On series of dilated functions

TL;DR

This paper investigates the convergence properties of series involving dilated functions, using orthogonal function theory and GCD sums to derive new results and unify existing theories.

Contribution

It introduces new methods connecting orthogonal functions and number theory to analyze convergence of dilated function series, improving and simplifying prior results.

Findings

Established new convergence criteria for series of dilated functions.

Unified various existing results into a coherent theoretical framework.

Enhanced understanding of the interplay between analytic and number theoretic factors.

Abstract

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number theoretic factors, but precise results exist only in a few special cases. In this paper we use connections with orthogonal function theory and GCD sums to prove several new results, improve old ones and also to simplify and unify the theory.