Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

: We investigate L 2 -approximation problems in the worst case setting in the weighted Hilbert spaces H ( K R d , α , γ ) with weights R d , α , γ under parameters 1 ≥ γ 1 ≥ γ 2 ≥ · · · ≥ 0 and 1 < α 1 ≤ α 2 ≤ · · · . Several interesting weighted Hilbert spaces H ( K R d , α , γ ) appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of L 2 -approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ε − 1 . As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and ( t 1 , t 2 ) -weak tractability ( ( t 1 , t 2 ) -WT) for all t 1 > 1 and t 2 > 0 in terms of the introduced weights under the absolute error criterion or the normalized error criterion.


Introduction
We investigate multivariate approximation problems S d with large or even huge d.
Examples include these problems in statistics (see [1]), computational finance (see [2]) and physics (see [3]).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 ∈ {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 (see [4][5][6][7][8][9][10][11]), weighted Korobov spaces (see [7][8][9][12][13][14]) and weighted Gaussian ANOVA spaces (see [15]).
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 with APP d ( f ) = f for all f ∈ 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∈N and α = {α j } j∈N .The tractability and EC-tractability of such problem APP in weighted Korobov spaces with parameters 1 ≥ γ 1 ≥ γ 2 ≥ • • • ≥ 0 and 1 < α 1 = α 2 = • • • were discussed in [12][13][14][15] and in [16], respectively.Additionally, [15] 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 for the ABS or the NOR under the information class consisting of arbitrary continuous linear functionals.Especially, although these three weighted Hilbert spaces are different, we get the same compete sufficient and necessary condition for SPT or PT, and the same exponent of SPT by appropriate method.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 6.

Approximation in Hilbert Spaces
Let F d and G d be two sequences of Hilbert spaces.Consider a sequence of compact linear operators for all d ∈ N. We approximation S d by algorithm A n,d of the form where functions g i ∈ G d and continuous linear functionals The worst case error for the algorithm A n,d of the form (1) is defined as The n-th minimal worst-case error, for n ≥ 1, is defined by where the infimum is taken over all linear algorithms of the form (1).For n = 0, we use 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 ∈ {ABS, NOR} is defined by where CRI d := 1, for X = ABS, e(0, S d ), for X = NOR.
It is well known, see e.g., refs.[7,17], that the n-th 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 ) be the eigenpairs of W d , i.e., where the eigenvalues λ d,j are ordered, and the eigenvectors η d,j are orthonormal, Then the n-th minimal error is obtained for the algorithm Hence the information complexity is equal to with ε ∈ (0, 1) and d ∈ 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.

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∈N .For X ∈ {ABS, NOR}, we say S is • strongly polynomially tractable (SPT) iff there exist non-negative numbers C and p such that for all d ∈ N, ε ∈ (0, 1), 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 ∈ N, ε ∈ (0, 1), n X (ε, S d ) ≤ Cd q (ε −1 ) p .
• quasi-polynomially tractable (QPT) iff there exist two constants C, t > 0 such that for all d ∈ N, ε ∈ (0, 1), 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 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 ≤ ε 0 and infinitely many d ∈ N, • Exponential convergence-strongly polynomially tractable (EC-SPT) iff there exist non-negative numbers C and p such that for all d ∈ N, ε ∈ (0, 1), The exponent of SPT is defined to be the infimum of all p for which the above inequality holds.
Lemma 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 ≥ 0 and τ > 0 such that Expecially, S is SPT for NOR iff (3) holds with q = 0.The exponent of SPT is p str = inf{2τ|τ satisfies (3) with q = 0}.

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∈N and α = {α j } j∈N satisfying be a reproducing kernel Hilbert space.The reproducing kernel function is a universal weighted function.Here Fourier weight R α,γ : N 0 → R + be a summable function, i.e., ∑ k∈N 0 R α,γ (k) < ∞.We will consider weight R α,γ later on in some examples.
Then we have and the corresponding inner product and where We note that the kernel The weights are introduced to model the importance of the functions from the space.The idea can be seen in the reference [18] 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 cases.

A First Variant of the Korobov Space
Let α = {α j } j∈N and γ = {γ j } j∈N satisfy (4) and (5), respectively.We consider the reproducing kernel Hilbert space H(K R d,α,γ ) with kernel (6) and corresponding inner product (7) 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 2. For all j, k ∈ N we have r α j ,γ j (k) ≤ ψ α j ,γ j (k) ≤ ⌈α j ⌉ ⌈α j ⌉ r α j ,γ j (k).
For 1 ≤ k < ⌈α j ⌉ we have For k ≥ ⌈α j ⌉ we have We find for all k ∈ N that Next, for all j, k ∈ N we need to prove For 1 ≤ k < ⌈α j ⌉ we have For k ≥ ⌈α j ⌉ we have Hence for all j, k ∈ N we obtain This finishes the proof.
For 1 ≤ k < ⌈α j ⌉ we have For k ≥ ⌈α j ⌉ we have Hence for all j, k ∈ N we get and thus by Lemma 2 Next, for all j, k ∈ N we need to prove It follows from ( 8) that for all j, k ∈ N we have This proof is complete.
Remark 5.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 (9) 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.

L 2 -Approximation in Weighted Hilbert Spaces and Main Results
In this section we consider L 2 -approximation where ζ(•) is the Riemann zeta function.
From Section 2.1 the information complexity of APP d depends on the eigenvalues of the operator where the eigenvalues λ d,j are ordered, and the eigenvectors η d,j are orthonormal, Obviously, we have e(0, APP d ) = 1 (or see [13]).Hence the NOR and the ABS for the problem APP d coincide in the worst case setting.We abbreviate n X (ε, APP d ) as n(ε, APP d ), i.e., It is well known that the eigenvalues of the operator W d are R d,α,γ (k) with k ∈ N d ; see, e.g., ([7] p. 215).Hence by ( 2) we have 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∈N and α = {α j } j∈N satisfying have been solved by [12,14,15] and [16], respectively.The following conditions have been obtained therein: and the exponent of SPT is α .
Proof.(1) For the problem APP we have λ d,1 = 1.Assume that APP is PT.From Lemma 1 there exist q ≥ 0 and τ > 0 such that and (11) that We conclude that ln C τ,q + q ln d ≥ d where we used ln(1 + x) ≥ x 2 for all x ∈ [0, 1].We further get ln(ln Hence we obtain Note that if APP is SPT, then it is PT.It implies that if APP is SPT, then (14) holds and the exponent On the other hand, assume that (12) holds.For an arbitrary ε ∈ (0, δ 2 ), there exists an integer N > 0 such that for all j ≥ N we have It means that for all j ≥ N From (11) we get for any q ≥ 0 and τ > 1 ⌈α 1 ⌉ .Due to (15), we further have for any q ≥ 0 and τ > 1 ⌈α 1 ⌉ .It follows from Lemma 1 that APP is SPT or PT and the exponent p str ≤ 2τ.Setting ε → 0, we obtain we have where we used (13).Set R d,α,γ = r d,α,γ .Assume that lim j→∞ γ j < 1.Then we have from (17) that where in the last inequality, we use ln(1 + x) ≤ x for all x ≥ 0. We will consider two cases: • Case lim j→∞ 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 ≥ J.
We have from that (18) ln n(ε, Noting that and setting τ → ∞, we obtain lim This implies WT.
On the other hand, it suffices to show that WT yields lim j→∞ γ j < 1. Assume on the contrary that lim j→∞ γ j = 1.It yields that γ j ≡ 1 for all j ∈ N. It follows that Hence APP suffers from the curse of dimensionality.We cannot have WT.
Remark 8. Indeed, SPT and PT are not equivalent under some conditions in the worst case setting; see [8] on Page 344.
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 ≥ γ 1 ≥ γ 2 ≥ • • • ≥ 0 and 1 < α 1 ≤ α 2 ≤ • • • .We get the matching necessary and sufficient condition lim inf on 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. The weights in weighted Hilbert spaces are very important for multivariate approximation problems, so we plan to further investigate the tractability notions and EC-tractability notions and hope to find out more effective method to solve such problems.