A constrained optimization problem for the Fourier transform: Existence

TL;DR

This paper investigates the existence of functions with maximal Fourier transform norms among those bounded by indicator functions of measure-one sets, using additive combinatorics to establish partial results on maximizing sequences.

Contribution

It introduces a new approach employing additive combinatorics to prove partial existence results for extremal functions in Fourier analysis.

Findings

Partial proof of existence of extremal functions

Use of additive combinatorics techniques

Conditional precompactness of maximizing sequences

Abstract

Among functions majorized by indicator functions of sets with measure one, which functions have maximal Fourier transforms in the $L^q$ norm? We partially prove the existence of such functions using techniques from additive combinatorics to establish a conditional precompactness for maximizing sequences.