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arXiv · 1904.11328

Uncertainty principles for eventually constant sign bandlimited functions

D. V. Gorbachev·V. I. Ivanov·S. Yu. Tikhonov
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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.

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D. V. Gorbachev, V. I. Ivanov, S. Yu. Tikhonov. 2019-04-25. Uncertainty principles for eventually constant sign bandlimited functions. https://arxiv.org/abs/1904.11328

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