A constrained optimization problem for the Fourier transform: Quantitative analysis

TL;DR

This paper investigates which functions bounded by indicator functions maximize the ratio of their Fourier transform norm to the measure of the set, providing a quantitative characterization for certain exponents.

Contribution

It offers a quantitative analysis of maximizers for the Fourier transform ratio among functions majorized by indicator functions, extending previous existence results.

Findings

Identifies conditions under which maximizers exist for specific exponents.

Provides a quantitative description of these maximizers.

Extends understanding of Fourier transform optimization problems.

Abstract

Among functions $f$ majorized by indicator functions $1_E$, which functions have maximal ratio $\|\widehat{f}\|_q/|E|^{1/p}$? We establish a quantitative answer to this question for exponents $q$ sufficiently close to even integers, building on previous work proving the existence of such maximizers.