Perturbing PLA

TL;DR

This paper demonstrates the limits of perturbation smoothness for representing measurable functions on the circle as power series, showing the necessity of non-smooth perturbations and exploring lacunary spectrum cases.

Contribution

It establishes the sharpness of previous results by proving that perturbations cannot be smooth or H"older continuous, and discusses related lacunary spectrum problems.

Findings

Perturbations cannot be made smooth or H"older for the series representation.

The result is essentially sharp, indicating the optimality of earlier theorems.

Explores similar issues for lacunary spectrum perturbations.

Abstract

We proved earlier that every measurable function on the circle, after a uniformly small perturbation, can be written as a power series (i.e. a series of exponentials with positive frequencies), which converges almost everywhere. Here we show that this result is basically sharp: the perturbation cannot be made smooth or even H\"older. We discuss also a similar problem for perturbations with lacunary spectrum.