A class of extremal functions for the Fourier transform

Abstract

We determine a class of real valued, integrable functions $f(x)$ and corresponding functions ${M_f}(x)$ such that $f(x) \leqslant {M_f}(x)$ for all $x$, the Fourier transform ${\hat M_f}(t)$ is zero when $\left | t \right | \geqslant 1$, and the value of ${\hat M_f}(0)$ is minimized. Several applications of these functions to number theory and analysis are given.

Locations

• Transactions of the American Mathematical Society - View - PDF