Algorithmic randomness in harmonic analysis

TL;DR

This paper explores the connection between algorithmic randomness and harmonic analysis, showing that certain convergence properties of Fourier series and Poisson integrals correspond precisely to Martin-Löf and Schnorr randomness notions.

Contribution

It establishes new characterizations of algorithmic randomness in the context of harmonic analysis, linking convergence of Fourier series and Poisson integrals to specific randomness notions.

Findings

Fourier series convergence at Martin-Löf random reals for computable functions.

Radial limits of Poisson integrals match function values at Schnorr and Martin-Löf random reals.

Provides effective versions of classical harmonic analysis theorems related to randomness.

Abstract

Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute characterized Schnorr randomness using an effective version of Carleson's Theorem. We show here that, for computable $1<p<\infty$, the reals at which the Fourier series of a weakly computable vector in $L^p[-\pi,\pi]$ converges are precisely the Martin-L\"{o}f random reals. Furthermore, we show that radial limits of the Poisson integral of an $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Schnorr random reals and that radial limits of the Poisson integral of a weakly $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Martin-L\"{o}f random reals.