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Weighted Beurling Density and Shift-Invariant Gabor Systems

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1914)

This chapter is concerned with the study of shift-invariant Gabor systems using the notion of weighted Beurling density. We introduce this new notion of Beurling density for collections of weighted sequences in ℝd and prove a useful reinterpretation for it. Then we derive a fundamental relationship for weighted Gabor frames with finitely many generators between the weighted Beurling density of the sequences of time-frequency indices, the frame bounds, and the norms of the generators. Finally, we study shift-invariant weighted Gabor systems and prove necessary density conditions for the sequences of time-frequency indices of a Gabor system and its shift-invariant counterpart.

Keywords

  • Weight Function
  • Weighted Sequence
  • Tight Frame
  • Gabor Frame
  • Fundamental Relationship

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© 2007 Springer-Verlag Berlin Heidelberg

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(2007). Weighted Beurling Density and Shift-Invariant Gabor Systems. In: Affine Density in Wavelet Analysis. Lecture Notes in Mathematics, vol 1914. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72949-5_7

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