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A Battle-Lemarié Frame Characterization of Besov and Triebel-Lizorkin Spaces

Andrew Haar

TL;DR

The paper develops a framework for characterizing Besov and Triebel-Lizorkin spaces using an oversampled Battle-Lemarié spline frame, establishing norm equivalences for smoothness up to $s < n+1$ and treating the endpoint $s=n+1$. The approach blends Littlewood-Paley theory, maximal-function techniques, and a crucial link between B-splines and Battle-Lemarié systems (via Ushakova localisation) to overcome noncompact support and lack of closed-form expressions. This work generalizes the Haar ($n=0$) case and connects to the unconditional-basis ranges studied by Srivastava, providing a robust spline-frame characterization of smoothness spaces in settings with spline-based frame oversampling. The endpoint analysis completes the picture, delivering a comprehensive description of how oversampling against Battle-Lemarié systems characterizes $B^s_{pq}$ and $F^s_{pq}$ across admissible parameter ranges and clarifies optimality constraints.

Abstract

In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, $n$. We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to $s < n+1$, which includes values of $s$ where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case $s=n+1$. This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case $n=0$, i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces.

A Battle-Lemarié Frame Characterization of Besov and Triebel-Lizorkin Spaces

TL;DR

The paper develops a framework for characterizing Besov and Triebel-Lizorkin spaces using an oversampled Battle-Lemarié spline frame, establishing norm equivalences for smoothness up to and treating the endpoint . The approach blends Littlewood-Paley theory, maximal-function techniques, and a crucial link between B-splines and Battle-Lemarié systems (via Ushakova localisation) to overcome noncompact support and lack of closed-form expressions. This work generalizes the Haar () case and connects to the unconditional-basis ranges studied by Srivastava, providing a robust spline-frame characterization of smoothness spaces in settings with spline-based frame oversampling. The endpoint analysis completes the picture, delivering a comprehensive description of how oversampling against Battle-Lemarié systems characterizes and across admissible parameter ranges and clarifies optimality constraints.

Abstract

In this paper, we investigate a spline frame generated by oversampling against the well-known Battle-Lemarié wavelet system of nonnegative integer order, . We establish a characterization of the Besov and Triebel-Lizorkin (quasi-) norms for the smoothness parameter up to , which includes values of where the Battle-Lemarié system no longer provides an unconditional basis; we, additionally, prove a result for the endpoint case . This builds off of earlier work by G. Garrigós, A. Seeger, and T. Ullrich, where they proved the case , i.e. that of the Haar wavelet, and work of R. Srivastava, where she gave a necessary range for the Battle-Lemarié system to give an unconditional basis of the Triebel-Lizorkin spaces.

Paper Structure

This paper contains 20 sections, 26 theorems, 210 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Prerequisite Material
  3. Splines and Wavelets
  4. Multiresolution Analyses for Spaces of Distributions
  5. Besov and Triebel-Lizorkin Spaces
  6. Maximal Functions
  7. Duality Considerations
  8. Proofs for the Main Theorems
  9. The Weaker Direction
  10. Some lemmata
  11. Proving the Weaker Inequalities
  12. The Stronger Direction
  13. The Relation of the B-splines to the Battle-Lemarié System
  14. Setup for the Proof
  15. Proving the Stronger Direction
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\{\psi,\Psi\}$ be a Battle-Lemarié system. Suppose $\frac{1}{2(n+1)} < p \leqslant \infty$, $0<q\leqslant \infty$, and Then for $f\in\mathcal{B}$, is equivalent to the (quasi-) norm of $B^s_{pq}(\mathbb{R})$ given in eq:normDefs.

Figures (1)

  • Figure 1: The parameter ranges for the Battle-Lemarié spline system to be an unconditional basis of $B^s_{pq}$ (left) and $F^s_{pq}$ (right, when $1<q<\infty$) are in pink, see Theorem \ref{['thm:SplineCharac']}. The extension provided by the frame in Theorems \ref{['thm:mainBesov']} and \ref{['thm:mainTriebel']} is in purple.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Remark 1.5
  • Definition 2.1: Battle-Lemarié Wavelet System
  • Lemma 2.2: Lemma 3.1 from srivastava2023orthogonal
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 43 more