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Tight Gabor Sets on Discrete Periodic Sets

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Abstract

This paper addresses Gabor analysis on a discrete periodic set. Such a scenario can potentially find its applications in signal processing where signals may present on a union of disconnected discrete index sets. We focus on the Gabor systems generated by characteristic functions. A sufficient and necessary condition for a set to be a tight Gabor set in discrete periodic sets is obtained; discrete periodic sets admitting a tight Gabor set are also characterized; the perturbation of tight Gabor sets is investigated; an algorithm to determine whether a set is a tight Gabor set is presented. Furthermore, we prove that an arbitrary Gabor frame set can be represented as the union of a tight Gabor set and a Gabor Bessel set.

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Correspondence to Qiao-Fang Lian.

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Supported by the National Natural Science Foundation of China (Grant No. 10671008), PHR (IHLB), the Project-sponsored by SRF for ROCS, SEM of China.

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Li, YZ., Lian, QF. Tight Gabor Sets on Discrete Periodic Sets. Acta Appl Math 107, 105–119 (2009). https://doi.org/10.1007/s10440-008-9413-1

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  • DOI: https://doi.org/10.1007/s10440-008-9413-1

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