Fourier majorants that match norms

TL;DR

This paper explores Fourier majorants in $L^p$ spaces, demonstrating how to construct functions that dominate given Fourier coefficients while maintaining the same $L^p$ norm, extending the concept of exact majorants from $L^2$ to other spaces.

Contribution

It introduces a method to derive Fourier majorants in $L^p$ spaces from $L^{2j}$ variants of exact majorants, broadening the understanding of Fourier coefficient domination.

Findings

Existence of Fourier majorants in specific $L^p$ spaces.

Construction of majorants with preserved $L^p$ norms.

Extension of the concept of exact majorants from $L^2$ to other $L^p$ spaces.

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 $n$, but that also satisfies $\|F\|_p \le \|f\|_p$. Rescaling $F$ suitably then gives a majorant with the same $L^p$ norm as $f$. We show how that majorant comes from a variant in $L^{2j}$ of the notion of exact majorant in $L^2$.