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Entropy of Compact Operators with Applications to Landau-Pollak-Slepian Theory and Sobolev Spaces

Thomas Allard, Helmut Bölcskei

TL;DR

The paper establishes a sharp connection between the metric entropy of a compact operator and the asymptotic behavior of its eigenvalues, enabling precise two-term expansions for $H(\varepsilon; T)$ and $\varepsilon_m(T)$ under polynomial decay of eigenvalues. By viewing the operator image of the unit ball as an ellipsoid with semi-axes given by $\{\lambda_n\}$, the authors obtain first- and second-order entropy terms that tighten classical bounds and extend to an eigenvalue-counting framework via $M_T(\gamma)$. These results are then applied to key problems in analysis: the entropy rate of the Landau-Pollak-Slepian operator is characterized exactly in the large-scale limit, and the metric entropy of unit balls in Sobolev spaces is determined to first order with a sharp second-order correction under mild regularity. The combination of ellipsoid coverings, Weyl laws, and eigenvalue counting yields both improved non-asymptotic statements and concrete constants, with implications for sampling theory and function-space complexity.

Abstract

We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known characterizations of the entropy of the Landau-Pollak-Slepian operator and the metric entropy of unit balls in Sobolev spaces.

Entropy of Compact Operators with Applications to Landau-Pollak-Slepian Theory and Sobolev Spaces

TL;DR

The paper establishes a sharp connection between the metric entropy of a compact operator and the asymptotic behavior of its eigenvalues, enabling precise two-term expansions for and under polynomial decay of eigenvalues. By viewing the operator image of the unit ball as an ellipsoid with semi-axes given by , the authors obtain first- and second-order entropy terms that tighten classical bounds and extend to an eigenvalue-counting framework via . These results are then applied to key problems in analysis: the entropy rate of the Landau-Pollak-Slepian operator is characterized exactly in the large-scale limit, and the metric entropy of unit balls in Sobolev spaces is determined to first order with a sharp second-order correction under mild regularity. The combination of ellipsoid coverings, Weyl laws, and eigenvalue counting yields both improved non-asymptotic statements and concrete constants, with implications for sampling theory and function-space complexity.

Abstract

We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known characterizations of the entropy of the Landau-Pollak-Slepian operator and the metric entropy of unit balls in Sobolev spaces.
Paper Structure (6 sections, 10 theorems, 81 equations, 1 figure)

This paper contains 6 sections, 10 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Entropy of compact operators
  4. Applications
  5. Entropy of the Landau-Pollak-Slepian operator
  6. Metric entropy of unit balls in Sobolev spaces

Key Result

Theorem 3

Let $\mathcal{H}$ be a separable real Hilbert space and let $T\colon \mathcal{H}\to \mathcal{H}$ be a positive self-adjoint compact operator with eigenvalues $\{\lambda_n\}_{n\in\mathbb{N}^*}$ satisfying where $c_1\in\mathbb{R}_+^*$, $c_2\in\mathbb{R}$, and $\alpha_1, \alpha_2 \in\mathbb{R}_+^*$ are such that Then, the entropy of $T$ satisfies which can equivalently be expressed in terms of ent

Figures (1)

  • Figure 1: Eigenvalue distribution of the Landau-Pollak-Slepian operator.

Theorems & Definitions (22)

  • Definition 1: Metric entropy and entropy numbers of sets
  • Definition 2: Entropy and entropy numbers of compact linear operators
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • ...and 12 more