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Spectral deviation of concentration operators for the short-time Fourier transform

Felipe Marceca, José Luis Romero

Abstract

Time-frequency concentration operators restrict the integral analysis-synthesis formula for the short-time Fourier transform to a given compact domain. We estimate how much the corresponding eigenvalue counting function deviates from the Lebesgue measure of the time-frequency domain. For window functions in the Gelfand-Shilov class, the bounds almost match known asymptotics, with the advantage of being effective for concrete domains and spectral thresholds. As such our estimates allow for applications where the spectral threshold depends on the geometry of the time-frequency concentration domain. We also consider window functions that decay only polynomially in time and frequency.

Spectral deviation of concentration operators for the short-time Fourier transform

Abstract

Time-frequency concentration operators restrict the integral analysis-synthesis formula for the short-time Fourier transform to a given compact domain. We estimate how much the corresponding eigenvalue counting function deviates from the Lebesgue measure of the time-frequency domain. For window functions in the Gelfand-Shilov class, the bounds almost match known asymptotics, with the advantage of being effective for concrete domains and spectral thresholds. As such our estimates allow for applications where the spectral threshold depends on the geometry of the time-frequency concentration domain. We also consider window functions that decay only polynomially in time and frequency.

Paper Structure

This paper contains 19 sections, 11 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Spectrum of concentration operators
  3. Robust versus non-robust error terms
  4. Results
  5. The dilation regime
  6. Polynomial time-frequency decay
  7. Some consequences
  8. Related literature and organization
  9. Preliminaries
  10. Notation
  11. Operators
  12. Perimeters and level sets
  13. Toeplitz and Hankel operators on the range of the STFT
  14. Estimates for Hankel operators on the range of the STFT
  15. Eigenvalue estimates
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega\subseteq\mathbb{R}^{2d}$ be a compact set with regular boundary at scale $\eta >0$. Let $g \in L^2(\mathbb{R}^d)$ satisfy $\lVert g \rVert_2=1$ and the following Gelfand-Shilov-type condition with parameter $\beta \geq 1/2$: there exist $C_g,A>0$ such that for every $n \in \mathbb{N}_0$: For $\delta\in(0,1)$ set $\tau=\max\left\{\tfrac{1}{\delta}, \tfrac{1}{1-\delta}\right\}$. Then wh

Theorems & Definitions (23)

  • Theorem 1.1: Threshold-robust spectral bounds, non-dilated domains
  • Remark 1.2
  • Corollary 1.3: Threshold-robust spectral bounds, the dilation regime
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 13 more