Orthonormal Sequences in L
2(R
d
) and Time Frequency Localization |
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| Authors: | Eugenia Malinnikova |
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| Institution: | 1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway
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| Abstract: | We prove that there does not exist an orthonormal basis {b
n
} for L
2(R) such that the sequences {μ(b
n
)},
{m(^(bn)])}\{\mu(\widehat{b_{n}})\}
, and
{D(bn)D(^(bn)])}\{\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
)},
{m(^(bn)])}\{\mu(\widehat{b_{n}})\}
, and
{D(bn)D(^(bn)])}\{\Delta(b_{n})\Delta(\widehat{b_{n}})\}
are bounded. |
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| Keywords: | |
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