Time-Frequency Mean and Variance Sequences of Orthonormal Bases

Abstract

We show that there exists an orthonormal basis {bWe also show that there does not exist any orthonormal basis forbeing bounded sequences. This is motivated by a question posed by H.S. Shapiro on the mean and variance sequences associated to orthonormal bases.

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

“…The results complement those in [13,14,18]; our approach is simple and works in R d for any d. We consider the operator that first time-limits the function and then frequency-limits it, following [20]. However we don't need the theory of Prolate Spheroidal Wave Functions and the celebrated 2W T approximation theorem that was used in [14].…”

“…This result was generalized recently by J. Benedetto and A. Powell [2]. The technique was also used by A. Powell to construct orthonormal bases with other properties, see [18]. The result of J. Bourgain implies that for each > 0 there is an orthonormal basis such that sup n (b n ) ( b n ) < 1 4π + , so inequality (1) can not be improved for an orthonormal basis.…”

“…A number of results on time-frequency localization of orthonormal sequences and bases have been obtained by J. Benedetto in [1] and A. Powell in [18]. In particular, another example of a condition on means and dispersions which can be satisfied by an infinite orthonormal sequence but never by an orthonormal basis is due to A. Powell.…”

“…In particular, another example of a condition on means and dispersions which can be satisfied by an infinite orthonormal sequence but never by an orthonormal basis is due to A. Powell. It is proved in [18] that there is no orthonormal basis with bounded (both) dispersions and bounded time means. Theorem 1 can be derived from the Mean Dispersion inequality.…”

Phase space localization of orthonormal sequences in Lα2(R+)

“…Some other results on time-frequency localization of orthonormal sequences and bases have been obtained by Benedetto [2] and Powell [18] and the quantitative version of Shapiro's result has been proved by Jaming and Powell [14] which states that, if { f n } +∞ n=0 is an orthonormal sequence in L 2 (R) then for all N ≥ 0:…”