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Oversampling and Donoho-Logan type theorems in model spaces

Anton Baranov, Philippe Jaming, Karim Kellay, Michael Speckbacher

Abstract

The aim of this paper is to extend two results from the Paley--Wiener setting to more generalmodel spaces. The first one is an analogue of the oversampling Shannon sampling formula. The second one is a version of the Donoho--Logan Large Sieve Theorem which is a quantitative estimate of the embedding of the Paley--Wiener space into an $L^2(\R,μ)$ space.

Oversampling and Donoho-Logan type theorems in model spaces

Abstract

The aim of this paper is to extend two results from the Paley--Wiener setting to more generalmodel spaces. The first one is an analogue of the oversampling Shannon sampling formula. The second one is a version of the Donoho--Logan Large Sieve Theorem which is a quantitative estimate of the embedding of the Paley--Wiener space into an space.
Paper Structure (8 sections, 9 theorems, 87 equations)

This paper contains 8 sections, 9 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemma
  3. Oversampling and the Large Sieve in model spaces
  4. Background on model spaces
  5. Enlarging model spaces and oversampling
  6. Proof of Theorem \ref{['th:intro2']} based on oversampling
  7. An $L^p$-version of Theorem \ref{['th:intro2']} and Bernstein-type inequalities
  8. A Donoho--Logan theorem for one component inner functions

Key Result

Theorem 1.1

Let $c,\delta>0$ and $\mu$ be a positive $\sigma$-finite measure. Then for every $f\in PW_c^2({\mathbb{R}})$, Moreover, for every $f\in PW_c^1({\mathbb{R}})$,

Theorems & Definitions (18)

  • Theorem 1.1: Donoho--Logan
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 4.1
  • Theorem 5.1
  • ...and 8 more