On convergence and growth of sums $\sum c_k f(kx)$

TL;DR

This paper investigates the conditions under which series of the form (n_k x) converge almost everywhere, establishing new convergence criteria involving logarithmic factors and providing bounds on their growth for functions with bounded variation.

Contribution

It introduces a new convergence criterion based on the decay of coefficients involving _k^2 (\u03bb ext)^ for functions with bounded variation, and constructs examples showing the sharpness of these conditions.

Findings

Series converge a.e. if _k^2 ( (\u03bb)^< for >4.

Constructed example shows condition is not sufficient for <2.

Provides bounds on growth of partial sums differing from classical results by a loglog factor.

Abstract

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^\gamma <\infty$, $\gamma>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k f(n_kx)$ for any increasing sequence $(n_k)$ of integers. We also construct an example showing that the previous condition is not sufficient for $\gamma<2$. Finally we give an a.e. bound for the growth of sums $\sum_{k=1}^N f(n_kx)$ differing from the corresponding optimal result for trigonometric sums by a loglog factor.