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Inversion formula for the windowed Fourier transform, II

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Abstract

In this paper, we study the approximation of the inversion of windowed Fourier transforms using Riemannian sums. We show that for certain window functions, the Riemannian sums are well defined on L p(ℝ), 1 < p < ∞, and tend to the function to be reconstructed as the sampling density tends to infinity.

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References

  1. Allen, J.B., Rabiner, L.R.: A unified approach to short-time Fourier analysis and synthesis. Proc. IEEE 65, 1558–1564 (1977)

    Article  Google Scholar 

  2. Benedetto, J.J., Zimmermann, G.: Sampling multipliers and the Poisson summation formula. J. Fourier Anal. Appl. 3, 505–523 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Butzer, P.L., Stens, R.L.: The Poisson summation formula, Whittaker’s cardinal series and approximate integration. In: Second Edmonton Conference on Approximation Theory (Edmonton, Alta., 1982), CMS Conf. Proc., vol. 3, pp. 19–36. American Mathematics Society, Providence (1983)

    Google Scholar 

  4. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2003)

    MATH  Google Scholar 

  5. Chui, C.K.: An Introduction to Wavelets. Academic Press, Boston (1992)

    MATH  Google Scholar 

  6. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  7. Feichtinger, H.G., Strohmer, T. (eds.): Gabor Analysis and Algorithms: Theory and Applications. Birkhäuser, Boston (1998)

    MATH  Google Scholar 

  8. Feichtinger, H.G., Weisz, F.: Inversion formulas for the short-time Fourier transform. J. Geom. Anal. 16, 507–521 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feichtinger, H.G., Weisz, F.: Gabor analysis on Wiener amalgams. Sampl. Theory Signal Image Process. 6, 129–150 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Gröchenig, K.: Foundations of Time-Frenquency Analysis. Birkhäuser, Boston (2001)

    Google Scholar 

  11. Sun, W.: Asymptotic properties of Gabor frame operators as sampling density tends to infinity. J. Funct. Anal. 258, 913–932 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sun, W.: Inversion formula for the windowed Fourier transform. http://arxiv.org/abs/1006.3967 (2010)

  13. Weisz, F.: Inversion of the short-time Fourier transform using Riemannian sums. J. Fourier Anal. Appl. 13, 357–368 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Science, Hohai University, Nanjing, 210098, China

    Xudong Sun

  2. Department of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Wenchang Sun

Authors
  1. Xudong Sun
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  2. Wenchang Sun
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Corresponding author

Correspondence to Wenchang Sun.

Additional information

Communicated by Qiyu Sun.

This work was supported partially by the National Natural Science Foundation of China (10971105 and 10990012) and the Natural Science Foundation of Tianjin (09JCYBJC01000).

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Sun, X., Sun, W. Inversion formula for the windowed Fourier transform, II. Adv Comput Math 38, 21–34 (2013). https://doi.org/10.1007/s10444-011-9223-2

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  • DOI: https://doi.org/10.1007/s10444-011-9223-2

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