An uncertainty inequality involving $L^1$ norms
HTML articles powered by AMS MathViewer
- 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.References
- William Beckner, Geometric inequalities in Fourier anaylsis, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991) Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 36–68. MR 1315541
- Eric A. Carlen and Michael Loss, Sharp constant in Nash’s inequality, Internat. Math. Res. Notices 7 (1993), 213–215. MR 1230297, DOI 10.1155/S1073792893000224
- I. Dreier, On the uncertainty principle for positive definite densities, Z. Anal. Anwendungen 15 (1996), no. 4, 1015–1023. MR 1422654, DOI 10.4171/ZAA/743
- Gerald B. Folland and Alladi Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238. MR 1448337, DOI 10.1007/BF02649110
- T. Gneiting , On the uncertainty relation for positive definite probability densities, to appear in Statistics.
- Elliott H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), no. 2, 349–374. MR 717827, DOI 10.2307/2007032
- Elliott H. Lieb and Michael Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997. MR 1415616, DOI 10.2307/3621022
- D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and its Applications (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1190927, DOI 10.1007/978-94-011-3562-7
- Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
- H.-J. Rossberg, Positive definite probability densities and probability distributions, J. Math. Sci. 76 (1995), no. 1, 2181–2197. MR 1356657, DOI 10.1007/BF02363232
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