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Sampling on infinite-dimensional Paley-Wiener spaces on graphs

Filippo Giannoni

Abstract

We prove a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs which allows for stable frame reconstruction. We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where a Poincaré-type inequality holds), thereby providing a sufficient condition for stable sampling and reconstruction on graphs such as $\mathbb{Z}^n$-lattices and radial trees with finite geometry.

Sampling on infinite-dimensional Paley-Wiener spaces on graphs

Abstract

We prove a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs which allows for stable frame reconstruction. We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where a Poincaré-type inequality holds), thereby providing a sufficient condition for stable sampling and reconstruction on graphs such as -lattices and radial trees with finite geometry.

Paper Structure

This paper contains 5 sections, 7 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Overview and motivations
  3. Results in the continuous case
  4. Pesenson's approach in PW1 and main results
  5. Sampling theorems for infinite-dimensional Paley-Wiener spaces

Key Result

Theorem 2.1

(Shannon). If $f \in PW_\omega(\mathbb{R})$, then where convergence is understood in the $L^2$-norm. Moreover, if we consider the entire class representative $F$ of $f$, we have and the series converges uniformly.

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Lemma 2.7
  • Theorem 2.8
  • Theorem 3.1
  • proof
  • ...and 2 more