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 $min mathbb{Z}_{+}$, find [ sup{|x|colon (-1)^{m}f(x)>0}cdot sup {|x|colon xin 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 $mge 1$. We also obtain the uncertainty principle for bandlimited functions related to the recent result by Bourgain, Clozel, and Kahane.