Fourier Series for Singular Measures

TL;DR

This paper establishes Fourier series representations for functions in L^2 spaces with respect to singular measures using the Kaczmarz algorithm, and introduces a Shannon-type sampling theorem for μ-bandlimited functions.

Contribution

It extends Fourier series theory to singular measures and provides explicit coefficient computation methods, along with a novel sampling theorem for μ-bandlimited functions.

Findings

Fourier series exist for all L^2(μ) functions with singular measures.

Coefficients can be explicitly computed from μ-Fourier transforms.

A Shannon-type sampling theorem is established for μ-bandlimited functions.

Abstract

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu$ on $[0,1)$, every $f\in L^2(\mu)$ possesses a Fourier series of the form $f(x)=\sum_{n=0}^{\infty}c_ne^{2\pi inx}$. We show that the coefficients $c_{n}$ can be computed in terms of the quantities $\hat{f}(n) = \int_{0}^{1} f(x) e^{-2\pi i n x} d \mu(x)$. We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $\mu$-bandlimited.