Orthonormal sequences in $L^2(R^d)$ and time frequency localization

Details Orthonormal sequences in $L^2(R^d)$ and time frequency localization Orthonormal sequences in $L^2(R^d)$ and time frequency localization

2009-03-23

arxiv.org/abs/0903.3763v1

J. Fourier Anal. Appl., 16 (2010), no 6, 983-1006

Mathematics

Classical Analysis and ODEs

18 pages

Scientific paper

10.1007/s00041-009-9114-9

We study uncertainty principles for orthonormal bases and sequences in $L^2(\R^d)$. As in the classical Heisenberg inequality we focus on the product of the dispersions of a function and its Fourier transform. In particular we prove that there is no orthonormal basis for $L^2(\R)$ for which the time and frequency means as well as the product of dispersions are uniformly bounded. The problem is related to recent results of J. Benedetto, A. Powell, and Ph. Jaming. Our main tool is a time frequency localization inequality for orthonormal sequences in $L^2(\R^d)$. It has various other applications.

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