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Eigenvalue estimates for Fourier concentration operators on two domains

Felipe Marceca, José Luis Romero, Michael Speckbacher

Abstract

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. Their spectral profiles describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers for the first time non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel's work on one dimensional intervals [arXiv: 1502.04404v1]. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.

Eigenvalue estimates for Fourier concentration operators on two domains

Abstract

We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. Their spectral profiles describe the number of prominent degrees of freedom in problems where functions are assumed to be supported on a certain domain and their Fourier transforms are known or measured on a second domain. We derive eigenvalue estimates that quantify the extent to which Fourier concentration operators deviate from orthogonal projectors, by bounding the number of eigenvalues that are away from 0 and 1 in terms of the geometry of the spatial and frequency domains, and a factor that grows at most poly-logarithmically on the inverse of the spectral margin. The estimates are non-asymptotic in the sense that they are applicable to concrete domains and spectral thresholds, and almost match asymptotic benchmarks. Our work covers for the first time non-convex and non-symmetric spatial and frequency concentration domains, as demanded by numerous applications that exploit the expected approximate low dimensionality of the modeled phenomena. The proofs build on Israel's work on one dimensional intervals [arXiv: 1502.04404v1]. The new ingredients are the use of redundant wave-packet expansions and a dyadic decomposition argument to obtain Schatten norm estimates for Hankel operators.
Paper Structure (23 sections, 14 theorems, 145 equations)

This paper contains 23 sections, 14 theorems, 145 equations.

Table of Contents

  1. Introduction and results
  2. The Euclidean space
  3. Discretization of continuous domains
  4. The discrete Fourier transform
  5. One sided estimates
  6. Significance
  7. Methods and related literature
  8. Preliminaries
  9. Notation
  10. Boundary regularity
  11. Israel's argument revisited
  12. Israel's lemma
  13. Local trigonometric frames
  14. Energy estimates
  15. General domain vs. rectangle
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $E,F\subseteq\mathbb{R}^d$, $d\geq 2$, be compact domains with maximally Ahlfors regular boundaries with constants $\kappa_{\partial E},\kappa_{\partial F}$ respectively, and assume that that $|\partial E||\partial F|\ge 1$. Consider the concentration operator eq_S and its eigenvalues $\{\lambda

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • ...and 17 more