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 bounded sequences. We also show that there does not exist any orthonormal basis for \(L^2 (\mathbb{R})$ with $\{ \Delta^2 (b_n)\}_{n=1}^{\infty}\), \(\{ \Delta^2(\widehat{b_n})\}_{n=1}^{\infty}\) 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 variance sequences associated to orthonormal bases.
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Powell, A. Time-Frequency Mean and Variance Sequences of Orthonormal Bases. J Fourier Anal Appl 11, 375–387 (2005). https://doi.org/10.1007/s00041-005-3082-5
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DOI: https://doi.org/10.1007/s00041-005-3082-5