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Frame-like Fourier expansions for finite Borel measures on $\mathbb{R}$

Chad Berner

Abstract

It is known that if a finite Borel measure $μ$ on $[0,1)$ possesses a frame of exponential functions for $L^{2}(μ)$, then $μ$ is of pure type. In this paper, we prove the existence of a class of finite Borel measures $μ$ on $[0,1)$ that are not of pure type that possess frame-like Fourier expansions for $L^{2}(μ)$. We also show properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for $L^{2}(μ)$ where $μ$ is a singular Borel probability measure on $\mathbb{R}$. Finally, we show measures $μ$ on $[0,1)$ that possess these frame-like Fourier expansions for $L^{2}(μ)$ have all $f\in L^{2}(μ)$ as $L^{2}(μ)$ limits of harmonic functions with frame-like coefficients. We also discuss when the inner products of these expansions coincide with model spaces and subspaces of harmonic functions on the disk.

Frame-like Fourier expansions for finite Borel measures on $\mathbb{R}$

Abstract

It is known that if a finite Borel measure on possesses a frame of exponential functions for , then is of pure type. In this paper, we prove the existence of a class of finite Borel measures on that are not of pure type that possess frame-like Fourier expansions for . We also show properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for where is a singular Borel probability measure on . Finally, we show measures on that possess these frame-like Fourier expansions for have all as limits of harmonic functions with frame-like coefficients. We also discuss when the inner products of these expansions coincide with model spaces and subspaces of harmonic functions on the disk.
Paper Structure (17 sections, 29 theorems, 80 equations)

This paper contains 17 sections, 29 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dextrodual observations
  4. Known results
  5. Measures on the torus with frame-like Fourier expansions
  6. Properties of measures with frame-like Fourier expansions
  7. Classes of measures with frame-like Fourier expansions
  8. A Fourier dextrodual example for the real line
  9. The case for the torus
  10. Two dimensional case
  11. Main result
  12. Subspaces of $h^{2}(\mathbb{D})$ generated by Bessel sequences.
  13. Correspondence with model spaces
  14. Examples
  15. Correspondence with subspaces of $h^{2}(\mathbb{D})$
  16. ...and 2 more sections

Key Result

Lemma 2.3

If a sequence $\{g_{n}\}$ is a Riesz basis in a Hilbert space $H$ and is dextrodual to sequence $\{x_{n}\}$, then $\{x_{n}\}$ is the canonical dual frame of $\{g_{n}\}$. In particular, $\{x_{n}\}$ is a Riesz basis.

Theorems & Definitions (59)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 2.5: He, Xing-Gang and Lai, Chun-Kit and Lau, Ka-Sing
  • Theorem 2.6: Herr and Weber
  • Theorem 2.7: Lai, Chun-Kit
  • Lemma 3.1
  • proof
  • ...and 49 more