  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.
