclusions

In this paper we discuss the EC-WT and EC-$(t,1)$-WT with $t<1$ for the approximation problem APP in weighted Hilbert spaces $H_{R_{d,\mathit{\alpha },\mathit{\gamma }}}$ for $R_{d,\mathit{\alpha },\mathit{\gamma }}\in \{r_{d,\mathit{\alpha },\mathit{\gamma }},{\psi }_{d,\mathit{\alpha },\mathit{\gamma }},{\omega }_{d,\mathit{\alpha },\mathit{\gamma }}\}$ with parameters $1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0$ and $1<{\alpha }_1\le {\alpha }_2\le \cdots$. We obtain the matching necessary and sufficient condition

$$\underset{j\to \infty }{lim}{\gamma }_j=0$$

on EC-WT, and the matching necessary and sufficient condition

$$\underset{j\to \infty }{lim}\frac{lnj}{ln({\gamma }_j^{-1})}=0$$

on EC-$(t,1)$-WT with $t<1$. The weights are used to model the importance of the functions from the weighted Hilbert spaces, so we will further research the other EC-tractability notions such as EC-SPT, EC-PT, EC-QWT, and EC-UWT.