Fourier majorants that match norms

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 $\Vert F{\Vert }_p\le \Vert f{\Vert }_p$. 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 $L^2$.

13 pages