A Paley-Wiener Type Theorem for Singular Measures on $\mathbb{T}$

TL;DR

This paper extends Paley-Wiener type theorems to singular measures on the unit circle, providing characterizations of Fourier transforms of functions in L^2 spaces with respect to such measures.

Contribution

It introduces new criteria for sampling and interpolation of Fourier transforms associated with singular measures on the circle, utilizing the Kaczmarz algorithm and Cauchy transform techniques.

Findings

Characterizations of Fourier transforms for L^2(μ) functions

Sampling criteria for entire functions as Fourier transforms

Interpolation criteria for bounded sequences by Fourier transforms

Abstract

For a fixed singular Borel probability measure $\mu$ on $\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(\mu)$. The first characterization is given in terms of criteria for sampling functions of the form $\hat{f}$ when $f \in L^2(\mu)$. The second characterization is given in terms of criteria for interpolation of bounded sequences on $\mathbb{N}_{0}$ by $\hat{f}$. Both characterizations use the construction of Fourier series for $f \in L^2(\mu)$ demonstrated in Herr and Weber via the Kaczmarz algorithm and classical results concerning the Cauchy transform of $\mu$.