Convergence of series of dilated functions and spectral norms of GCD matrices

TL;DR

This paper links the convergence of series of dilated functions with specific Fourier decay rates to the spectral norms of GCD matrices, providing sharp conditions for convergence in $L^2$ and almost everywhere.

Contribution

It introduces a novel connection between Fourier coefficient decay, GCD matrix spectral norms, and convergence criteria for dilated function series.

Findings

Established bounds for spectral norms of GCD matrices.

Derived sharp $L^2$ convergence conditions.

Provided criteria for almost everywhere convergence.

Abstract

We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-\alpha})$ for some $\alpha \in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.