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Spectral deviation of concentration operators on reproducing kernel Hilbert spaces

Felipe Marceca, José Luis Romero, Michael Speckbacher, Lisa Valentini

Abstract

We study the eigenvalue profile of concentration operators (multiplication by an indicator function followed by projection) acting on reproducing kernel Hilbert spaces. The spectral profile of such operators provides a useful notion of local degrees of freedom. We formalize this idea by estimating the number of eigenvalues that lie away from 0 and 1, commonly referred to as the plunge region. Our main motivation is to treat discrete and continuous settings simultaneously and uniformly, and to be able to argue that approximations arising from discretization schemes reflect, in a non-asymptotic sense, the spectral profile of their continuous counterparts. As a case in point, we show that Gabor multipliers computed on sufficiently fine grids obey spectral deviation estimates similar to those available for the short-time Fourier transform (STFT) with bounds that are uniform in the discretization step. Concretely, this means that the theoretical localization properties of the STFT are observable in practice.

Spectral deviation of concentration operators on reproducing kernel Hilbert spaces

Abstract

We study the eigenvalue profile of concentration operators (multiplication by an indicator function followed by projection) acting on reproducing kernel Hilbert spaces. The spectral profile of such operators provides a useful notion of local degrees of freedom. We formalize this idea by estimating the number of eigenvalues that lie away from 0 and 1, commonly referred to as the plunge region. Our main motivation is to treat discrete and continuous settings simultaneously and uniformly, and to be able to argue that approximations arising from discretization schemes reflect, in a non-asymptotic sense, the spectral profile of their continuous counterparts. As a case in point, we show that Gabor multipliers computed on sufficiently fine grids obey spectral deviation estimates similar to those available for the short-time Fourier transform (STFT) with bounds that are uniform in the discretization step. Concretely, this means that the theoretical localization properties of the STFT are observable in practice.
Paper Structure (43 sections, 17 theorems, 193 equations)

This paper contains 43 sections, 17 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Concentration operators
  3. Time-frequency localization
  4. Spectral stability under discretization
  5. Further applications
  6. Results
  7. Setup
  8. Assumptions
  9. Poincaré perimeter
  10. Off-diagonal decay of the reproducing kernel
  11. Main result
  12. Applicability
  13. Applications
  14. Gabor multipliers
  15. Spectral stability under discretization with frames
  16. ...and 28 more sections

Key Result

Theorem 2.2

Let $\delta\in(0,1)$ and $\tau \coloneqq \max\{\frac{1}{\delta},\frac{1}{1-\delta}\}$. Assume Conditions norm2 and stripe hold. Then If in addition Condition doubling holds, then one also has Here, the constant implied in eq:last_common is absolute, while the one in eq:last_common2 is $O(C_X^4)$.

Theorems & Definitions (43)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3: Exponential decay
  • Corollary 2.4
  • Lemma 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • Proposition 4.3: Geometric perimeter controls the Poincaré perimeter
  • proof
  • ...and 33 more