Approximations in $L^1$ with convergent Fourier series

TL;DR

This paper demonstrates that for any integrable function on a measure space, one can find a nearly full subset where the function can be approximated by another function with a Fourier series that converges, extending classical Fourier analysis results.

Contribution

It introduces a universal subset of the measure space where functions can be approximated with convergent Fourier series, applicable to homogeneous spaces and spheres.

Findings

Existence of a universal subset with small complement for approximation

Construction of approximants with convergent Fourier series

Extension to homogeneous spaces and spheres

Abstract

For a separable finite diffuse measure space $\mathcal{M}$ and an orthonormal basis $\{\varphi_n\}$ of $L^2(\mathcal{M})$ consisting of bounded functions $\varphi_n\in L^\infty(\mathcal{M})$, we find a measurable subset $E\subset\mathcal{M}$ of arbitrarily small complement $|\mathcal{M}\setminus E|<\epsilon$, such that every measurable function $f\in L^1(\mathcal{M})$ has an approximant $g\in L^1(\mathcal{M})$ with $g=f$ on $E$ and the Fourier series of $g$ converges to $g$, and a few further properties. The subset $E$ is universal in the sense that it does not depend on the function $f$ to be approximated. Further in the paper this result is adapted to the case of $\mathcal{M}=G/H$ being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of $n$-spheres with spherical harmonics is discussed. The construction of the subset $E$ and approximant $g$ is sketched briefly at the end of the paper.