The divergence of Mock Fourier series for spectral measures

TL;DR

This paper investigates divergence phenomena of Mock Fourier series on Cantor-type fractal measures, providing conditions for divergence and an example where convergence fails almost everywhere.

Contribution

It establishes a sufficient condition for divergence of Mock Fourier series on spectral measures and presents an explicit example with non-convergent sums.

Findings

Mock Fourier series can diverge on non-zero sets for certain spectral measures

A specific quarter Cantor measure example shows non-almost everywhere convergence

Provides criteria for divergence of Fourier series on fractal measures

Abstract

In this paper, we study divergence properties of Fourier series on Cantor-type fractal measure, also called Mock Fourier series. We give a sufficient condition under which the Mock Fourier series for doubling spectral measure is divergent on non-zero set. In particularly, there exists an example of the quarter Cantor measure whose Mock Fourier sums is not almost everywhere convergent.