Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2023-08-22 DOI:10.1016/j.jat.2023.105958

Morten Nielsen

引用次数: 0

Abstract

We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel–Lizorkin spaces in an anisotropic setting on $R^{d}$ . The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel–Lizorkin spaces.

In the second part of the paper we study nonlinear $m$ -term approximation with the constructed basis in the mixed-norm setting, where the behavior, in general, for $d \geq 2$ , is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for $m$ -term approximation can still be derived.

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齐次混合范数triiebel - lizorkin空间的稳定分解

我们在Rd上的各向异性设置中构造了与齐次混合范数Triebel–Lizorkin空间兼容的光滑局部化正交正规基。该构造基于所谓的单变量brushlet函数的张量积，该函数是使用频域中的局部三角基构造的。证明了关联分解系统构成齐次混合范数Triebel–Lizorkin空间的无条件基。在本文的第二部分中，我们研究了在混合范数设置中具有构造基的非线性m项近似，其中，对于d≥2，通常的行为与未混合的情况有根本的不同。然而，m项近似的Jackson和Bernstein不等式仍然可以导出。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.