Fourier coefficients of continuous functions with sparse spectrum

Abstract

Let $(r_k)$ be an increasing sequence and $(w_k)$ a positive sequence. We study the following question: is it true that for every sequence $(a_k)$ satisfying ${\sum }_{k=0}^{\infty }|a_k{|}^2{w}_k^2<\infty$ there exists a function $f\in C(\mathbb{T})$ such that $\hat{f}(2^k)=a_k$ and $\hat{f}(n)=0$ for $n\notin {\cup }_k[2^k-r_k,2^k+r_k]$? We show that this is possible if and only if ${\sup }_{k\in \mathbb{N}}{\sum }_{n=[{\log }_2{r}_k]}^k{w}_k^{-2}<\infty$.