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An uncertainty inequality involving norms

Authors:	Enrico Laeng Carlo Morpurgo

Institution:	Dipartimento di Matematica, Politecnico di Milano, 20133 Milano, Italy ; Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milano, Italy

Abstract:	We derive a sharp uncertainty inequality of the form $\begin{equation}\\|x^{2} f\\|_{1}^{} \,\\|\xi \,\hat {f}\\|_{2}^{2}\ge {\frac{\Lambda _{0}}{4\pi ^{2}}}\, \\|f\\|_{1}^{}\,\\|f\\|_{2}^{2},\end{equation}$ with $\Lambda _{0}=0.428368\dots$ . As a consequence of this inequality we derive an upper bound for the so-called Laue constant, that is, the infimum $\lambda _{0}^{}$ of the functional $\lambda (p)=4\pi ^{2} \\|x^{2} p\\|_{1}^{}\\|x^{2} \hat p\\|_{1}^{}/(p(0)\hat p(0))$ , taken over all $p\ge 0$ with $\hat p\ge 0$ ( $p\not \equiv 0$ ). Precisely, we obtain that $\lambda _{0}^{}\le 2\Lambda _{0}=0.85673673\dots ,$ which improves a previous bound of T. Gneiting.

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