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 $\|F\|_p \le \|f\|_p$. 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 $L^2$.