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Time-Frequency Mean and Variance Sequences of Orthonormal Bases

Authors:	Alexander M. Powell

Affiliation:	(1) Department of Mathematics, Vanderbilt University Nashville TN, 37240, USA

Abstract:	We show that there exists an orthonormal basis ${b_n}_{n=1}^{infty}$ for $L^2 (mathbb{R})$ such that ${ Delta^2 (b_n)}_{n=1}^{infty}, { mu( b_n ) }_{n=1}^{infty}$ and ${ mu(widehat{b_n})}_{n=1}^{infty}$ are boundedsequences. We also show that there does not exist any orthonormal basisfor $L^2 (mathbb{R})$ with ${ Delta^2 (b_n)}_{n=1}^{infty}$ , ${ Delta^2(widehat{b_n})}_{n=1}^{infty}$ and ${ mu( b_n ) }_{n=1}^{infty}$ being bounded sequences.This is motivated by a question posed by H.S. Shapiro on the mean and variancesequences associated to orthonormal bases.

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