Uncertainty principles for eventually constant sign bandlimited functions

Abstract

We study the uncertainty principles related to the generalized Logan problem in ${\mathbb{R}}^d$. Our main result provides the complete solution of the following problem: for a fixed $m\in {\mathbb{Z}}_{+}$, find \[ \sup\{|x|\colon (-1)^{m}f(x)>0\}\cdot \sup \{|x|\colon x\in \mathrm{supp}\,\widehat{f}\,\}\to \inf, \] where the infimum is taken over all nontrivial positive definite bandlimited functions such that ${\int }_{{\mathbb{R}}^d}|x{|}^{2k}f(x)dx=0$ for $k=0,\dots ,m-1$ if $m\ge 1$. We also obtain the uncertainty principle for bandlimited functions related to the recent result by Bourgain, Clozel, and Kahane.

30 pages