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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An uncertainty inequality involving $L^1$ norms
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by Enrico Laeng and Carlo Morpurgo PDF
Proc. Amer. Math. Soc. 127 (1999), 3565-3572 Request permission

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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Additional Information
  • Enrico Laeng
  • Affiliation: Dipartimento di Matematica, Politecnico di Milano, 20133 Milano, Italy
  • MR Author ID: 295007
  • Email: enrlae@mate.polimi.it
  • Carlo Morpurgo
  • Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milano, Italy
  • Email: morpurgo@dsdipa.mat.unimi.it
  • Received by editor(s): February 13, 1998
  • Published electronically: May 17, 1999
  • Additional Notes: The second author was partially supported by NSF grant DMS-9622891.
  • Communicated by: Christopher D. Sogge
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 3565-3572
  • MSC (1991): Primary 26D15, 42A82
  • DOI: https://doi.org/10.1090/S0002-9939-99-05022-4
  • MathSciNet review: 1621969