Validity of WH-frame bound conditions depends on Lattice parameters

H.G. Feichtinger, A.J.E.M. Janssen

Research output: Contribution to journalArticleAcademicpeer-review

21 Citations (Scopus)

Abstract

In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time-frequency lattice used. Indeed, for any irrational a>0 we can construct a smooth g¿ L2(R) such that for any two rationals a >0 and b >0 the collection (gna, mb)n, m¿ Zof time-frequency translates of ghas a finite frame upper bound, while for any ß>0 and any rational c> 0 the collection (gnca, mß)n, m¿ Zhas no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gröchenig, that for any nonzero g¿ L2(R) which is sufficiently well behaved, there exist ac>0, bc>0 such that (gn a, m b)n, m¿ Zis a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a nonzero g¿ L2(R), bounded and supported by (0, 1), for which such numbers ac, bcdo not exist. In the first one of these examples, the frame bound equals 0 for all a >0, b >0, b <1. In the second example, the frame lower bound equals 0 for all aof the form l· 3- kwith l, k¿ N and all b, 0 < b <1, while the frame lower bound is at least 1 for all aof the form (2 m)- 1with m¿ N and all b, 0 < b <1.
Original languageEnglish
Pages (from-to)104-112
Number of pages9
JournalApplied and Computational Harmonic Analysis
Volume8
Issue number1
DOIs
Publication statusPublished - 2000

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Lattice constants
Lower bound
Upper bound
Theorem
Form

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@article{aebb975ebc3c4087ad0db2411fc71899,
title = "Validity of WH-frame bound conditions depends on Lattice parameters",
abstract = "In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time-frequency lattice used. Indeed, for any irrational a>0 we can construct a smooth g¿ L2(R) such that for any two rationals a >0 and b >0 the collection (gna, mb)n, m¿ Zof time-frequency translates of ghas a finite frame upper bound, while for any {\ss}>0 and any rational c> 0 the collection (gnca, m{\ss})n, m¿ Zhas no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gr{\"o}chenig, that for any nonzero g¿ L2(R) which is sufficiently well behaved, there exist ac>0, bc>0 such that (gn a, m b)n, m¿ Zis a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a nonzero g¿ L2(R), bounded and supported by (0, 1), for which such numbers ac, bcdo not exist. In the first one of these examples, the frame bound equals 0 for all a >0, b >0, b <1. In the second example, the frame lower bound equals 0 for all aof the form l· 3- kwith l, k¿ N and all b, 0 < b <1, while the frame lower bound is at least 1 for all aof the form (2 m)- 1with m¿ N and all b, 0 < b <1.",
author = "H.G. Feichtinger and A.J.E.M. Janssen",
year = "2000",
doi = "10.1006/acha.2000.0281",
language = "English",
volume = "8",
pages = "104--112",
journal = "Applied and Computational Harmonic Analysis",
issn = "1063-5203",
publisher = "Academic Press Inc.",
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}

Validity of WH-frame bound conditions depends on Lattice parameters. / Feichtinger, H.G.; Janssen, A.J.E.M.

In: Applied and Computational Harmonic Analysis, Vol. 8, No. 1, 2000, p. 104-112.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Validity of WH-frame bound conditions depends on Lattice parameters

AU - Feichtinger, H.G.

AU - Janssen, A.J.E.M.

PY - 2000

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N2 - In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time-frequency lattice used. Indeed, for any irrational a>0 we can construct a smooth g¿ L2(R) such that for any two rationals a >0 and b >0 the collection (gna, mb)n, m¿ Zof time-frequency translates of ghas a finite frame upper bound, while for any ß>0 and any rational c> 0 the collection (gnca, mß)n, m¿ Zhas no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gröchenig, that for any nonzero g¿ L2(R) which is sufficiently well behaved, there exist ac>0, bc>0 such that (gn a, m b)n, m¿ Zis a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a nonzero g¿ L2(R), bounded and supported by (0, 1), for which such numbers ac, bcdo not exist. In the first one of these examples, the frame bound equals 0 for all a >0, b >0, b <1. In the second example, the frame lower bound equals 0 for all aof the form l· 3- kwith l, k¿ N and all b, 0 < b <1, while the frame lower bound is at least 1 for all aof the form (2 m)- 1with m¿ N and all b, 0 < b <1.

AB - In the study of Weyl-Heisenberg frames the assumption of having a finite frame upper bound appears recurrently. In this article it is shown that it actually depends critically on the time-frequency lattice used. Indeed, for any irrational a>0 we can construct a smooth g¿ L2(R) such that for any two rationals a >0 and b >0 the collection (gna, mb)n, m¿ Zof time-frequency translates of ghas a finite frame upper bound, while for any ß>0 and any rational c> 0 the collection (gnca, mß)n, m¿ Zhas no such bound. It follows from a theorem of I. Daubechies, as well as from the general atomic theory developed by Feichtinger and Gröchenig, that for any nonzero g¿ L2(R) which is sufficiently well behaved, there exist ac>0, bc>0 such that (gn a, m b)n, m¿ Zis a frame whenever 0 < a < ac, 0 < b < bc. We present two examples of a nonzero g¿ L2(R), bounded and supported by (0, 1), for which such numbers ac, bcdo not exist. In the first one of these examples, the frame bound equals 0 for all a >0, b >0, b <1. In the second example, the frame lower bound equals 0 for all aof the form l· 3- kwith l, k¿ N and all b, 0 < b <1, while the frame lower bound is at least 1 for all aof the form (2 m)- 1with m¿ N and all b, 0 < b <1.

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SN - 1063-5203

IS - 1

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