Strong Equivalences of Approximation Numbers and Tractability of Weighted Anisotropic Sobolev Embeddings
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
STRONG EQUIVALENCES OF APPROXIMATION NUMBERS AND TRACTABILITY OF WEIGHTED ANISOTROPIC SOBOLEV EMBEDDINGS∗
ÏGÀ)
Jidong HAO (
Ú²)
Heping WANG (
†
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China E-mail : [email protected]; [email protected] Abstract In this article, we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences a = {aj }j≥1 and b = {bj }j≥1 of positive numbers. We obtain strong equivalences of the approximation numbers, and necessary and sufficient conditions on a, b to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings. Key words
strong equivalences; tractability; approximation numbers; weighted anisotropic spaces; analytic Korobov spaces
2010 MR Subject Classification
1
41A25; 41A63; 65D15; 65Y20
Introduction
This article is devoted to investigating sharp constants of approximation numbers and the tractability of embeddings of weighted anisotropic Sobolev spaces on [0, 1]d into L2 ([0, 1]d ). The approximation numbers of a bounded linear operator T : X → Y between two Banach spaces are defined as an (T : X → Y ) : = =
inf
sup kT x − AxkY
rankA 0. k≥1
(2.2)
The weighted anisotropic Sobolev space W2a,b ([0, 1]d ) is defined by n o b W2a,b ([0, 1]d ) = f ∈ L2 ([0, 1]d ) : Dj j f ∈ L2 ([0, 1]d ), j = 1, 2, · · · , d ,
with the norm
d X
f a,b = f 2 + W 2 2
j=1
1/2 aj
Dbj f 2 . j 2 (2π)2bj
Clearly, W2a,b ([0, 1]d ) is a Hilbert space. We remark that b is a smoothness parameter sequence, and a is a (regulated) scaling parameter sequence with respect to the sequence b.
1768
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
It follows from (2.1) that
f
W2a,b
=
X
1+
d X j=1
k∈Zd
12 aj |kj |2bj |fˆ(k)|2 .
(2.3)
If aj = (2π)2bj , j ∈ N, then W2a,b ([0, 1]d ) reduces to the usual anisotropic Sobolev spaces W2b ([0, 1]d ) on the torus [0, 1]d . We emphasize that the anisotropic Sobolev spaces given in [1] are defined on the torus Td = [0, 2π]d , not on [0, 1]d . It is easily seen that d d 12 X X
bj 2 1/2 X
f b = f 2 +
D f = 1 + |2πkj |2bj |fˆ(k)|2 , j W 2 2 2
j=1
k∈Zd
(2.4)
j=1
˜ d ˜ = {˜bj }, ˜bj = (2π)2bj , j ∈ N, then W b ([0, 1]d ) = W b,b so if we write b 2 2 ([0, 1] ).
2.2
Analytic Korobov spaces
Let a = {aj }j≥1 and b = {bj }j≥1 be sequences satisfying (2.2). Fix ω ∈ (0, 1). We define the analytic Korobov kernel Kd,a,2b by X ωk e2πik(x−y) , for all x, y ∈ [0, 1]d, Kd,a,2b(x, y) = k∈Zd
where ωk = ω
d P
aj |kj |2bj
j=1
, for all k ∈ Zd .
Denote by H(Kd,a,2b ) the analytic Korobov space which is a reproducing kernel Hilbert space with the reproducing kernel Kd,a,2b. The inner product of the space H(Kd,a,2b) is given by X fˆ(k)ˆ hf, giH(Kd,a,2b ) = g (k)ωk−1 , f, g ∈ H(Kd,a,2b), k∈Zd
where fˆ(k), gˆ(k), k ∈ Zd are the Fourier coefficients of the functions f and g. The norm of a function f in H(Kd,a,2b) is given by
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