Fourier coefficients of continuous functions with sparse spectrum

TL;DR

This paper investigates the conditions under which a continuous function on the circle can have Fourier coefficients with a sparse spectrum, characterized by specific support and weighted square-summability.

Contribution

It establishes a necessary and sufficient condition relating the sparsity pattern and weights for the existence of such functions.

Findings

The existence of the function depends on a boundedness condition involving the weights and the sparsity parameters.

The paper characterizes precisely when Fourier coefficients with sparse support can be realized by continuous functions.

Provides a criterion linking the growth of the sparsity sequence and the weight sequence.

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 $\sup_{k\in\mathbb{N}}\sum_{n=[\log_2 r_k]}^k w_k^{-2} < \infty$.