Minimal Fourier majorants in $L^p$

TL;DR

This paper investigates minimal Fourier majorants in $L^p$ spaces for specific $p$, providing new insights into their structure and existence, especially for cases where explicit constructions are not previously known.

Contribution

The paper refines existence proofs for minimal Fourier majorants in $L^p$, revealing their structure and uniqueness for certain $p$ values where explicit forms were unknown.

Findings

Existence of minimal Fourier majorants in $L^p$ for specific $p$ values.

Uniqueness of these minimal majorants when $j > 1$.

Enhanced understanding of the form of these majorants.

Abstract

Denote the coefficients in the complex form of the Fourier series of a function $f$ on the interval $[-\pi, \pi)$ by $\hat f(n)$. It is known that if $p = 2j/(2j-1)$ for some integer $j>0$, then for each function $f$ in $L^p$ there exists another function $F$ in $L^p$ that majorizes $f$ in the sense that $\hat F(n) \ge |\hat f(n)|$ for all $n$, and for which $\|F\|_p \le \|f\|_p$. When $j > 1$, the existence proofs for such small majorants do not provide constructions of them, but there is a unique majorant of minimal $L^p$ norm. We modify previous existence proofs to say more about the form of that majorant.