Abstract
We prove that there does not exist an orthonormal basis {b n } for L 2(R) such that the sequences {μ(b n )}, \(\{\mu(\widehat{b_{n}})\}\) , and \(\{\Delta(b_{n})\Delta(\widehat{b_{n}})\}\) are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main tool is a time-frequency localization inequality for orthonormal sequences in L 2(R d). On the other hand, for d>1 we construct a basis {b n } for L 2(R d) such that the sequences {μ(b n )}, \(\{\mu(\widehat{b_{n}})\}\) , and \(\{\Delta(b_{n})\Delta(\widehat{b_{n}})\}\) are bounded.
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Communicated by John J. Benedetto.
The author is supported by the Research Council of Norway, grants 160192/V30 and 177355/V30.
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Malinnikova, E. Orthonormal Sequences in L 2(R d) and Time Frequency Localization. J Fourier Anal Appl 16, 983–1006 (2010). https://doi.org/10.1007/s00041-009-9114-9
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DOI: https://doi.org/10.1007/s00041-009-9114-9