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Wave packet systems and connections to spectral analysis of limiting operators

Kevin Hughes, Arie Israel, Azita Mayeli

TL;DR

The paper advances the design of wave packet systems that achieve strong phase-space concentration by linking their concentration properties to the spectral behavior of limiting operators $B_SP_FB_S$. It unifies Gabor and wavelet concepts into a general wave packet framework $\mathcal{P}(\Theta,\mathcal{A},\Xi)$ and analyzes concentration/packing via energy bounds and a packing lemma that connects to eigenvalue counts. It then constructs concentrated bases on the unit cube using the Coifman–Meyer local sine basis and a tensorization approach, providing explicit bounds on the number of eigenvalues in the mid-spectrum and on the residual size of the basis. These results have practical implications for stable extrapolation of bandlimited functions and for signal processing tasks in imaging, geophysics, and astronomy, with open questions on extending the framework to balls in higher dimensions.

Abstract

We discuss the design of ``wave packet systems'' that admit strong concentration properties in phase space. We make a connection between this problem and topics in signal processing related to the spectral behavior of spatial and frequency-limiting operators. The results have engineering applications in medical imaging, geophysics, and astronomy.

Wave packet systems and connections to spectral analysis of limiting operators

TL;DR

The paper advances the design of wave packet systems that achieve strong phase-space concentration by linking their concentration properties to the spectral behavior of limiting operators . It unifies Gabor and wavelet concepts into a general wave packet framework and analyzes concentration/packing via energy bounds and a packing lemma that connects to eigenvalue counts. It then constructs concentrated bases on the unit cube using the Coifman–Meyer local sine basis and a tensorization approach, providing explicit bounds on the number of eigenvalues in the mid-spectrum and on the residual size of the basis. These results have practical implications for stable extrapolation of bandlimited functions and for signal processing tasks in imaging, geophysics, and astronomy, with open questions on extending the framework to balls in higher dimensions.

Abstract

We discuss the design of ``wave packet systems'' that admit strong concentration properties in phase space. We make a connection between this problem and topics in signal processing related to the spectral behavior of spatial and frequency-limiting operators. The results have engineering applications in medical imaging, geophysics, and astronomy.
Paper Structure (6 sections, 5 theorems, 21 equations, 1 figure)

This paper contains 6 sections, 5 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Methodology and Main Results
  3. Wave Packet Systems
  4. Concentrated wave packet systems on the unit cube.
  5. Coifman-Meyer local sine basis
  6. Tensor systems

Key Result

Theorem 1.1

Let $Q = [0,1]^d$, and let $S \subset B(0,1)$ be a compact coordinate-wise symmetric convex set in $\mathbb{R}^d$. There exists a dimensional constant $C_d > 0$ (independent of $S$) such that, for any $\epsilon \in (0,1/2)$ and $r \geq 1$, where $E_d(\epsilon,r) := \max \{ r^{d-1} \log(r/\epsilon)^{5/2}, \log(r/\epsilon)^{5d/2} \}$.

Figures (1)

  • Figure 1: A decomposition of the square into rectangles $\mathbf L$ in dimension $d=2$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: $\epsilon$-concentrated system
  • Definition 2.3: $\epsilon$-packing
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof : Proof of Lemma \ref{['conc_frame:lem']}
  • Proposition 2.6
  • Proposition 2.7
  • ...and 1 more