Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

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Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

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Submitted:

28 December 2023

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02 January 2024

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Abstract

We investigate \texorpdfstring{$L_2$}{}-approximation problems in the worst case setting in the weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} with weights \texorpdfstring{$R_{d,\a,{\bm \ga}}$}{} under parameters \texorpdfstring{$1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $1<\az_1\le \az_2\le \cdots$}{}. Several interesting weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. Multivariate approximation; information complexity; tractability; weighted Hilbert spacesWe discuss tractability of \texorpdfstring{$L_2$}{}-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on \texorpdfstring{$d$}{} and \texorpdfstring{$\va^{-1}$}{}. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and \texorpdfstring{$(t_1,t_2)$}{}-weak tractability (\texorpdfstring{$(t_1,t_2)$}{}-WT) for all \texorpdfstring{$t_1>1$}{} and \texorpdfstring{$t_2>0$}{} in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

We investigate multivariate approximation problems

S_{d}

with large or even huge d. Examples include these problems in statistics, computational finance and physics. In order to solve these problems we usually consider algorithms using finitely many evaluations of arbitrary continuous linear functionals. We use either the absolute error criterion (ABS) or the normalized error criterion (NOR). For

X \in {ABS, NOR}

we define the information complexity

n^{X} (ε, S_{d})

to be the minimal number of linear functionals which are needed to find an algorithm whose worst case error is at most

ε

. The behavior of the information complexity

n^{X} (ε, S_{d})

is the major concern when the accuracy

ε

of approximation goes to zero and the number d of variables goes to infinity. For small

ε

and large d, tractability is aimed at studying how the information complexity

n^{X} (ε, S_{d})

behaves as a function of d and

ε^{- 1}

, while the exponential convergence-tractability (EC-tractability) is aimed at studying how the information complexity

n^{X} (ε, S_{d})

behaves as a function of d and

(1 + ln (ε^{- 1}))

. Recently the study of tractability and EC-tractability in the worst case setting has attracted much interest in analytic Korobov spaces, weighted Korobov spaces and weighted Gaussian ANOVA spaces; see [1,2,3,4,5,6,7,8,9,10,11,12] and the references therein.

Weighted multivariate approximation of functions on space

{[0, 1]}^{d}

are studied in many problems. We are interested in weighted Hilbert spaces of functions in this paper. We present three examples of weighted Hilbert spaces, which are similar but also different. We devote to discussing worst case tractability of

L_{2}

-approximation problem

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

with

{APP}_{d} (f) = f

for all

f \in H (K_{R_{d, α, γ}})

in weighted Hilbert spaces

H (K_{R_{d, α, γ}})

with three weights

R_{d, α, γ}

under positive parameter sequences

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

. The tractability and EC-tractability of such problem APP in weighted Korobov spaces with parameters

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

and

1 < α_{1} = α_{2} = \dots

were discussed in [2,4,6,11] and in [13], respectively. Additionally, [4] considered the tractability of the

L_{2}

-approximation in several weighted Hilbert spaces for permissible information class consisting of arbitrary continuous linear functionals and consisting of functions evaluations.

In this paper we study SPT, PT, WT and

(t_{1}, t_{2})

-WT for all

t_{1} > 1

and

t_{2} > 0

of the above problem APP with parameters

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

and

1 < α_{1} \leq α_{2} \leq \dots

for the ABS or the NOR under the information class consisting of arbitrary continuous linear functionals. Especially, we get a compete sufficient and necessary condition for SPT, PT and WT, respectively, and the exponent of SPT.

The paper is organized as follows. In Section 2 we give preliminaries about multivariate approximation problems in Hilbert spaces for information class consisting of arbitrary continuous linear functionals in the worst case setting, and definitions of tractability. In Section 3 we present several examples of weighted Hilbert spaces and study some facts and relations between them. In Section 4 we discuss the tractability properties of

L_{2}

-approximation problems in the above weighted Hilbert spaces, then state out main result Theorem 4.1.

2. Approximation and tractability in Hilbert spaces

2.1. Approximation in Hilbert spaces

Let

F_{d}

and

G_{d}

be two sequences of Hilbert spaces. Consider a sequence of compact linear operators

S_{d} : F_{d} \to G_{d}

for all

d \in N

. We approximation

S_{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 S_{d},

(2.1)

where functions

g_{i} \in G_{d}

and continuous linear functionals

T_{i} \in F_{d}^{*}

for

i = 1, \dots, n

. The worst case error for the algorithm

A_{n, d}

of the form (2.1) is defined as

e (A_{n, d}) : = sup_{f \in F_{d} {, | | f | |}_{F_{d}} \leq 1} | | S_{d} (f) - A_{n, d} (f) {| |}_{G_{d}} .

The nth minimal worst-case error, for

n \geq 1

, is defined by

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

where the infimum is taken over all linear algorithms of the form (2.1). For

n = 0

, we use

A_{0, d} = 0

. We call

e (0, S_{d}) = sup_{f \in F_{d} {, | | f | |}_{F_{d}} \leq 1} | | S_{d} (f) {| |}_{G_{d}}

the initial error of the problem

S_{d}

The information complexity for

S_{d}

can be studied using either the absolute error criterion (ABS), or the normalized error criterion (NOR). The information complexity

n^{X} (ε, S_{d})

for

X \in {ABS, NOR}

is defined by

n^{X} (ε, S_{d}) : = min {n \in N_{0} : e (n, S_{d}) \leq ε {CRI}_{d}},

where

{CRI}_{d} : = \{\begin{matrix} 1, for X = ABS, \\ e (0, S_{d}), for X = NOR . \end{matrix}

Here,

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

and

N = {1, 2, \dots}

It is well known, see e.g., [7,14], that the nth minimal worst case errors

e (n, S_{d})

and the information complexity

n^{X} (ε, S_{d})

depend on the eigenvalues of the continuously linear operator

W_{d} = S_{d}^{*} S_{d} : F_{d} \to F_{d}

. 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} 〉}_{F_{d}} = δ_{i, j} for all i, j \in N .

Then the nth minimal error is obtained for the algorithm

A_{n, d}^{⋄} f = \sum_{j = 1}^{n} {〈 f, η_{d, j} 〉}_{F_{d}} η_{d, j} for all f \in F_{d},

and

e (n, S_{d}) = e (A_{n, d}^{⋄}) = \sqrt{λ_{d, n + 1}} for all n \in N_{0} .

Hence the information complexity is equal to

\begin{matrix} n^{X} (ε, S_{d}) & = min {n \in N_{0} : \sqrt{λ_{d, n + 1}} \leq ε {CRI}_{d}} \\ = min {n \in N_{0} : λ_{d, n + 1} \leq ε^{2} {CRI}_{d}^{2}} \\ = | {n \in N : λ_{d, n} > ε^{2} {CRI}_{d}^{2}} |, \end{matrix}

(2.2)

with

ε \in (0, 1)

and

d \in N

. We focus on the rate of the information complexity when the error threshold

ε

tends to 0 and the problem dimension d grows to infinity.

2.2. Tractability

In order to characterize the dependency of the information complexity

n^{X} (ε, S_{d})

for the absolute error criterion and the normalized error criterion on the dimension d and the error threshold

ε

, we will briefly recall some of the basic tractability and exponential convergence-tractability (EC-tractability) notions.

Let

S = {S_{d}}_{d \in N}

. For

X \in {ABS, NOR}

, we say S is

• strongly polynomially tractable (SPT) iff there exist non-negative numbers C and p such that for all

d \in N

ε \in (0, 1)

n^{X} (ε, S_{d}) \leq C {(ε^{- 1})}^{p} .

The exponent

p^{str}

of SPT is defined to be the infimum of all p for which the above inequality holds.

• polynomially tractable (PT) iff there exist non-negative numbers C, p and q such that for all

d \in N

ε \in (0, 1)

n^{X} (ε, S_{d}) \leq C d^{q} {(ε^{- 1})}^{p} .

• quasi-polynomially tractable (QPT) iff there exist two constants

C, t > 0

such that for all

d \in N, ε \in (0, 1)

n^{X} (ε, S_{d}) \leq C exp (t (1 + ln ε^{- 1}) (1 + ln d)) .

The exponent

t^{pol}

of QPT is defined to be the infimum of all t for which the above inequality holds.

• uniformly weakly tractable (UWT) iff for all

t_{1}, t_{2} > 0

lim_{ε^{- 1} + d \to \infty} \frac{ln n^{X} (ε, S_{d})}{d^{t_{1}} + {(ε^{- 1})}^{t_{2}}} = 0;

• weakly tractable (WT) iff

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

•

(t_{1}, t_{2})

-weakly tractable (

(t_{1}, t_{2})

-WT) for fixed positive

t_{1}

and

t_{2}

iff

lim_{ε^{- 1} + d \to \infty} \frac{ln n^{X} (ε, S_{d})}{d^{t_{1}} + {(ε^{- 1})}^{t_{2}}} = 0 .

We call that S suffers from the curse of dimensionality if there exist positive numbers

C_{1}

C_{2}

ε_{0}

such that for all

0 < ε \leq ε_{0}

and infinitely many

d \in N

n (ε, d) \geq C_{1} {(1 + C_{2})}^{d} .

• Exponential convergence-strongly polynomially tractable (EC-SPT) iff there exist non-negative numbers C and p such that for all

d \in N

ε \in (0, 1)

n^{X} (ε, S_{d}) \leq C {(1 + ln (ε^{- 1}))}^{p} .

The exponent of SPT is defined to be the infimum of all p for which the above inequality holds.

• Exponential convergence-polynomially tractable (EC-PT) iff there exist non-negative numbers C, p and q such that for all

d \in N

ε \in (0, 1)

n^{X} (ε, S_{d}) \leq C d^{q} {(1 + ln (ε^{- 1}))}^{p} .

• Exponential convergence-uniformly weakly tractable (EC-UWT) iff for all

t_{1}, t_{2} > 0

lim_{ε^{- 1} + d \to \infty} \frac{ln n^{X} (ε, S_{d})}{d^{t_{1}} + {(1 + ln (ε^{- 1}))}^{t_{2}}} = 0 .

• Exponential convergence-weakly tractable (EC-WT) iff

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

• Exponential convergence-

(t_{1}, t_{2})

-weakly tractable (EC-

(t_{1}, t_{2})

-WT) for fixed positive

t_{1}

and

t_{2}

iff

lim_{ε^{- 1} + d \to \infty} \frac{ln n^{X} (ε, S_{d})}{d^{t_{1}} + {(1 + ln (ε^{- 1}))}^{t_{2}}} = 0 .

Clearly, (1,1)-WT is the same as WT, and EC-(1,1)-WT is the same as EC-WT. Obviously, in the definitions of SPT, PT, QPT, UWT, WT and

(t_{1}, t_{2})

-WT, if we replace

ε^{- 1}

(1 + ln (ε^{- 1}))

, we get the definitions of EC-SPT, EC-PT, EC-QPT, EC-UWT, EC-WT and EC-

(t_{1}, t_{2})

-WT, respectively. We also have

SPT ⟹ PT ⟹ QPT ⟹ UWT ⟹ WT,

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

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

and

EC - (t_{1}, t_{2}) - WT ⟹ (t_{1}, t_{2}) - WT, EC - UWT ⟹ UWT, EC - WT ⟹ WT .

We can learn more information about tractability of multivariate problems in the volumes [7,8,9] by Novak and Woźniakowski.

Lemma 2.1.

([7] Theorem 5.2) Consider the non-zero problem

S = {S_{d}}

for compact linear problems

S_{d}

defined over Hilbert spaces. Then S is PT for NOR iff there exist

q \geq 0

and

τ > 0

such that

C_{τ, q} : = sup_{d \in N} {(\sum_{j = 1}^{\infty} {(\frac{λ_{d, j}}{λ_{d, 1}})}^{τ})}^{\frac{1}{τ}} d^{- q} < \infty .

(2.3)

Expecially, S is SPT for NOR iff (2.3) holds with q=0. The exponent of SPT is

p^{str} = inf {2 τ | τ satisfies (2.3) with q = 0} .

3. Weighted Hilbert spaces

Let the space

H (K_{R_{d, α, γ}})

with weight

R_{d, α, γ}

under positive parameter sequences

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,

(3.1)

and

1 < α_{1} \leq α_{2} \leq \dots .

(3.2)

be a reproducing kernel Hilbert space. The 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 given by

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

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

, where

K_{R_{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]

is a universal weighted function. Here Fourier weight

R_{α, γ} : N_{0} \to R^{+}

be a summable function, i.e.,

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

. We will consider weight

R_{α, γ}

later on in some examples.

Then we have

K_{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},

(3.3)

and the corresponding 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)

(3.4)

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 .

We note that the kernel

K_{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_{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

. If

γ_{1} = γ_{2} = \dots = 1

and

α_{1} = α_{2} = \dots > 1

then the space is called unweighted space.

The weights are introduced to model the importance of the functions from the space. The idea can be seen in the reference [15] by Sloan and Woźniakowski. There are various ways to introduce weighted Hilbert spaces. We consider possible choices for Fourier weights

R_{d, α, γ}

on three examples.

3.1. A Korobov space

Let

α = {α_{j}}_{j \in N}

and

γ = {γ_{j}}_{j \in N}

satisfy (3.1) and (3.2), respectively. We are interesting in the weighted Korobov space

H (K_{R_{d, α, γ}})

defined by Irrgeher and Leobacher (see [16]) with kernel (3.3) and corresponding inner product (3.4), where weight

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

with

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]

. Note that we have

r_{α, γ} (k) \in (0, 1]

for all

k \in N_{0}

The space

H (K_{R_{d, α, γ}}) : = H (K_{r_{d, α, γ}})

is a reproducing kernel Hilbert space with parameter sequences

α = {α_{j}}_{j \in N}

and

γ = {γ_{j}}_{j \in N}

3.2. A first variant of the Korobov space

Let

α = {α_{j}}_{j \in N}

and

γ = {γ_{j}}_{j \in N}

satisfy (3.1) and (3.2), respectively. We consider the reproducing kernel Hilbert space

H (K_{R_{d, α, γ}})

with kernel (3.3) and corresponding inner product (3.4) determined by

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

with

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]

The following lemma gives the upper bound and the lower bound of the weight

ψ_{α, γ} (k)

, which shows that

ψ_{α, γ} (k)

has the same decay rate as the weight

r_{α, γ} (k)

of the Korobov space

H (K_{r_{d, α, γ}})

under the same parameter sequences

α

and

γ

Lemma 3.1.

For all

j, k \in N

we have

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

Proof.

First for all

j, k \in N

we want to prove

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

For

1 \leq k < ⌈ α_{j} ⌉

we have

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

For

k \geq ⌈ α_{j} ⌉

we have

\begin{matrix} ψ_{α_{j}, γ_{j}} (k) & = \frac{γ_{j} (k - ⌈ α_{j} ⌉)!}{k!} = \frac{γ_{j}}{k (k - 1) \dots (k - ⌈ α_{j} ⌉ + 1)} \\ \leq \frac{γ_{j}}{{(k - ⌈ α_{j} ⌉ + 1)}^{⌈ α_{j} ⌉}} = \frac{γ_{j}}{k^{⌈ α_{j} ⌉} {(1 - \frac{⌈ α_{j} ⌉ - 1}{k})}^{⌈ α_{j} ⌉}} \\ \leq \frac{γ_{j}}{k^{⌈ α_{j} ⌉} {(1 - \frac{⌈ α_{j} ⌉ - 1}{⌈ α_{j} ⌉})}^{⌈ α_{j} ⌉}} = \frac{{⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} γ_{j}}{k^{⌈ α_{j} ⌉}} . \end{matrix}

We find for all

k \in N

that

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

Next, for all

j, k \in N

we need to prove

ψ_{α_{j}, γ_{j}} (k) \geq r_{α_{j}, γ_{j}} (k) .

For

1 \leq k < ⌈ α_{j} ⌉

we have

ψ_{α_{j}, γ_{j}} (k) = \frac{γ_{j}}{k!} \geq \frac{γ_{j}}{k^{k}} \geq \frac{γ_{j}}{k^{⌈ α_{j} ⌉}} .

For

k \geq ⌈ α_{j} ⌉

we have

ψ_{α_{j}, γ_{j}} (k) = \frac{γ_{j} (k - ⌈ α_{j} ⌉)!}{k!} = \frac{γ_{j}}{k (k - 1) \dots (k - ⌈ α_{j} ⌉ + 1)} \geq \frac{γ_{j}}{k^{⌈ α_{j} ⌉}} .

Hence for all

j, k \in N

we obtain

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

This finishes the proof. □

3.3. A second variant of the Korobov space

In [17], the reproducing kernel Hilbert space

H (K_{R_{d, α, γ}})

was considered with kernel (3.3) and corresponding inner product (3.4). Here

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

was defined as

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

for

α > 1

and

γ \in (0, 1]

, where

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

Note that for

k \in N

we have

\sum_{l = 1}^{⌈ α ⌉} θ_{l} (k) \leq 2 k^{⌈ α ⌉} .

(3.5)

Indeed, for

k = 1

we have

\sum_{l = 1}^{⌈ α ⌉} θ_{l} (k) = 1 \leq 2 k^{⌈ α ⌉},

for

2 \leq k \leq ⌈ α ⌉

we have

\sum_{l = 1}^{⌈ α ⌉} θ_{l} (k) = \sum_{l = 1}^{k} \frac{k!}{(k - l)!} = k! \sum_{l = 0}^{k - 1} \frac{1}{l!} \leq k! \sum_{l = 0}^{\infty} \frac{1}{l!} \leq k! e \leq 2 k^{k} \leq 2 k^{⌈ α ⌉},

and for

k > ⌈ α ⌉

we have

\sum_{l = 1}^{⌈ α ⌉} θ_{l} (k) = \sum_{l = 1}^{⌈ α ⌉} \frac{k!}{(k - l)!} \leq \sum_{l = 1}^{⌈ α ⌉} k^{l} = k^{⌈ α ⌉} + \frac{k^{⌈ α ⌉} - k}{k - 1} \leq 2 k^{⌈ α ⌉} .

Lemma 3.2.

For all

j, k \in N

we have

\frac{1}{3} r_{α_{j}, γ_{j}} (k) \leq ω_{α_{j}, γ_{j}} (k) \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k) .

Proof.

First for all

j, k \in N

we want to prove

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

For

1 \leq k < ⌈ α_{j} ⌉

we have

ω_{α_{j}, γ_{j}} (k) = {(1 + {\frac{1}{γ}}_{j} \sum_{l = 1}^{⌈ α_{j} ⌉} θ_{l} (k))}^{- 1} = {(1 + {\frac{1}{γ}}_{j} \sum_{l = 1}^{k} θ_{l} (k))}^{- 1} \leq {({\frac{1}{γ}}_{j} θ_{k} (k))}^{- 1} = \frac{γ_{j}}{k!} .

For

k \geq ⌈ α_{j} ⌉

we have

ω_{α_{j}, γ_{j}} (k) = {(1 + {\frac{1}{γ}}_{j} \sum_{l = 1}^{⌈ α_{j} ⌉} θ_{l} (k))}^{- 1} \leq {({\frac{1}{γ}}_{j} θ_{⌈ α_{j} ⌉} (k))}^{- 1} = \frac{γ_{j} (k - ⌈ α_{j} ⌉)!}{k!} .

Hence for all

j, k \in N

we get

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

and thus by Lemma 3.1

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

holds.

Next, for all

j, k \in N

we need to prove

ω_{α_{j}, γ_{j}} (k) \geq \frac{1}{3} r_{α_{j}, γ_{j}} (k) .

It follows from (3.5) that for all

j, k \in N

we have

ω_{α_{j}, γ_{j}} (k) = {(1 + {\frac{1}{γ}}_{j} \sum_{l = 1}^{⌈ α_{j} ⌉} θ_{l} (k))}^{- 1} \geq {(1 + \frac{2 k^{⌈ α_{j} ⌉}}{γ_{j}})}^{- 1} \geq {(\frac{3 k^{⌈ α_{j} ⌉}}{γ_{j}})}^{- 1} = \frac{1}{3} r_{α_{j}, γ_{j}} (k) .

This proof is complete. □

Remark 3.3.

Set

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

for all

j, k \in N

. From Lemma 3.1 and Lemma 3.2 we have for all

j, k \in N

\frac{1}{3} r_{α_{j}, γ_{j}} (k) \leq R_{α_{j}, γ_{j}} (k) \leq {⌈ α_{j} ⌉}^{⌈ α_{j} ⌉} r_{α_{j}, γ_{j}} (k) .

(3.6)

Note that for all

j, k \in N

we have

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

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

and

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

which means that

R_{α_{j}, γ_{j}} (k) \leq R_{α_{1}, γ_{j}} (k), for all j, k \in N .

(3.7)

Combining with (3.6) and (3.7), we conclude

\frac{1}{3} r_{α_{j}, γ_{j}} (k) \leq R_{α_{j}, γ_{j}} (k) \leq R_{α_{1}, γ_{j}} (k) \leq {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} r_{α_{1}, γ_{j}} (k),

(3.8)

for all

j, k \in N

Remark 3.4.

The weight

R_{d, α, γ}

are used to describe the importance of the different coordinates for the functions from the space

H (K_{R_{d, α, γ}})

. According to (3.6) we have the weight

ψ_{d, α, γ}

and the weight

ω_{d, α, γ}

have the same decay rate as the weight

r_{d, α, γ}

of the Korobov space

H (K_{r_{d, α, γ}})

. Hence the above reproducing kernel Hilbert spaces

H (K_{r_{d, α, γ}})

H (K_{ψ_{d, α, γ}})

and

H (K_{ω_{d, α, γ}})

are different but also similar.

4. $L_{2}$ -approximation in weighted Hilbert spaces and main results

In this section we consider

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 spaces

H (K_{R_{d, α, γ}})

with weights

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

. It is well known from [6] that this embedding

{APP}_{d}

is compact with

1 < α_{1} \leq α_{2} \leq \dots

. The kernel

K_{R_{d, α, γ}} (x, y)

is well defined for

α_{1} > 1

and for all

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

, since by (3.7)

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

where

ζ (\cdot)

is the Riemann zeta function.

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

L_{2}

-approximation problems

S_{d}

with

G_{d} = L_{2} ({[0, 1]}^{d})

were investigated in analytic Korobov spaces and weighted Korobov spaces; see [1,2,3,6,10,11,12,13]. Additionally, [2,6,11,13] discussed tractability and EC-tractability in weighted Korobov spaces.

From subsection 2.1 the information complexity of

{APP}_{d}

depends on the eigenvalues of the 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}

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 .

Obviously, we have

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

(or see [6]). Hence the NOR and the ABS for the problem

{APP}_{d}

coincide in the worst case setting. We abbreviate

n^{X} (ε, {APP}_{d})

n (ε, {APP}_{d})

, i.e.,

n (ε, {APP}_{d}) : = n^{X} (ε, {APP}_{d}) .

It is well known that the eigenvalues of the operator

W_{d}

are

R_{d, α, γ} (k)

with

k \in N^{d}

; see, e.g., [7, p. 215]. Hence by (2.2) we have

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

Tractability such as SPT, PT, WT, and

(t_{1}, t_{2})

-WT for

t_{1} > 1

, and EC-tractability such as EC-WT and EC-

(t_{1}, 1)

-WT for

t_{1} < 1

of the above problem

APP = {{APP}_{d}}

with the parameter sequences

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 [2,4,11] and [13], respectively. The following conditions have been obtained therein:

• For

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

γ_{I} < \infty

implies QPT.

In those cases the exponent of QPT is

t^{pol} : = \{\begin{matrix} 2 max (\frac{1}{α}, \frac{1}{log γ_{I}^{- 1}}), for γ_{I} \neq 0, \\ \frac{2}{α}, for γ_{I} = 0 . \end{matrix}

• For

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

and

t_{1} > 1

(t_{1}, t_{2})

-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} < 1

, EC-

(t_{1}, 1)

-WT holds iff

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

We will research the worst case tractability of the problem APP with sequences satisfying (3.1) and (3.2).

Theorem 4.1.

Let the sequences

γ = {γ_{j}}_{j \in N}

and

α = {α_{j}}_{j \in N}

satisfy (3.1) and (3.2). Consider the

L_{2}

-approximation APP for the weighted Hilbert spaces

H_{R_{d, α, γ}}

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

. Then we have the following tractability results:

(1) SPT and PT are equivalent and hold iff

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

(4.1)

The exponent of SPT is

p^{str} = 2 max \{\frac{1}{δ}, \frac{1}{⌈ α_{1} ⌉}\} .

(2) For

R_{d, α, γ} = r_{d, α, γ}

, WT holds iff

lim_{j \to \infty} γ_{j} < 1 .

(3) For

t_{1} > 1

(t_{1}, t_{2})

-WT holds.

Proof. (1) For the problem

APP

we have

λ_{d, 1} = 1

. Assume that APP is PT. From Lemma 2.1 there exist

q \geq 0

and

τ > 0

such that

C_{τ, q} : = sup_{d \in N} {(\sum_{j = 1}^{\infty} λ_{d, j}^{τ})}^{\frac{1}{τ}} d^{- q} < \infty .

It follows from

\sum_{j = 1}^{\infty} λ_{d, j}^{τ} = \prod_{j = 1}^{d} (\sum_{k = 0}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{τ}) = \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{τ}),

(4.2)

and (3.8) that

\begin{matrix} \infty > C_{τ, q} & \geq {(\sum_{j = 1}^{\infty} λ_{d, j}^{τ})}^{\frac{1}{τ}} d^{- q} \\ \geq \prod_{j = 1}^{d} {(1 + \sum_{k = 1}^{\infty} {(\frac{1}{3} r_{α_{j}, γ_{j}} (k))}^{τ})}^{\frac{1}{τ}} d^{- q} \\ = {(\prod_{j = 1}^{d} (1 + \frac{1}{3^{τ}} γ_{j}^{τ} ζ (⌈ α_{j} ⌉ τ)))}^{\frac{1}{τ}} d^{- q} \\ \geq (\prod_{j = 1}^{d} {(1 + \frac{γ_{j}^{τ}}{3^{τ}})}^{\frac{1}{τ}}) d^{- q} \\ \geq {(1 + \frac{γ_{d}^{τ}}{3^{τ}})}^{\frac{d}{τ}} d^{- q} . \end{matrix}

We conclude that

ln C_{τ, q} + q ln d \geq \frac{d}{τ} ln (1 + \frac{γ_{d}^{τ}}{3^{τ}}) \geq \frac{d}{2 τ} \cdot \frac{γ_{d}^{τ}}{3^{τ}},

where we used

ln (1 + x) \geq \frac{x}{2}

for all

x \in [0, 1]

. We further get

ln (ln C_{τ, q} + q ln d) \geq ln d - τ ln γ_{d}^{- 1} - ln (2 τ \cdot 3^{τ}),

i.e.,

\frac{ln γ_{d}^{- 1}}{ln d} \geq \frac{ln d - ln (ln C_{τ, q} + q ln d) - ln (2 τ \cdot 3^{τ})}{τ \cdot ln d} .

Hence we obtain

δ = {lim inf}_{d \to \infty} \frac{ln γ_{d}^{- 1}}{ln d} \geq \frac{1}{τ} > 0 .

(4.3)

Note that if APP is SPT, then it is PT. It implies that if APP is SPT, then (4.3) holds and the exponent

p^{str} \geq 2 max {\frac{1}{δ}, \frac{1}{⌈ α_{1} ⌉}} .

On the other hand, assume that (4.1) holds. For an arbitrary

ε \in (0, \frac{δ}{2})

, there exists an integer

N > 0

such that for all

j \geq N

we have

\frac{ln γ_{j}^{- 1}}{ln j} \geq δ - ε .

It means that for all

j \geq N

γ_{j} \leq j^{- (δ - ε)} .

Choosing

τ = \frac{1}{δ - 2 ε}

and noting that

\frac{δ - ε}{δ - 2 ε} > 1

, we have

\sum_{j = N}^{\infty} γ_{j}^{τ} \leq \sum_{j = N}^{\infty} j^{- (δ - ε) τ} = \sum_{j = N}^{\infty} j^{- \frac{δ - ε}{δ - 2 ε}} < \infty,

which yields that

\sum_{j = 1}^{\infty} γ_{j}^{τ} < \infty .

(4.4)

From (3.8) we get

\begin{matrix} {(\sum_{j = 1}^{\infty} λ_{d, j}^{τ})}^{\frac{1}{τ}} d^{- q} & = \prod_{j = 1}^{d} {(1 + \sum_{k = 1}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{τ})}^{\frac{1}{τ}} d^{- q} \\ \leq \prod_{j = 1}^{d} {(1 + \sum_{k = 1}^{\infty} {({⌈ α_{1} ⌉}^{⌈ α_{1} ⌉} r_{α_{1}, γ_{j}} (k))}^{τ})}^{\frac{1}{τ}} d^{- q} \\ = d^{- q} \cdot exp \{ln (\prod_{j = 1}^{d} {(1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ τ} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ))}^{\frac{1}{τ}})\} \\ = d^{- q} \cdot exp \{\frac{1}{τ} \sum_{j = 1}^{d} ln (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ τ} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ))\} \\ \leq d^{- q} \cdot exp \{\frac{1}{τ} \sum_{j = 1}^{d} {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ τ} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)\} \\ = d^{- q} \cdot exp \{\frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ τ} ζ (⌈ α_{1} ⌉ τ)}{τ} \cdot \sum_{j = 1}^{d} γ_{j}^{τ}\} \\ \leq d^{- q} \cdot exp \{\frac{{⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ τ} ζ (⌈ α_{1} ⌉ τ)}{τ} \cdot \sum_{j = 1}^{\infty} γ_{j}^{τ}\} \\ < \infty \end{matrix}

for any

q \geq 0

and

τ > \frac{1}{⌈ α_{1} ⌉}

. We further get

\begin{matrix} C_{τ, q} & = sup_{d \in N} {(\sum_{j = 1}^{\infty} λ_{d, j}^{τ})}^{\frac{1}{τ}} d^{- q} < \infty \end{matrix}

for any

q \geq 0

and

τ > \frac{1}{⌈ α_{1} ⌉}

. It follows from Lemma 2.1 that APP is SPT or PT and the exponent

p^{str} \leq 2 τ

. Setting

ε \to 0

, we obtain

p^{str} \leq 2 τ \leq 2 max \{\frac{1}{δ}, \frac{1}{⌈ α_{1} ⌉}\} .

Hence the exponent of SPT is

p^{str} = 2 max \{\frac{1}{δ}, \frac{1}{⌈ α_{1} ⌉}\} .

(2) Let

τ > 0

. Due to

n λ_{d, n}^{τ} \leq \sum_{j = 1}^{n} λ_{d, j}^{τ} \leq \sum_{j = 1}^{\infty} λ_{d, j}^{τ},

we have

λ_{d, n} \leq \frac{{(\sum_{j = 1}^{\infty} λ_{d, j}^{τ})}^{\frac{1}{τ}}}{n^{\frac{1}{τ}}} .

Noting that

λ_{d, n} \leq ε^{2}

holds for

n = ⌈ \sum_{j = 1}^{\infty} λ_{d, j}^{τ} ε^{- 2 τ} ⌉,

we get

\begin{matrix} n (ε, {APP}_{d}) & = min {n | λ_{d, n + 1} \leq ε^{2}} \\ \leq ⌈ \sum_{j = 1}^{\infty} λ_{d, j}^{τ} ε^{- 2 τ} ⌉ \\ \leq 1 + ε^{- 2 τ} \sum_{j = 1}^{\infty} λ_{d, j}^{τ} \\ = 1 + ε^{- 2 τ} \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{τ}) \\ \leq 2 ε^{- 2 τ} \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{τ}), \end{matrix}

(4.5)

where we used (4.2).

Set

R_{d, α, γ} = r_{d, α, γ}

. Assume that

{lim}_{j \to \infty} γ_{j} < 1

. Then we have from (4.5) that

\begin{matrix} \frac{ln n (ε, {APP}_{d})}{d + ε^{- 1}} & \leq \frac{ln (2 ε^{- 2 τ} \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{τ}))}{d + ε^{- 1}} \\ = \frac{ln (2 ε^{- 2 τ} \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(r_{α_{j}, γ_{j}} (k))}^{τ}))}{d + ε^{- 1}} \\ = \frac{ln (2 ε^{- 2 τ} \prod_{j = 1}^{d} (1 + γ_{j}^{τ} ζ (⌈ α_{j} ⌉ τ)))}{d + ε^{- 1}} \\ \leq \frac{ln (2 ε^{- 2 τ} \prod_{j = 1}^{d} (1 + γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)))}{d + ε^{- 1}} \\ \leq \frac{ln 2 + 2 τ ln (ε^{- 1})}{ε^{- 1}} + \frac{\sum_{j = 1}^{d} ln (1 + γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)))}{d} \\ \leq \frac{ln 2 + 2 τ ln (ε^{- 1})}{ε^{- 1}} + \frac{\sum_{j = 1}^{d} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)}{d} . \end{matrix}

(4.6)

We will consider two cases:

Case ${lim}_{j \to \infty} γ_{j} = 0$ : It means that for any $δ > 0$ there exists a positive integer $J = J (δ)$ such that

$γ_{j} < δ for all j \geq J .$

Then we conclude from (4.6) that

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

which deduces that

\begin{matrix} lim_{d + ε^{- 1} \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ε^{- 1}} \leq δ^{τ} ζ (⌈ α_{1} ⌉ τ) . \end{matrix}

Setting

δ \to 0

, we have

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

. This yields WT.

Case ${lim}_{j \to \infty} γ_{j} \in (0, 1)$ : Then, for every ${lim}_{j \to \infty} γ_{j} < γ_{*} < 1$ there exists a positive integer $J_{0} = J (γ_{*})$ such that

$γ_{j} < γ_{*} for all j \geq J_{0} .$

We have from (4.6) that

\begin{matrix} \frac{ln n (ε, {APP}_{d})}{d + ε^{- 1}} & \leq \frac{ln 2 + 2 τ ln (ε^{- 1})}{ε^{- 1}} + \frac{\sum_{j = 1}^{d} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)}{d} \\ \leq \frac{ln 2 + 2 τ ln (ε^{- 1})}{ε^{- 1}} + \frac{\sum_{j = 1}^{J_{0} - 1} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)}{d} \\ + \frac{\sum_{j = J_{0}}^{max (J_{0}, d)} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)}{d} \\ \leq \frac{ln 2 + 2 τ ln (ε^{- 1})}{ε^{- 1}} + \frac{(J_{0} - 1) ζ (⌈ α_{1} ⌉ τ)}{d} \\ + \frac{\sum_{j = J_{0}}^{max (J_{0}, d)} γ_{*}^{τ} ζ (⌈ α_{1} ⌉ τ)}{d}, \end{matrix}

which means

lim_{d + ε^{- 1} \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ε^{- 1}} \leq γ_{*}^{τ} ζ (⌈ α_{1} ⌉ τ) .

Noting that

ζ (α) = 1 + \sum_{k = 2}^{\infty} \frac{1}{k^{α}} \leq 1 + \int_{1}^{\infty} \frac{1}{x^{α}} d x = 1 + \frac{1}{α - 1} for all α > 1,

and setting

τ \to \infty

, we obtain

lim_{d + ε^{- 1} \to \infty} \frac{ln n (ε, {APP}_{d})}{d + ε^{- 1}} \leq lim_{τ \to \infty} γ_{*}^{τ} ζ (⌈ α_{1} ⌉ τ) = 0 .

This implies WT.

On the other hand, it suffices to show that WT yields

{lim}_{j \to \infty} γ_{j} < 1 .

Assume on the contrary that

{lim}_{j \to \infty} γ_{j} = 1 .

It yields that

γ_{j} \equiv 1

for all

j \in N

. It follows that

1 = r_{d, α, γ} (k) > ε^{2}

for all

k \in {0, 1}^{d}

. Then we have

n (ε, {APP}_{d}) = | {k \in N_{0}^{d} : r_{d, α, γ} (k) > ε^{2}} | \geq 2^{d} .

Hence APP suffers from the curse of dimensionality. We cannot have WT.

(3) Let

τ > 0

. Due to (4.5) and (3.8) we have

\begin{matrix} n (ε, {APP}_{d}) & \leq 2 ε^{- 2 τ} \prod_{j = 1}^{d} (1 + \sum_{k = 1}^{\infty} {(R_{α_{j}, γ_{j}} (k))}^{τ}) \\ \leq 2 ε^{- 2 τ} \prod_{j = 1}^{d} (1 + {⌈ α_{1} ⌉}^{⌈ α_{1} ⌉ τ} γ_{j}^{τ} ζ (⌈ α_{1} ⌉ τ)) . \end{matrix}

It follows that

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

We obtain for all

t_{1} > 1

and

t_{2} > 0

lim_{ε^{- 1} + d \to \infty} \frac{ln n (ε, {APP}_{d})}{d^{t_{1}} + ε^{- t_{2}}} = 0,

which means APP is

(t_{1}, t_{2})

-WT for all

t_{1} > 1

and

t_{2} > 0

□

In this paper we consider the SPT, PT, WT and

(t_{1}, t_{2})

-WT for all

t_{1} < 1

and

t_{2} > 0

for worst case

L_{2}

-approximation in weighted Hilbert spaces

H_{R_{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 get the matching necessary and sufficient conditions on SPT or PT for

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

and WT for

R_{d, α, γ} = r_{d, α, γ}

. In particular, it is

(t_{1}, t_{2})

-WT for all

t_{1} > 1

and

t_{2} > 0

Author Contributions

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

Funding

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

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. The authors were supported by the Scientific and Technological Innovation Project of Colleges and Universities in Shanxi Province (Project no. 2022L438), National Natural Science Foundation of China (Project no. 12001342), Basic Youth Research Found Project of Shanxi Datong University (Project no. 2022Q10), and Doctoral Foundation Project of Shanxi Datong University (Project no. 2021-B-17 and 2019-B-10).

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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Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

Abstract

1. Introduction

2. Approximation and tractability in Hilbert spaces

2.1. Approximation in Hilbert spaces

2.2. Tractability

3. Weighted Hilbert spaces

3.1. A Korobov space

3.2. A first variant of the Korobov space

3.3. A second variant of the Korobov space

4. L 2 -approximation in weighted Hilbert spaces and main results

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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4. $L_{2}$ -approximation in weighted Hilbert spaces and main results