Orthonormal Sequences in L2(Rd) and Time Frequency Localization

By Eugenia Malinnikova. 2009

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AbstractWe prove that there does not exist an orthonormal basis {bn} for L2(R) such that the sequences {μ(bn)}, $\{\mu(\widehat{b_{n}})\}$ , 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 L2(Rd). On the other hand, for d>1 we construct a basis {bn} for L2(Rd) such that the sequences {μ(bn)}, $\{\mu(\widehat{b_{n}})\}$ , and $\{\Delta(b_{n})\Delta(\widehat{b_{n}})\}$ are bounded.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/2a5ec6de544b010638428308af06559cd0bacbad