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On Dual Gabor Frame Pairs Generated by Polynomials

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We provide explicit constructions of particularly convenient dual pairs of Gabor frames. We prove that arbitrary polynomials restricted to sufficiently large intervals will generate Gabor frames, at least for small modulation parameters. Unfortunately, no similar function can generate a dual Gabor frame, but we prove that almost any such frame has a dual generated by a B-spline. Finally, for frames generated by any compactly supported function φ whose integer-translates form a partition of unity, e.g., a B-spline, we construct a class of dual frame generators, formed by linear combinations of translates of φ. This allows us to chose a dual generator with special properties, for example, the one with shortest support, or a symmetric one in case the frame itself is generated by a symmetric function. One of these dual generators has the property of being constant on the support of the frame generator.

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Correspondence to Rae Young Kim.

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Communicated by Akram Aldroubi.

This research was supported by the Yeungnam University research grants in 2007.

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Christensen, O., Kim, R.Y. On Dual Gabor Frame Pairs Generated by Polynomials. J Fourier Anal Appl 16, 1–16 (2010).

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Mathematics Subject Classification (2000)