Convergence of series of dilated functions and spectral norms of GCD matrices
Volume 168 / 2015
Acta Arithmetica 168 (2015), 221-246
MSC: 42A16, 42A20, 42A61, 42B05, 11A05, 15A18, 26A45.
DOI: 10.4064/aa168-3-2
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.
