Some counterexamples in the theory of Weyl-Heisenberg frames

A.J.E.M. Janssen

doi:10.1109/18.485730

Some counterexamples in the theory of Weyl-Heisenberg frames

A.J.E.M. Janssen

Control Systems

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Abstract

We present an example of a positive function g with a positive Fourier transform <i and reasonable smoothness and decay properties such that (-1)^nmexp(p'tm)g(t-n), n,m eZ does not constitute a frame for L2(R). We also give counterexamples for the statement that one can tell (in)definiteness of a Weyl-Heisenberg frame operator from (in)deflniteness of its Weyl symbol. Index Terms-Weyl-Heisenberg frame, Zak transform, Weyl symbol.

Original language	English
Pages (from-to)	621-623
Number of pages	3
Journal	IEEE Transactions on Information Theory
Volume	42
Issue number	2
DOIs	https://doi.org/10.1109/18.485730
Publication status	Published - 1996

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title = "Some counterexamples in the theory of Weyl-Heisenberg frames",

abstract = "We present an example of a positive function g with a positive Fourier transform nmexp(p'tm)g(t-n), n,m eZ does not constitute a frame for L2(R). We also give counterexamples for the statement that one can tell (in)definiteness of a Weyl-Heisenberg frame operator from (in)deflniteness of its Weyl symbol. Index Terms-Weyl-Heisenberg frame, Zak transform, Weyl symbol.",

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AB - We present an example of a positive function g with a positive Fourier transform nmexp(p'tm)g(t-n), n,m eZ does not constitute a frame for L2(R). We also give counterexamples for the statement that one can tell (in)definiteness of a Weyl-Heisenberg frame operator from (in)deflniteness of its Weyl symbol. Index Terms-Weyl-Heisenberg frame, Zak transform, Weyl symbol.

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JF - IEEE Transactions on Information Theory

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Some counterexamples in the theory of Weyl-Heisenberg frames

Abstract

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