Journal of Fourier Analysis and Applications

, Volume 16, Issue 6, pp 983–1006 | Cite as

Orthonormal Sequences in L 2(R d ) and Time Frequency Localization

  • Eugenia Malinnikova


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.

Uncertainty principle Orthonormal basis Time frequency localization 

Mathematics Subject Classification (2000)

42B10 42C25 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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