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Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space

Karlheinz Gröchenig, Joaquim Ortega-Cerdà

Abstract

We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted $L^2$-space and sampling theorems for entire functions in the Fock space and the dual relation between uniform interpolating families for polynomials and interpolating sequences. As a consequence we obtain a description of signal subspaces spanned by Hermite functions by means of Gabor frames.

Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space

Abstract

We study the relation between Marcinkiewicz-Zygmund families for polynomials in a weighted -space and sampling theorems for entire functions in the Fock space and the dual relation between uniform interpolating families for polynomials and interpolating sequences. As a consequence we obtain a description of signal subspaces spanned by Hermite functions by means of Gabor frames.

Paper Structure

This paper contains 8 sections, 17 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Fock space
  3. Asymptotics of the incomplete Gamma function
  4. Sampling implies Marcinkie-wicz--Zyg-mund inequalities
  5. Marcinkie-wicz--Zyg-mund inequalities imply sampling
  6. Uniform Interpolation
  7. Gabor frames for subspaces spanned by Hermite functions
  8. Appendix

Key Result

Theorem 1.1

(i) Assume that $\Lambda \subseteq \mathbb{C}$ is a sampling set for $\mathcal{ F} ^2$. For $\tau >0$ set $\rho _n$ such that $\pi \rho _n^2 = n + \sqrt{n} \tau$ and let $B_{\rho _n}$ be the centered disk of radius $\rho _n$. Then for $\tau >0$ large enough, the sets $\Lambda _n = \Lambda \cap B_{\

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 21 more