1. Introduction
We investigate multivariate approximation problems
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
we define the information complexity
to be the minimal number of linear functionals which are needed to find an algorithm whose worst case error is at most
. The behavior of the information complexity
is the major concern when the accuracy
of approximation goes to zero and the number
d of variables goes to infinity. For small
and large
d, tractability is aimed at studying how the information complexity
behaves as a function of
d and
, while the exponential convergence-tractability (EC-tractability) is aimed at studying how the information complexity
behaves as a function of
d and
. 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
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
-approximation problem
with
for all
in weighted Hilbert spaces
with three weights
under positive parameter sequences
and
. The tractability and EC-tractability of such problem APP in weighted Korobov spaces with parameters
and
were discussed in [
12,
13,
14,
15] and in [
16], respectively. Additionally, [
15] considered the tractability of the
-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
-WT for all
and
of the above problem APP with parameters
and
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
-approximation problems in the above weighted Hilbert spaces, then state out main result Theorem 6.
3. Weighted Hilbert Spaces
Let the space
with weight
under positive parameter sequences
and
satisfying
and
be a reproducing kernel Hilbert space. The reproducing kernel function
of the space
is given by
, where
is a universal weighted function. Here Fourier weight
be a summable function, i.e.,
. We will consider weight
later on in some examples.
Then we have
and the corresponding inner product
and
where
and
We note that the kernel
is well defined for
and for all
, since
. If
and
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 [
18] by Sloan and Woźniakowski. There are various ways to introduce weighted Hilbert spaces. We consider possible choices for Fourier weights
on three cases.
3.1. A Korobov Space
Let
and
satisfy (
4) and (
5), respectively. We are interesting in the weighted Korobov space
defined by Irrgeher and Leobacher (see [
19]) with kernel (
6) and corresponding inner product (
7), where weight
with
for
and
. Note that we have
for all
.
The space is a reproducing kernel Hilbert space with parameter sequences and .
3.2. A First Variant of the Korobov Space
Let
and
satisfy (
4) and (
5), respectively. We consider the reproducing kernel Hilbert space
with kernel (
6) and corresponding inner product (
7) determined by
with
for
and
.
The following lemma gives the upper bound and the lower bound of the weight , which shows that has the same decay rate as the weight of the Korobov space under the same parameter sequences and .
Lemma 2. For all we have Proof. First for all
we want to prove
For
we have
For
we have
We find for all
that
Next, for all
we need to prove
For
we have
For
we have
Hence for all
we obtain
This finishes the proof. □
3.3. A Second Variant of the Korobov Space
In [
20], the reproducing kernel Hilbert space
was considered with kernel (
6) and corresponding inner product (
7). Here
was defined as
for
and
, where
Note that for
we have
Indeed, for
we have
for
we have
and for
we have
Lemma 3. For all we have Proof. First for all
we want to prove
For
we have
For
we have
Hence for all
we get
and thus by Lemma 2
holds.
Next, for all
we need to prove
It follows from (
8) that for all
we have
This proof is complete. □
Remark 4. Set for all . From Lemmas 2 and 3 we have for all ,Note that for all we have and which means thatCombining with (9) and (10), we concludefor all . Remark 5. The weight are used to describe the importance of the different coordinates for the functions from the space . According to (9) we have the weight and the weight have the same decay rate as the weight of the Korobov space . Hence the above reproducing kernel Hilbert spaces , and are different but also similar. 4. -Approximation in Weighted Hilbert Spaces and Main Results
In this section we consider
-approximation
with
for all
in Hilbert spaces
with weights
. It is well known from [
13] that this embedding
is compact with
. The kernel
is well defined for
and for all
, since by (
10)
where
is the Riemann zeta function.
In the worst case setting the tractability and EC-tractability of
-approximation problems
with
were investigated in analytic Korobov spaces and weighted Korobov spaces; see [
4,
5,
10,
11,
12,
13,
14,
16]. Additionally, refs. [
12,
13,
14,
16] discussed tractability and EC-tractability in weighted Korobov spaces.
From
Section 2.1 the information complexity of
depends on the eigenvalues of the operator
. Let
be the eigenpairs of
,
where the eigenvalues
are ordered,
and the eigenvectors
are orthonormal,
Obviously, we have
(or see [
13]). Hence the NOR and the ABS for the problem
coincide in the worst case setting. We abbreviate
as
, i.e.,
It is well known that the eigenvalues of the operator
are
with
; see, e.g., ([
7] p. 215). Hence by (
2) we have
Tractability such as SPT, PT, WT, and
-WT for
, and EC-tractability such as EC-WT and EC-
-WT for
of the above problem
with the parameter sequences
and
satisfying
and
have been solved by [
12,
14,
15] and [
16], respectively. The following conditions have been obtained therein:
For
, PT holds iff SPT holds iff
and the exponent of SPT is
For
, QPT, UWT and WT are equivalent and hold iff
For
,
implies QPT.
In those cases the exponent of QPT is 1.3
For and , -WT holds for all .
For
, EC-WT holds iff
For
and
, EC-
-WT holds iff
We will research the worst case tractability of the problem APP with sequences satisfying (
4) and (
5).
Theorem 6. Let the sequences and satisfy (4) and (5). Consider the -approximation APP for the weighted Hilbert spaces , . Then we have the following tractability results: - (1)
SPT and PT are equivalent and hold iff - (2)
For , WT holds iff - (3)
For , -WT holds.
Proof. (1) For the problem
we have
. Assume that APP is PT. From Lemma 1 there exist
and
such that
It follows from
and (
11) that
We conclude that
where we used
for all
. We further get
i.e.,
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
, there exists an integer
such that for all
we have
It means that for all
Choosing
and noting that
, we have
which yields that
From (
11) we get
for any
and
. Due to (
15), we further have
for any
and
. It follows from Lemma 1 that APP is SPT or PT and the exponent
. Setting
, we obtain
Hence the exponent of SPT is
(2) Let
. Due to
we have
Noting that
holds for
we get
where we used (
13).
Set
. Assume that
. Then we have from (
17) that
where in the last inequality, we use
for all
. We will consider two cases:
On the other hand, it suffices to show that WT yields
Assume on the contrary that
It yields that
for all
. It follows that
for all
. Then we have
Hence APP suffers from the curse of dimensionality. We cannot have WT.
(3) Let
. Due to (
17) and (
11) we have
It follows that
We obtain for all
and
,
which means APP is
-WT for all
and
. □
Example 7. Examples for SPT, PT, WT and -WT for .
Assume that and for all . We consider the above weighted Hilbert spaces , .
SPT and PT for .
Obviously, satisfying (12). By (16) we have for any and . Choosing , we further get for any and which yields that APP is SPT or PT from Lemma 1.
WT for .
Obviously, . By (18) and choosing we have which means Hence WT holds.
-WT with for .
From the proof (3) of Lemma 6, we can easily obtain that -WT holds for and .
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
-WT for all
and
for worst case
-approximation in weighted Hilbert spaces
with parameters
and
. We get the matching necessary and sufficient condition
on SPT or PT for
, and the matching necessary and sufficient condition
on WT for
. In particular, it is
-WT for all
and
. 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.