Uncertainty principles for eventually constant sign bandlimited functions

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 $$\sup \{\mathrm{\mid }x\mathrm{\mid }\text{\hspace{0.17em}⁣}:(-1{)}^{m}f(x)>0\}\cdot \sup \{\mathrm{\mid }x\mathrm{\mid }\text{\hspace{0.17em}⁣}:x\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\widehat{f}\}\to \inf ,$$ where the infimum is taken over all nontrivial positive definite bandlimited functions such that ${\int }_{{\mathbb{R}}^d}\mathrm{\mid }x{\mathrm{\mid }}^{2k}f(x)dx=0$ 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.