Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Painless Construction of Unconditional Bases for Anisotropic Modulation and Triebel-Lizorkin Type Spaces

Morten Nielsen

TL;DR

This work delivers an explicit, anisotropy-aware construction of tensor brushlet bases that form unconditional bases for the full family of $\alpha$-modulation and $\alpha$-Triebel–Lizorkin spaces on $\mathbb{R}^d$, including their Besov-type variants. By leveraging a polynomial-type frequency decomposition and a careful frequency-space partition, the authors obtain universal norm characterizations and demonstrate nonredundancy to enable sharp inverse estimates in nonlinear $m$-term approximation. They establish Jackson and Bernstein inequalities within this framework, connecting approximation rates to smoothness in $\alpha$-modulation and $\alpha$-Triebel–Lizorkin spaces, and show the resulting bases are well-suited for sparse representations of anisotropic functions. The results provide a flexible, explicit tool for discretising and analyzing anisotropic smoothness spaces, with potential impact on numerical analysis, signal processing, and sparse approximation in high dimensions.

Abstract

We construct smooth localised orthonormal bases compatible with anisotropic Triebel-Lizorkin and Besov type spaces on $\mathbb{R}^d$. The construction is based on tensor products of so-called univariate brushlet functions that are based on local trigonometric bases in the frequency domain, and the construction is painless in the sense that all parameters for the construction are explicitly specified. It is shown that the associated decomposition system form unconditional bases for the full family of Triebel-Lizorkin and Besov type spaces, including for the so-called $α$-modulation and $α$-Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear $m$-term approximation with the constructed bases, where direct Jackson and Bernstein inequalities for $m$-term approximation with the tensor brushlet system in $α$-modulation and $α$-Triebel-Lizorkin spaces are derived. The inverse Bernstein estimates rely heavily on the fact that the constructed system is non-redundant.

Painless Construction of Unconditional Bases for Anisotropic Modulation and Triebel-Lizorkin Type Spaces

TL;DR

This work delivers an explicit, anisotropy-aware construction of tensor brushlet bases that form unconditional bases for the full family of -modulation and -Triebel–Lizorkin spaces on , including their Besov-type variants. By leveraging a polynomial-type frequency decomposition and a careful frequency-space partition, the authors obtain universal norm characterizations and demonstrate nonredundancy to enable sharp inverse estimates in nonlinear -term approximation. They establish Jackson and Bernstein inequalities within this framework, connecting approximation rates to smoothness in -modulation and -Triebel–Lizorkin spaces, and show the resulting bases are well-suited for sparse representations of anisotropic functions. The results provide a flexible, explicit tool for discretising and analyzing anisotropic smoothness spaces, with potential impact on numerical analysis, signal processing, and sparse approximation in high dimensions.

Abstract

We construct smooth localised orthonormal bases compatible with anisotropic Triebel-Lizorkin and Besov type spaces on . The construction is based on tensor products of so-called univariate brushlet functions that are based on local trigonometric bases in the frequency domain, and the construction is painless in the sense that all parameters for the construction are explicitly specified. It is shown that the associated decomposition system form unconditional bases for the full family of Triebel-Lizorkin and Besov type spaces, including for the so-called -modulation and -Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear -term approximation with the constructed bases, where direct Jackson and Bernstein inequalities for -term approximation with the tensor brushlet system in -modulation and -Triebel-Lizorkin spaces are derived. The inverse Bernstein estimates rely heavily on the fact that the constructed system is non-redundant.

Paper Structure

This paper contains 14 sections, 14 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The anisotropic setup and preliminaries
  3. Anisotropic tensor product brushlet orthonormal bases
  4. Decomposition of the frequency space $\mathbb{R}^d$
  5. Tensor-product brushlet bases
  6. Modulation and Triebel-Lizorkin type smoothness spaces
  7. Characterizations of ${F}_{p,q}^{s,\alpha}(\vec{a})$ and ${M}_{p,q}^{s,\alpha}(\vec{a})$
  8. Nonlinear approximation with anisotropic brushlet bases
  9. Direct and inverse estimates for $m$-term approximation with tensor product brushlet systems
  10. Jackson inequality
  11. Bernstein inequalities
  12. Additional Jackson and Bernstein estimates
  13. Univariate brushlets
  14. Some Estimates for Maximal Functions

Key Result

Lemma 3.1

The following holds true about the partitions $\mathcal{A}_j^i$ defined in Eq. eq:partt.

Figures (1)

  • Figure 1: The corridors ${K}_0, {K}_1$, ${K}_2$, and ${K}_3$ in $\mathbb{R}^2$ for $\beta=1.1$ and $\vec{a}=(\sqrt{3},\frac{3}{2})$. The fine grid represents the sets in the partition $\mathcal{K}_3$ of $K_3$. Notice that $\lceil 3^{\sqrt{3}-1}\rceil=3$, while $\lceil 3^{3/2-1}\rceil=2$, leading to the illustrated $3\times 2$ sub-grid for the corner set $[3^{1.1\cdot \sqrt{3}}, 4^{1.1\cdot \sqrt{3}})\times [3^{1.1\cdot 1.5},4^{1.1\cdot 1.5})$, which is compatible with the anisotropy $\vec{a}$.

Theorems & Definitions (37)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 4.1
  • Definition 4.2
  • Remark 4.3
  • Proposition 4.4
  • proof
  • Remark 4.5
  • ...and 27 more