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

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

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Submitted:

17 May 2024

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20 May 2024

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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 $\ga_j$ and $\az_j$ for $j=1,\,2,\dots,\,d$. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss the exponential convergence-(t,s)-weak tractability (EC-(t,s)-WT) of these L_2-approximation problems under the absolute or normalized error criterion. In particular, we obtain the sufficient and necessary conditions for EC-(1,1)-WT and EC-(t,1)-WT with t

Keywords:

Subject: Computer Science and Mathematics - Mathematics

MSC: 41A81; 47A58; 47B02

1. Introduction

We study multivariate approximation problems

APP = {{APP}_{d}}_{d \in N}

of functions defined over Hilbert spaces with large or huge d in the worst case setting. Such problems appear in statistics (see [1]), computational finance (see [2]) and physics (see [3]). We consider algorithms that use finitely many continuous linear functionals. The information complexity

n (ε, {APP}_{d})

is defined to be the minimal number of linear functionals for which the approximation error of some algorithm is at most

ε

. Tractability describes the growth rate of the information complexity

n (ε, {APP}_{d})

when the error threshold

ε

tends to 0 and the dimension d tends to infinity. There are two kinds of tractability, classical tractability based on polynomial convergence and exponential convergence-tractability (EC-tractability) based on exponential convergence. Recently many authors are interested in classical tractability and EC-tractability in weighted Hilbert spaces, such as classical tractability and EC-tractability in analytic Korobov spaces (see [4,5,6,7,8,9]), classical tractability and EC-tractability in weighted Korobov spaces (see [10,11,12,13,14,15,16,17]), and classical tractability in weighted Gaussian ANOVA spaces (see [18]).

This paper is devoted to discussing EC-tractability of

L_{2}

-approximation problems from the weighted Hilbert spaces in the worst case setting. Let

H (K_{R_{d, α, γ}})

be a Hilbert space with weight

R_{d, α, γ}

, where

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

are two positive sequences satisfying

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0

and

1 < α_{1} \leq α_{2} \leq \dots .

We consider the

L_{2}

-approximation problem

{APP}_{d} : H (K_{R_{d, α, γ}}) \to L_{2} ({[0, 1]}^{d}) with {APP}_{d} (f) = f .

In the worst case setting the classical tractability of the problem

APP = {{APP}_{d}}

in weighted Korobov spaces such as strong polynomial tractability and polynomial tractability were discussed in [13,14,18]; quasi-polynomial tractability, uniform weak tractability, weak tractability and

(t, s)

-weak tractability were investigated in [15,18]. Additionally, [18] also discussed classical tractability in several weighted Hilbert spaces including weighted Korobov spaces and weighted Gaussian ANOVA spaces. The EC-tractability of the problem

APP = {{APP}_{d}}

in weighted Korobov spaces such as EC-

(t, 1)

-weak tractability for

0 < t \leq 1

were studied in [17]. However, the above weighted Hilbert spaces

H (K_{R_{d, α, γ}})

with weights

R_{d, α, γ}

satisfy

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0

and

1 < α_{1} = α_{2} = \dots .

In this paper we present three cases of weighted Hilbert spaces

H (K_{R_{d, α, γ}})

with weights

R_{d, α, γ}

for

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0

and

1 < α_{1} \leq α_{2} \leq \dots

, that appear in the reference [16]. These weighted Hilbert spaces are similar but also different. [16] studied the strong polynomial tractability, polynomial tractability, weak tractability and

(t, s)

-weak tractability for

t > 1

of the problems

APP = {{APP}_{d}}

in these three weighted Hilbert spaces. However, EC-tractability have not yet been considered for the approximation problems

APP = {{APP}_{d}}

for the above three weighted Hilbert spaces. We will investigate EC-

(t, s)

-weak tractability for some

t > 0

s > 0

and get the sufficient and necessary conditions for the EC-

(1, 1)

-weak tractability (EC-weak tractability) and EC-

(t, 1)

-weak tractability with

t < 1

The paper is organized as follows. In Section 2 we present three cases of weighted Hilbert spaces. In Section 3 we give preliminaries about the

L_{2}

-approximation problem in the weight Hilbert space. Section 4.1 are devoted to recall some notions about the tractability such as classical tractability and exponential convergence-tractability and state out the main results. In Section 4.2 we give the proof of Theorem 10.

2. Weighted Reproducing Kernel Hilbert Spaces

In this section we introduce multivariate approximation problems in weighted reproducing kernel Hilbert spaces in the worst case setting.

In this paper, let

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

be two positive sequences of the reproducing kernel Hilbert space

H (K_{R_{d, α, γ}})

with weight

R_{d, α, γ}

satisfying

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0,

(1)

and

1 < α_{1} \leq α_{2} \leq \dots .

(2)

Assume that the weighted reproducing kernel function

K_{R_{d, α, γ}} : {[0, 1]}^{d} \times {[0, 1]}^{d} \to C

of the space

H (K_{R_{d, α, γ}})

is of product form

K_{R_{d, α, γ}} (x, y) : = \prod_{k = 1}^{d} K_{R_{1, α_{k}, γ_{k}}} (x_{k}, y_{k}),

where

K_{R_{1, α, γ}} : [0, 1] \times [0, 1] \to C

is a universal weighted function,

K_{R_{1, α, γ}} (x, y) : = \sum_{k \in N_{0}} R_{α, γ} (k) exp (2 π i k \cdot (x - y)), x, y \in [0, 1] .

Here, let weight

R_{α, γ} : N_{0} \to R^{+}

be a summable function, i.e.,

\sum_{k \in N_{0}} R_{α, γ} (k) < \infty

. Then we have

K_{R_{d, α, γ}} (x, y) = \sum_{k \in N_{0}^{d}} R_{d, α, γ} (k) exp (2 π i k \cdot (x - y)), x, y \in {[0, 1]}^{d},

(3)

and the inner product

{〈 f, g 〉}_{H (K_{R_{d, α, γ}})} = \sum_{k \in \in N_{0}^{d}} \frac{1}{R_{d, α, γ} (k)} \hat{f} (k) \bar{\hat{g}} (k)

(4)

and

{| | f | |}_{H (K_{R_{d, α, γ}})} = \sqrt{{〈 f, f 〉}_{H (K_{R_{d, α, γ}})}},

where

R_{d, α, γ} (k) : = \prod_{j = 1}^{d} R_{α_{j}, γ_{j}} (k_{j}), k = (k_{1}, k_{2}, \dots, k_{d}) \in N_{0}^{d},

x \cdot y : = \sum_{k = 1}^{d} x_{k} \cdot y_{k}, x = (x_{1}, x_{2}, \dots, x_{d}), y = (y_{1}, y_{2}, \dots, y_{d}) \in {[0, 1]}^{d},

and

\hat{f} (k) = \int_{{[0, 1]}^{d}} f (x) exp (- 2 π i k \cdot x) d x .

Note that

K_{R_{d, α, γ}} (x, y)

is well defined for

1 < α_{1} \leq α_{2} \leq \dots

and for all

x, y \in {[0, 1]}^{d}

, since

| K_{R_{d, α, γ}} (x, y) | \leq \sum_{k \in N_{0}^{d}} R_{d, α, γ} (k) = \prod_{j = 1}^{d} (\sum_{k \in N_{0}} R_{α_{j}, γ_{j}} (k)) < \infty .

γ_{1} = γ_{2} = \dots = 1

and

1 < α_{1} = α_{2} = \dots

, then the space

H (K_{R_{d, α, γ}})

is called unweighted space. Here,

N_{0} = {0, 1, \dots}

and

N = {1, 2, \dots}

There are many ways for introducing weighted reproducing kernel Hilbert spaces with weights

R_{d, α, γ}

. In this paper we consider three weights like the cases in the reference [16].

2.1. A Weighted Korobov Space

Let

α = {α_{j}}_{j \in N}

and

γ = {γ_{j}}_{j \in N}

satisfy (1) and (2). We consider a weighted Korobov space

H (K_{R_{d, α, γ}})

with weight

R_{d, α, γ} (k) = r_{d, α, γ} (k) : = \prod_{j = 1}^{d} r_{α_{j}, γ_{j}} (k_{j}),

where

r_{α, γ} (k) : = \{\begin{matrix} 1, for k = 0, \\ \frac{γ}{k^{⌈ α ⌉}}, for k \geq 1, \end{matrix}

for

α > 1

and

γ \in (0, 1]

. We can see the case in the references [16,19]. Then we have the kernel function (3) and the inner product (4) as follows:

K_{R_{d, α, γ}} (x, y) = K_{r_{d, α, γ}} (x, y) = \sum_{k \in N_{0}^{d}} r_{d, α, γ} (k) exp (2 π i k \cdot (x - y)), x, y \in {[0, 1]}^{d},

and

{〈 f, g 〉}_{H (K_{R_{d, α, γ}})} = {〈 f, g 〉}_{H (K_{r_{d, α, γ}})} = \sum_{k \in \in N_{0}^{d}} \frac{1}{r_{d, α, γ} (k)} \hat{f} (k) \bar{\hat{g}} (k) .

Remark 1.

Obviously, the kernel

K_{r_{d, α, γ}} (x, y)

is well defined for

α

and

γ

satisfying (1) and (2), due to

| K_{r_{d, α, γ}} (x, y) | \leq \sum_{k \in N_{0}^{d}} r_{d, α, γ} (k) = \prod_{j = 1}^{d} (1 + ζ (⌈ α_{j} ⌉) γ_{j}) < \infty,

where

ζ (\cdot)

is the Riemann zeta function.

2.2. A First Variant of the Weighted Korobov Space

Let

α = {α_{j}}_{j \in N}

and

γ = {γ_{j}}_{j \in N}

satisfy (1) and (2), respectively. We consider a first variant of the weighted Korobov space with weight

R_{d, α, γ} (k) = ψ_{d, α, γ} (k) : = \prod_{j = 1}^{d} ψ_{α_{j}, γ_{j}} (k_{j}),

where 1.5

ψ_{α, γ} (k) : = \{\begin{matrix} 1, & for k = 0, \\ \frac{γ}{k!}, & for 1 \leq k < ⌈ α ⌉, \\ \frac{γ (k - ⌈ α ⌉)!}{k!}, & for k \geq ⌈ α ⌉, \end{matrix}

for

α > 1

and

γ \in (0, 1]

Then we have the kernel function (3) and the inner product (4) as follows:

K_{R_{d, α, γ}} (x, y) = K_{ψ_{d, α, γ}} (x, y) = \sum_{k \in N_{0}^{d}} ψ_{d, α, γ} (k) exp (2 π i k \cdot (x - y)), x, y \in {[0, 1]}^{d},

and

{〈 f, g 〉}_{H (K_{R_{d, α, γ}})} = {〈 f, g 〉}_{H (K_{ψ_{d, α, γ}})} = \sum_{k \in \in N_{0}^{d}} \frac{1}{ψ_{d, α, γ} (k)} \hat{f} (k) \bar{\hat{g}} (k) .

Lemma 2.

([16] Lemma 2) For all

j, k \in N

we have

ψ_{α_{j}, γ_{j}} (k) \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k) .

Remark 3.

From Lemma 2 and

1 < α_{1} \leq α_{2} \leq \dots

we obtain

\begin{matrix} | K_{ψ_{d, α, γ}} (x, y) | & \leq \sum_{k \in N_{0}^{d}} ψ_{d, α, γ} (k) = \prod_{j = 1}^{d} (1 + \sum_{k \in N} ψ_{α_{j}, γ_{j}} (k)) \\ \leq \prod_{j = 1}^{d} (1 + \sum_{k \in N} {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k)) \\ = \prod_{j = 1}^{d} (1 + {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} ζ (⌈ α_{j} ⌉) γ_{j}) \\ < \infty . \end{matrix}

Hence the kernel

K_{ψ_{d, α, γ}} (x, y)

is well defined.

2.3. A Second Variant of the Weighted Korobov Space

Let

α = {α_{j}}_{j \in N}

and

γ = {γ_{j}}_{j \in N}

satisfy (1) and (2), respectively. We consider a second variant of the weighted Korobov space

H (K_{R_{d, α, γ}})

(see the references [16,20]) with weight

R_{d, α, γ} (k) = ω_{d, α, γ} (k) : = \prod_{j = 1}^{d} ω_{α_{j}, γ_{j}} (k_{j}),

where

ω_{α, γ} (k) : = {(1 + \frac{1}{γ} \sum_{l = 1}^{⌈ α ⌉} θ_{l} (k))}^{- 1},

for

α > 1

and

γ \in (0, 1]

, and 1.5

θ_{l} (k) : = \{\begin{matrix} \frac{k!}{(k - l)!}, & for k \geq l, \\ 0, & for 0 \leq k < l . \end{matrix}

Then we have the kernel function (3) and the inner product (4) as follows:

K_{R_{d, α, γ}} (x, y) = K_{ω_{d, α, γ}} (x, y) = \sum_{k \in N_{0}^{d}} ω_{d, α, γ} (k) exp (2 π i k \cdot (x - y)), x, y \in {[0, 1]}^{d},

and

{〈 f, g 〉}_{H (K_{R_{d, α, γ}})} = {〈 f, g 〉}_{H (K_{ω_{d, α, γ}})} = \sum_{k \in \in N_{0}^{d}} \frac{1}{ω_{d, α, γ} (k)} \hat{f} (k) \bar{\hat{g}} (k) .

Lemma 4.

([16] Lemma 3) For all

j, k \in N

we have

ω_{α_{j}, γ_{j}} (k) \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k) .

Remark 5.

We note that the kernel

K_{R_{d, α, γ}} (x, y)

is also well defined. Indeed, it follows from Lemma 4 and

1 < α_{1} \leq α_{2} \leq \dots

that

\begin{matrix} | K_{ω_{d, α, γ}} (x, y) | & \leq \sum_{k \in N_{0}^{d}} ω_{d, α, γ} (k) = \prod_{j = 1}^{d} (1 + \sum_{k \in N} ω_{α_{j}, γ_{j}} (k)) \\ \leq \prod_{j = 1}^{d} (1 + \sum_{k \in N} {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k)) \\ = \prod_{j = 1}^{d} (1 + {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} ζ (⌈ α_{j} ⌉) γ_{j}) \\ < \infty . \end{matrix}

Lemma 6.

Let

R_{α_{j}, γ_{j}} \in \{r_{α_{j}, γ_{j}}, ψ_{α_{j}, γ_{j}}, ω_{α_{j}, γ_{j}}\}

for all

j \in N

. Then we have for all

j \in N

k \in N_{0}

R_{α_{j}, γ_{j}} (k) \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k) .

Especially, we have for all

j \in N

k \in N_{0}

R_{α_{j}, γ_{j}} (k) \leq {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} r_{α_{1}, γ_{j}} (k) .

Proof.

On the one hand, it is obvious from Lemma 2 and Lemma 4 that

R_{α_{j}, γ_{j}} (k) \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k)

(5)

for all

j, k \in N

. Since for all

j \in N

r_{α_{j}, γ_{j}} (0) = ψ_{α_{j}, γ_{j}} (0) = ω_{α_{j}, γ_{j}} (0) = 1,

we have

R_{α_{j}, γ_{j}} (0) = 1 \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} = {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (0) .

Thus we have for all

j \in N

k \in N_{0}

that

R_{α_{j}, γ_{j}} (k) \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k) .

On the other hand, noting for all

j, k \in N

r_{α_{j}, γ_{j}} (k) \leq r_{α_{1}, γ_{j}} (k), ψ_{α_{j}, γ_{j}} (k) \leq ψ_{α_{1}, γ_{j}} (k), ω_{α_{j}, γ_{j}} (k) \leq ω_{α_{1}, γ_{j}} (k),

and for all

j \in N

r_{α_{j}, γ_{j}} (0) = r_{α_{1}, γ_{j}} (0) = 1, ψ_{α_{j}, γ_{j}} (0) = ψ_{α_{1}, γ_{j}} (0) = 1, ω_{α_{j}, γ_{j}} (0) = ω_{α_{1}, γ_{j}} (0) = 1,

we have for all

j \in N

k \in N_{0}

that

R_{α_{j}, γ_{j}} (k) \leq R_{α_{1}, γ_{j}} (k) .

Hence by (5) we further get for all

j \in N

k \in N_{0}

that

R_{α_{j}, γ_{j}} (k) \leq R_{α_{1}, γ_{j}} (k) \leq {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} r_{α_{1}, γ_{j}} (k) .

□

Remark 7.

Let

R_{α_{j}, γ_{j}} \in \{r_{α_{j}, γ_{j}}, ψ_{α_{j}, γ_{j}}, ω_{α_{j}, γ_{j}}\}

for all

j \in N

. Then we obtain

R_{α_{j}, γ_{j}} (0) = 1 and R_{α_{j}, γ_{j}} (1) \geq \frac{γ_{j}}{2}

(6)

for all

j \in N

. Indeed, for all

j \in N

we have

ψ_{α_{j}, γ_{j}} (0) = r_{α_{j}, γ_{j}} (0) = ω_{α_{j}, γ_{j}} (0) = 1,

which means

R_{α_{j}, γ_{j}} (0) = 1

. Due to for all

j \in N

, we get

ψ_{α_{j}, γ_{j}} (1) = r_{α_{j}, γ_{j}} (1) = γ_{j} and ω_{α_{j}, γ_{j}} (1) = {(1 + \frac{1}{γ_{j}})}^{- 1} \geq \frac{γ_{j}}{2},

which yields

R_{α_{j}, γ_{j}} (1) \geq \frac{γ_{j}}{2} .

3. $L_{2}$ -Approximation in the Weighted Hilbert Spaces

In this paper we investigate the

L_{2}

-approximation

{APP}_{d} : H (K_{R_{d, α, γ}}) \to L_{2} ({[0, 1]}^{d}) with {APP}_{d} (f) = f,

for all

f \in H (K_{R_{d, α, γ}})

in Hilbert space

H (K_{R_{d, α, γ}})

with weight

R_{d, α, γ} \in {r_{d, α, γ}, ψ_{d, α, γ}, ω_{d, α, γ}}

. It is well known from Remark 1, Remark 3, Remark 5 and [14] that this

L_{2}

-approximation is compact for

1 < α_{1} \leq α_{2} \leq \dots

. We approximation

{APP}_{d}

by algorithm

A_{n, d}

of the form

A_{n, d} (f) = \sum_{i = 1}^{n} T_{i} (f) g_{i}, for f \in H (K_{R_{d, α, γ}}),

(7)

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, α, γ}})

. The worst case error for the algorithm

A_{n, d}

of the form (7) is defined as

e (A_{n, d}) : = sup_{{| | f | |}_{H (K_{R_{d, α, γ}})} \leq 1} | | {APP}_{d} (f) - A_{n, d} (f) {| |}_{L_{2}} .

The nth minimal worst-case error, for

n \geq 1

, is defined by

e (n, {APP}_{d}) : = inf_{A_{n, d}} e (A_{n, d}),

where the infimum is taken over all linear algorithms of the form (7). For

n = 0

, we use

A_{0, d} = 0

. We call

e (0, {APP}_{d}) = sup_{{| | f | |}_{H (K_{R_{d, α, γ}})} \leq 1} | | {APP}_{d} (f) {| |}_{L_{2}}

the initial error of the problem

{APP}_{d}

We are interested in how the worst case error for the algorithm

A_{n, d}

depend on the number n and d. To this end, we define the so-called information complexity as

n (ε, {APP}_{d}) : = min {n \in N_{0} : e (n, {APP}_{d}) \leq ε},

where

ε \in (0, 1)

and

d \in N

. Here,

N_{0} = {0, 1, \dots}

and

N = {1, 2, \dots}

It is well known, see e.g., [2,10], that the nth minimal worst case errors

e (n, {APP}_{d})

and the information complexity

n (ε, {APP}_{d})

depend on the eigenvalues of the continuously linear operator

W_{d} = {APP}_{d}^{*} {APP}_{d} : H (K_{R_{d, α, γ}}) \to H (K_{R_{d, α, γ}})

. Let

(λ_{d, j}, η_{d, j})

be the eigenpairs of

W_{d}

, i.e.,

W_{d} η_{d, j} = λ_{d, j} η_{d, j}, for all j \in N,

where the eigenvalues

λ_{d, j}

are ordered,

λ_{d, 1} \geq λ_{d, 2} \geq \dots \geq 0,

and the eigenvectors

η_{d, j}

are orthonormal,

{〈 η_{d, i}, η_{d, j} 〉}_{H (K_{R_{d, α, γ}})} = δ_{i, j}, for all i, j \in N .

Then the nth minimal worst-case error,

n \geq 1

, is obtained for the algorithm

A_{n, d}^{⋄} f = \sum_{j = 1}^{n} {〈 f, η_{d, j} 〉}_{H (K_{R_{d, α, γ}})} η_{d, j}, for all n \in N .

and

e (n, {APP}_{d}) = e (A_{n, d}^{⋄}) = \sqrt{λ_{d, n + 1}}, for all n \in N .

The initial error

e (0, {APP}_{d}) = \sqrt{λ_{d, 1}}

. Hence we have

e (n, {APP}_{d}) = \sqrt{λ_{d, n + 1}}

for all

n \in N_{0}

. The information complexity is

\begin{matrix} n (ε, {APP}_{d}) & = min \{n \in N_{0} : \sqrt{λ_{d, n + 1}} \leq ε\} = min \{n \in N_{0} : λ_{d, n + 1} \leq ε^{2}\} . \end{matrix}

(8)

Since the eigenvalues

λ_{d, j}

with

j \in N

of the operator

W_{d}

are

R_{d, α, γ} (k)

with

k \in N_{0}^{d}

(see [10], p. 215), by (8) the information complexity of

{APP}_{d}

from the space

H (K_{R_{d, α, γ}})

is equal to

\begin{matrix} n (ε, {APP}_{d}) & = min \{n \in N_{0} : λ_{d, n + 1} \leq ε^{2}\} = |\{n \in N : λ_{d, n} > ε^{2}\}| \\ = |\{h \in N_{0}^{d} : R_{d, α, γ} (h) > ε^{2}\}| = |\{h \in N_{0}^{d} : \prod_{j = 1}^{d} R_{α_{j}, γ_{j}} (h_{j}) > ε^{2}\}|, \end{matrix}

(9)

with

ε \in (0, 1)

and

d \in N

Note that for the

L_{2}

-approximation

{APP}_{d}

from the space

H (K_{R_{d, α, γ}})

we do not need to distinguish between the absolute error criterion and the normalized error criterion since the initial error

e (0, {APP}_{d}) = \sqrt{λ_{d, 1}} = 1

4. Tractability in Weighted Hilbert Spaces and Main Results

In this section we will consider the classical tractability the exponential convergence-tractability (EC-tractability) for the problem

APP = {{APP}_{d}}_{d \in N}

in the weighted Hilbert space

H_{d, α, γ}

4.1. Tractability and Main Results

We focus on the behaviours of the information complexity

n (ε, {APP}_{d})

depending on the dimension d and the error threshold

ε

. Hence we will recall several notions of the classical tractability and exponential convergence-tractability (EC-tractability) notions (see [4,5,7,8,9,10,11,12,17,21]).

Definition 8.

Let

APP = {{APP}_{d}}_{d \in N}

. We say we have:

Strong polynomial tractability (SPT) if there exist non-negative numbers C and p such that

$n (ε, {APP}_{d}) \leq C {(ε^{- 1})}^{p} for all d \in N, ε \in (0, 1) .$

In this case we define the exponent $p^{str}$ of SPT as

$p^{str} : = inf {p : \exists C > 0 such that n (ε, {APP}_{d}) \leq C {(ε^{- 1})}^{p}, \forall d \in N, ε \in (0, 1)} .$
Polynomial tractability (PT) if there exist non-negative numbers C, p and q such that

$n (ε, {APP}_{d}) \leq C d^{q} {(ε^{- 1})}^{p} for all d \in N, ε \in (0, 1) .$
Quasi-polynomial tractability (QPT) if there exist two constants $C, t > 0$ such that

$n (ε, {APP}_{d}) \leq C exp (t (1 + ln ε^{- 1}) (1 + ln d)) for all d \in N, ε \in (0, 1) .$
Uniform weak tractability (UWT) if for all $t, s > 0$ ,

$lim_{ε^{- 1} + d \to \infty} \frac{ln n (ε, {APP}_{d})}{d^{t} + {(ε^{- 1})}^{s}} = 0 .$
Weak tractability (WT) if

$lim_{ε^{- 1} + d \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ε^{- 1}} = 0 .$
$(t, s)$ -weak tractability ( $(t, s)$ -WT) for fixed positive t and s if

$lim_{ε^{- 1} + d \to \infty} \frac{ln n (ε, {APP}_{d})}{d^{t} + {(ε^{- 1})}^{s}} = 0 .$
${APP}_{d}$ suffers from the curse of the dimensionality if there exist positive numbers $C_{1}$ , $C_{2}$ , $ε_{0}$ such that

$n (ε, {APP}_{d}) \geq C_{1} {(1 + C_{2})}^{d} for all 0 < ε \leq ε_{0} and infinitely many d \in N .$

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

SPT ⟹ PT ⟹ QPT ⟹ UWT ⟹ WT .

In the above definitions about classical tractability, if we replace

ε^{- 1}

(1 + ln (ε^{- 1}))

we will get the following definitions about exponential convergence-tractability (EC-tractability).

Definition 9.

Let

APP = {{APP}_{d}}_{d \in N}

. We say we have:

Exponential convergence-strong polynomial tractability (EC-SPT) if there exist non-negative numbers C and p such that

$n (ε, {APP}_{d}) \leq C {(1 + ln (ε^{- 1}))}^{p} for all d \in N, ε \in (0, 1) .$

In this case we define the exponent of EC-SPT as

$inf {p : \exists C > 0 such that n (ε, {APP}_{d}) \leq C {(1 + ln (ε^{- 1}))}^{p}, \forall d \in N, ε \in (0, 1)} .$
Exponential convergence-polynomial tractability (EC-PT) if there exist non-negative numbers C, p and q such that

$n (ε, {APP}_{d}) \leq C d^{q} {(1 + ln (ε^{- 1}))}^{p} for all d \in N, ε \in (0, 1) .$
Exponential convergence-uniform weak tractability (EC-UWT) if for all $t, s > 0$

$lim_{ε^{- 1} + d \to \infty} \frac{ln n (ε, {APP}_{d})}{d^{t} + {(1 + ln (ε^{- 1}))}^{s}} = 0 .$
Exponential convergence-weak tractability (EC-WT) if

$lim_{ε^{- 1} + d \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} = 0 .$
Exponential convergence- $(t, s)$ -weak tractability (EC- $(t, s)$ -WT) for fixed positive t and s if

$lim_{ε^{- 1} + d \to \infty} \frac{ln n (ε, {APP}_{d})}{d^{t} + {(1 + ln (ε^{- 1}))}^{s}} = 0 .$

We note that EC-

(1, 1)

-WT is the same as EC-WT, and

EC - SPT ⟹ EC - PT ⟹ EC - QPT ⟹ EC - UWT ⟹ EC - WT .

If the problem

APP

has exponential convergence-tractability, then it has classical tractability. Hence we have

EC - SPT ⟹ SPT, EC - PT ⟹ PT, EC - QPT ⟹ QPT,

EC - (t, s) - WT ⟹ (t, s) - WT, EC - UWT ⟹ UWT, EC - WT ⟹ WT .

In the worst case setting the classical tractability and EC-tractability of the problem

APP = {{APP}_{d}}_{d \in N}

in the weighted Hilbert space

H_{d, α, γ}

with

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

satisfying

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0,

and

1 < α^{*} = α_{1} = α_{2} = \dots .

have been solved by [13,15,16] and [17], respectively. The following results have been obtained:

For $R_{d, α^{*}, γ} \in {r_{d, α^{*}, γ}, ψ_{d, α^{*}, γ}, ω_{d, α^{*}, γ}}$ , PT holds iff SPT holds iff

$s_{γ} : = inf \{κ > 0 : \sum_{j = 1}^{\infty} γ_{j}^{κ} < \infty\} < \infty,$

and the exponent of SPT is

$p^{str} = 2 max (s_{γ}, \frac{1}{α}) .$
For $R_{d, α^{*}, γ} = r_{d, α^{*}, γ}$ , QPT, UWT and WT are equivalent and hold iff

$γ_{I} : = inf_{j \in N} γ_{j} < 1 .$

For $R_{d, α^{*}, γ} \in {ψ_{d, α^{*}, γ}, ω_{d, α^{*}, γ}}$ ,

$γ_{I} < \infty$

implies QPT.
For $R_{d, α^{*}, γ} \in {r_{d, α^{*}, γ}, ψ_{d, α^{*}, γ}, ω_{d, α^{*}, γ}}$ and $t > 1$ , $(t, s)$ -WT holds for all $1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0$ .
For $R_{d, α^{*}, γ} = r_{d, α^{*}, γ}$ , EC-WT holds iff

$lim_{j \to \infty} γ_{j} = 0 .$
For $R_{d, α^{*}, γ} = r_{d, α^{*}, γ}$ and $t < 1$ , EC- $(t, 1)$ -WT holds iff

$lim_{j \to \infty} \frac{ln j}{ln (γ_{j}^{- 1})} = 0 .$

In the worst case setting the classical tractability such as SPT, PT and WT of the problem

APP = {{APP}_{d}}_{d \in N}

in the weighted Hilbert space

H_{d, α, γ}

with

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

satisfying (1) and (2), i.e.,

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0,

and

1 < α_{1} \leq α_{2} \leq \dots .

has been solved by [16] as follows:

For $R_{d, α, γ} \in {r_{d, α, γ}, ψ_{d, α, γ}, ω_{d, α, γ}}$ , SPT and PT are equivalent and hold iff

$δ : = \underset{j \to \infty}{lim inf} \frac{ln γ_{j}^{- 1}}{ln j} > 0 .$

The exponent of SPT is

$p^{str} = 2 max \{\frac{1}{δ}, \frac{1}{⌈ α_{1} ⌉}\} .$
For $R_{d, α, γ} = r_{d, α, γ}$ , WT holds iff

$lim_{j \to \infty} γ_{j} < 1 .$
For $R_{d, α, γ} \in {r_{d, α, γ}, ψ_{d, α, γ}, ω_{d, α, γ}}$ and $t > 1$ , $(t, s)$ -WT holds.

In this paper, we investigate the EC-tractability of the problem

APP = {{APP}_{d}}_{d \in N}

in the weighted Hilbert space

H_{d, α, γ}

with

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

satisfying (1) and (2). We obtain sufficient and necessary conditions for EC-

(t, 1)

-WT with

0 < t < 1

and

t = 1

Theorem 10.

Let the sequences

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

satisfy (1) and (2). Then the problem

APP = {{APP}_{d}}_{d \in N}

for the weighted Hilbert spaces

H_{R_{d, α, γ}}

with

R_{d, α, γ} \in {r_{d, α, γ}, ψ_{d, α, γ}, ω_{d, α, γ}}

(1) is EC-WT, if and only if

lim_{j \to \infty} γ_{j} = 0 .

(2) is EC-

(t, 1)

-WT with

t < 1

, if and only if

lim_{j \to \infty} \frac{ln j}{ln (γ_{j}^{- 1})} = 0 .

4.2. The Proof

In order to prove Theorem 10 we need the following Lemmas.

Lemma 11.

Let

η > 0

ε \in (0, 1)

. We have for any

d \in N

n (ε, {APP}_{d}) \leq ε^{- 2 η} \prod_{j = 1}^{d} (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}) .

Proof.

By Lemma 6 we have

\begin{matrix} \sum_{k = 1}^{\infty} λ_{d, k}^{η} & = \sum_{k \in N_{0}^{d}} {(R_{d, α, γ} (k))}^{η} = \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{η}) \\ \leq \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {({⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} r_{α_{1}, γ_{j}} (k))}^{η}) \\ = \prod_{j = 1}^{d} (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} \sum_{k = 1}^{\infty} {(r_{α_{1}, γ_{j}} (k))}^{η}) \\ = \prod_{j = 1}^{d} (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} \sum_{k = 1}^{\infty} {(\frac{γ_{j}}{k^{⌈ α_{1} ⌉}})}^{η}) \\ = \prod_{j = 1}^{d} (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}) . \end{matrix}

This yields

n λ_{d, n}^{η} \leq \sum_{k = 1}^{n} λ_{d, k}^{η} \leq \sum_{k = 1}^{\infty} λ_{d, k}^{η} \leq \prod_{j = 1}^{d} (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}),

which means

λ_{d, n} \leq \frac{\prod_{j = 1}^{d} {(1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η})}^{1 / η}}{n^{1 / η}} .

It follows from the above inequality and (8)

n (ε, {APP}_{d}) = min \{n \in N_{0} : λ_{d, n + 1} \leq ε^{2}\},

that

n (ε, {APP}_{d}) \leq ε^{- 2 η} \prod_{j = 1}^{d} (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}) .

This proof is complete. □

Lemma 12.

Let

ε \in (0, 1)

. We have for any

d \geq 2

n (ε, {APP}_{d}) \geq ⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉ .

Proof.

Set

H = H (ε, d, α) : = \{h \in N_{0} : h \leq ⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉ - 1\} .

h > ⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉ - 1

and

d \geq 2

, by Lemma 6 we have

\prod_{j = 1}^{d - 1} R_{α_{j}, γ_{j}} (h_{j}) R_{α_{d}, γ_{d}} (h) \leq R_{α_{d}, γ_{d}} (h) \leq {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} r_{α_{1}, γ_{d}} (h) = {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} \frac{γ_{d}}{h^{⌈ α_{1} ⌉}} \leq ε^{2}

for any

{h_{1}, \dots, h_{d - 1}} \in N_{0}^{d - 1}

, which means

\{h \in N_{0}^{d - 1} : \prod_{j = 1}^{d - 1} R_{α_{j}, γ_{j}} (h_{j}) R_{α_{d}, γ_{d}} (h) > ε^{2}\} = ⌀

(10)

for all

h > ⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉ - 1

. It follows from (9) and (10) that

\begin{matrix} n (ε, {APP}_{d}) & = |\{h \in N_{0}^{d} : \prod_{j = 1}^{d} R_{α_{j}, γ_{j}} (h_{j}) > ε^{2}\}| \\ = |\{h \in N_{0}^{d} : \prod_{j = 1}^{d - 1} R_{α_{j}, γ_{j}} (h_{j}) R_{α_{d}, γ_{d}} (h_{d}) > ε^{2}\}| \\ = \sum_{h \in N_{0}} |\{h \in N_{0}^{d - 1} : \prod_{j = 1}^{d - 1} R_{α_{j}, γ_{j}} (h_{j}) R_{α_{d}, γ_{d}} (h) > ε^{2}\}| \\ = \sum_{h \in H} |\{h \in N_{0}^{d - 1} : \prod_{j = 1}^{d - 1} R_{α_{j}, γ_{j}} (h_{j}) R_{α_{d}, γ_{d}} (h) > ε^{2}\}| \\ = \sum_{h \in (H ∖ {0})} |\{h \in N_{0}^{d - 1} : \prod_{j = 1}^{d - 1} R_{α_{j}, γ_{j}} (h_{j}) > ε^{2} R_{α_{d}, γ_{d}}^{- 1} (h)\}| \\ + |\{h \in N_{0}^{d - 1} : \prod_{j = 1}^{d - 1} R_{α_{j}, γ_{j}} (h_{j}) > ε^{2}\}| \\ = \sum_{h \in (H ∖ {0})} n (ε R_{α_{d}, γ_{d}}^{- 1 / 2} (h), {APP}_{d - 1}) + n (ε, {APP}_{d - 1}) \\ = \sum_{h = 1}^{⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉ - 1} n (ε R_{α_{d}, γ_{d}}^{- 1 / 2} (h), {APP}_{d - 1}) + n (ε, {APP}_{d - 1}) \\ \geq ⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉ . \end{matrix}

This finishes the proof. □

Lemma 13.

For

\prod_{j = 1}^{d} (\frac{γ_{j}}{2}) > ε^{2}

and

ε \in (0, 1)

we have

n (ε, {APP}_{d}) \geq 2^{d} .

Proof.

Set

A (ε, d) = \{h \in N_{0}^{d} : \prod_{j = 1}^{d} R_{α_{j}, γ_{j}} (h_{j}) > ε^{2}\} .

h = {h_{1}, h_{2}, \dots, h_{d}} \in {0, 1}^{d}

, we have from (6) that

R_{d, α, γ} (h) = \prod_{j = 1}^{d} R_{α_{j}, γ_{j}} (h_{j}) \geq \prod_{j = 1}^{d} (\frac{γ_{j}}{2}) .

Thus, we have

{0, 1}^{d} \in A (ε, d)

for

\prod_{j = 1}^{d} (\frac{γ_{j}}{2}) > ε^{2}

. Hence it follows from (9) that

\begin{matrix} n (ε, {APP}_{d}) & = | A (ε, d) | \geq |\{h \in {0, 1}^{d} : \prod_{j = 1}^{d} R_{α_{j}, γ_{j}} (h_{j}) > ε^{2}\}| = 2^{d} \end{matrix}

for

\prod_{j = 1}^{d} (\frac{γ_{j}}{2}) > ε^{2}

. This proof is complete. □

Proof of Theorem 10.

If there are infinitely many

γ_{j} = 0

for

j \in N

, the results are obviously true. Without lose of generality we consider only the situation when all the

γ_{j}

for

j \in N

are positive. (1) Let

δ > 0

and take

ε = \prod_{j = 1}^{d} {(\frac{γ_{j}}{2})}^{\frac{1 + δ}{2}}

, then we have

\prod_{j = 1}^{d} (\frac{γ_{j}}{2}) > ε^{2} .

It follows from Lemma 13 that

\begin{matrix} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} & \geq \frac{d ln 2}{d + \frac{1 + δ}{2} \cdot ln (\prod_{j = 1}^{d} (2 γ_{j}^{- 1}))} \\ \geq \frac{d ln 2}{d + \frac{1 + δ}{2} \cdot d \cdot ln (2 γ_{d}^{- 1})} \\ = \frac{ln 2}{1 + \frac{1 + δ}{2} \cdot (ln 2 + ln (γ_{d}^{- 1}))} . \end{matrix}

(11)

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

ε

lim_{d \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} = 0 .

Combing (11) and the above equality we have

0 = lim_{d \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} \geq lim_{d \to \infty} \frac{ln 2}{1 + \frac{1 + δ}{2} \cdot (ln 2 + ln (γ_{d}^{- 1}))} .

This implies

lim_{d \to \infty} γ_{d} = 0

On the other hand, assume that we have

lim_{d \to \infty} γ_{d} = 0

. For

η > 0

we obtain from the upper bound in Lemma 11 that

\begin{matrix} \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} & \leq \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} ln (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η})}{d + ln (ε^{- 1})} \\ \leq \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}}{d + ln (ε^{- 1})} \\ \leq 2 η + \underset{d \to \infty}{lim sup} \frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \sum_{j = 1}^{d} γ_{j}^{η}}{d} \\ = 2 η, \end{matrix}

where we used that

ln (1 + x) \leq x

for all

x \geq 0

and

\underset{d \to \infty}{lim sup} \frac{\sum_{j = 1}^{d} γ_{j}^{η}}{d} = 0

lim_{d \to \infty} γ_{d} = 0

. Setting

η \to 0

, we have

\underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} = 0,

which yields that ET-WT holds.

(2) Assume that EC-

(t, 1)

-WT for

t < 1

holds. First, we note that

lim_{d \to \infty} γ_{d} = 0

. Indeed, if

lim_{d \to \infty} γ_{d} \neq 0

, we deduce from Theorem 10 (1) that EC-WT doesn’t hold, i.e.,

0 < \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} \leq \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} .

This means APP is not EC-

(t, 1)

-WT for all

t < 1

Next, we want to prove

lim_{j \to \infty} \frac{ln j}{ln (γ_{j}^{- 1})} = 0

. Let

ε = ε_{d} \in (0, 1)

such that

ln {(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}} = d^{t}

for large

d \in N

. From the lower bound in Lemma 12 we obtain

\begin{matrix} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} & \geq \frac{ln ⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉}{d^{t} + ln (ε^{- 1})} \geq \frac{ln {(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}}{d^{t} + ln (ε^{- 1})} \\ = \frac{d^{t}}{d^{t} + ln (ε^{- 1})} = \frac{d^{t}}{d^{t} + ⌈ α_{1} ⌉ d^{t} / 2 + ln (γ_{d}^{- 1}) / 2 - ⌈ α_{1} ⌉ (ln ⌈ α_{1} ⌉) / 2} \\ = \frac{1}{1 + ⌈ α_{1} ⌉ / 2 + ln (γ_{d}^{- 1}) / (2 d^{t}) - ⌈ α_{1} ⌉ (ln ⌈ α_{1} ⌉) / (2 d^{t})} . \end{matrix}

It follows from the assumption that

\begin{matrix} 0 = \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} & \geq \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{1}{1 + ⌈ α_{1} ⌉ / 2 + ln (γ_{d}^{- 1}) / (2 d^{t}) - ⌈ α_{1} ⌉ (ln ⌈ α_{1} ⌉) / (2 d^{t})} \\ = \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{1}{1 + ⌈ α_{1} ⌉ / 2 + ln (γ_{d}^{- 1}) / (2 d^{t})}, \end{matrix}

which implies

lim_{d \to \infty} \frac{d^{t}}{ln (γ_{d}^{- 1})} = 0 .

Using the fact that

d^{t} \geq ln d^{t} = t ln d \geq 0

for large

d \in N

, we have

0 \leq lim_{d \to \infty} \frac{ln d}{ln (γ_{d}^{- 1})} \leq lim_{d \to \infty} \frac{d^{t}}{t ln (γ_{d}^{- 1})} = 0,

i.e.,

lim_{d \to \infty} \frac{ln d}{ln (γ_{d}^{- 1})} = 0 .

On the other hand, assume that

lim_{j \to \infty} \frac{ln j}{ln (γ_{j}^{- 1})} = 0

. Then we obtain that for all

δ > 0

there is a number

N_{δ} > 0

such that

\begin{matrix} γ_{j} \leq j^{- δ} for all j \geq N_{δ} . \end{matrix}

(12)

Let

η > 0

. We get from Lemma 11 that

\begin{matrix} ln n (ε, {APP}_{d}) & \leq 2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} ln (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}) \\ \leq 2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}, \end{matrix}

(13)

where we used that

ln (1 + x) \leq x

for all

x > 0

. Choose

δ = \frac{2}{η}

. By (12) and (13) we get

\begin{matrix} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} & \leq \frac{2 η ln (ε^{- 1}) + \sum_{j = 1}^{N_{2 / η} - 1} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η} + \sum_{j = N_{2 / η}}^{max {d, N_{2 / η}}} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}}{d^{t} + ln (ε^{- 1})} \\ \leq 2 η + \frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{1}^{η} (N_{2 / η} - 1) + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \sum_{j = N_{2 / η}}^{max {d, N_{2 / η}}} j^{- 2}}{d^{t} + ln (ε^{- 1})} . \end{matrix}

It follows that

\begin{matrix} \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} & \leq 2 η + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{\sum_{j = N_{2 / η}}^{max {d, N_{2 / η}}} j^{- 2}}{d^{t} + ln (ε^{- 1})} \\ \leq 2 η + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{\sum_{j = 1}^{\infty} j^{- 2}}{d^{t} + ln (ε^{- 1})} \\ = 2 η . \end{matrix}

Setting

η \to 0

, we have

\underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} = 0 .

Therefore Theorem 10 is proved. □

Example 14.

An example for EC-WT.

Assume that

γ_{j} = j^{- 2}

and

α_{j} = j + 1

for all

j \in N

. We consider the above weighted Hilbert spaces

H_{R_{d, α, γ}}

R_{d, α, γ} \in {r_{d, α, γ}, ψ_{d, α, γ}, ω_{d, α, γ}}

Obviously, we have

lim_{j \to \infty} γ_{j} = 0

. By Lemma 11 we get

\begin{matrix} \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} & \leq \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} ln (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η})}{d + ln (ε^{- 1})} \\ \leq \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}}{d + ln (ε^{- 1})} \\ \leq 2 η + \underset{d \to \infty}{lim sup} \frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \sum_{j = 1}^{d} γ_{j}^{η}}{d} \\ = 2 η + \underset{d \to \infty}{lim sup} \frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \sum_{j = 1}^{d} j^{- 2 η}}{d} \\ = 2 η, \end{matrix}

where we used

ln (1 + x) \leq x

for all

x \geq 0

and

\underset{d \to \infty}{lim sup} \frac{\sum_{j = 1}^{d} j^{- 2 η}}{d} = 0

. Setting

η \to 0

, we have

\underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} = 0 .

Hence APP is EC-WT.

Example 15.

An example for EC-

(t, 1)

-WT for

t < 1

Assume that

γ_{j} = 2^{- j}

and

α_{j} = 2 j

for all

j \in N

. We consider the above weighted Hilbert spaces

H_{R_{d, α, γ}}

R_{d, α, γ} \in {r_{d, α, γ}, ψ_{d, α, γ}, ω_{d, α, γ}}

Note that

lim_{j \to \infty} \frac{ln j}{ln (γ_{j}^{- 1})} = lim_{j \to \infty} \frac{ln j}{j ln 2} = 0

. It follows from Lemma 11 that

\begin{matrix} ln n (ε, {APP}_{d}) & \leq 2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} ln (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) γ_{j}^{η}) \\ = 2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} ln (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) 2^{- η j}) \\ \leq 2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) 2^{- η j}, \end{matrix}

where we used that

ln (1 + x) \leq x

for all

x > 0

. It yields that

\begin{matrix} \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} & \leq \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{2 η ln (ε^{- 1}) + \sum_{j = 1}^{d} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) 2^{- η j}}{d^{t} + ln (ε^{- 1})} \\ \leq 2 η + \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \sum_{j = 1}^{d} 2^{- η j}}{d^{t} + ln (ε^{- 1})} \\ \leq 2 η + \underset{d + ε^{- 1} \to \infty}{lim sup} \frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ η} ζ (⌈ α_{1} ⌉ η) \sum_{j = 1}^{\infty} 2^{- η j}}{d^{t} + ln (ε^{- 1})} \\ = 2 η . \end{matrix}

Setting

η \to 0

, we have

\underset{d + ε^{- 1} \to \infty}{lim sup} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} = 0 .

Hence APP is EC-

(t, 1)

-WT for

t < 1

Remark 16.

We note that for Example 14 with

γ_{j} = j^{- 2}

and

α_{j} = j + 1

for all

j \in N

, APP is EC-WT, but not EC-

(t, 1)

-WT for

t < 1

. Indeed, let

ε = ε_{d} \in (0, 1)

such that

ln {(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}} = d, i . e ., ε^{- 1} = \frac{d e^{d ⌈ α_{1} ⌉ / 2}}{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ / 2}},

for large

d \in N

. From Lemma 12 we have

\begin{matrix} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} & \geq \frac{ln ⌈{(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}⌉}{d^{t} + ln (ε^{- 1})} \geq \frac{ln {(ε^{- 2} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} γ_{d})}^{\frac{1}{⌈ α_{1} ⌉}}}{d^{t} + ln (ε^{- 1})} \\ = \frac{d}{d^{t} + ln (ε^{- 1})} = \frac{d}{d^{t} + ⌈ α_{1} ⌉ d / 2 + ln d - ⌈ α_{1} ⌉ (ln ⌈ α_{1} ⌉) / 2} \\ = \frac{1}{d^{t - 1} + ⌈ α_{1} ⌉ / 2 + ln d / d - ⌈ α_{1} ⌉ (ln ⌈ α_{1} ⌉) / (2 d)} . \end{matrix}

For the above fixed ε and

t < 1

we obtain

lim_{d \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} \geq lim_{d \to \infty} \frac{1}{d^{t - 1} + ⌈ α_{1} ⌉ / 2 + ln d / d - ⌈ α_{1} ⌉ (ln ⌈ α_{1} ⌉) / (2 d)} = \frac{2}{⌈ α_{1} ⌉} .

This means that APP is not EC-

(t, 1)

-WT for

t < 1

Remark 17.

Obviously, for Example 15 with

γ_{j} = 2^{- j}

and

α_{j} = 2 j

for all

j \in 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

lim_{d + ε^{- 1} \to \infty} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})} = 0 .

Since

0 \leq lim_{d + ε^{- 1} \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} \leq lim_{d + ε^{- 1} \to \infty} \frac{ln n (ε, {APP}_{d})}{d^{t} + ln (ε^{- 1})},

we further get

lim_{d + ε^{- 1} \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ln (ε^{- 1})} = 0,

which means that APP is EC-WT.

In this paper we discuss the EC-WT and EC-

(t, 1)

-WT with

t < 1

for the problem APP in weighted Hilbert spaces

H_{R_{d, α, γ}}

for

R_{d, α, γ} \in {r_{d, α, γ}, ψ_{d, α, γ}, ω_{d, α, γ}}

with parameters

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0

and

1 < α_{1} \leq α_{2} \leq \dots

. We obtain the matching necessary and sufficient condition

lim_{j \to \infty} γ_{j} = 0

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

lim_{j \to \infty} \frac{ln j}{ln (γ_{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 Hiblert spaces, so we plan to further study the other EC-tractability notions such as EC-UWT and EC-QWT.

Author Contributions

Conceptualization, Chen, J. and Yan, H.; methodology, Chen, J. and Yan, H.; validation, Chen, J.; formal analysis, Chen, J.; investigation, Yan, H.; resources, Yan, H.; data curation, Yan, H.; writing—original draft preparation, Chen, J.; writing—review and editing, Chen, J.; visualization, Chen, J.; supervision, Chen, J. and Yan, H.; project administration, Chen, J. and Yan, H. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by National Natural Science Foundation of China of found grant number 12001342, Scientific and Technological Innovation Project of Colleges and Universities in Shanxi Province of found grant number 2022L438, Basic Youth Research Found Project of Shanxi Datong University of found grant number 2022Q10, and Doctoral Foundation Project of Shanxi Datong University of found grant number 2019-B-10 and found grant number 2021-B-17.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The authors would like to thank all those for important and very useful comments on this paper.

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

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

Abstract

1. Introduction

2. Weighted Reproducing Kernel Hilbert Spaces

2.1. A Weighted Korobov Space

2.2. A First Variant of the Weighted Korobov Space

2.3. A Second Variant of the Weighted Korobov Space

3. L 2 -Approximation in the Weighted Hilbert Spaces

4. Tractability in Weighted Hilbert Spaces and Main Results

4.1. Tractability and Main Results

4.2. The Proof

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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3. $L_{2}$ -Approximation in the Weighted Hilbert Spaces