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Differential and Falsified Sampling Expansions

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Abstract

Differential and falsified sampling expansions \(\sum _{k\in {\mathbb {Z}}^d}c_k\varphi (M^jx+k)\), where M is a matrix dilation, are studied. In the case of differential expansions, \(c_k=Lf(M^{-j}\cdot )(-k)\), where L is an appropriate differential operator. For a large class of functions \(\varphi \), the approximation order of differential expansions was recently studied. Some smoothness of the Fourier transform of \(\varphi \) from this class is required. In the present paper, we obtain similar results for a class of band-limited functions \(\varphi \) with the discontinuous Fourier transform. In the case of falsified expansions, \(c_k\) is the mathematical expectation of random integral average of a signal f near the point \(M^{-j}k\). To estimate the approximation order of the falsified sampling expansions we compare them with the differential expansions. Error estimations in \(L_p\)-norm are given in terms of the Fourier transform of f.

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Acknowledgements

This research was supported by Volkswagen Foundation; the first author is also supported by H2020-MSCA-RISE-2014 Project Number 645672; the second and the third authors are also supported by Grants from RFBR # 15-01-05796-a, St. Petersburg State University # 9.38.198.2015.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Slov’yans’k, Ukraine

    Yu. Kolomoitsev

  2. Institut für Mathematik, Universität zu Lübeck, Lübeck, Germany

    Yu. Kolomoitsev

  3. St. Petersburg State University, Saint Petersburg, Russia

    A. Krivoshein & M. Skopina

Authors
  1. Yu. Kolomoitsev
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  2. A. Krivoshein
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  3. M. Skopina
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Correspondence to M. Skopina.

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Communicated by Yura Lyubarskii.

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Kolomoitsev, Y., Krivoshein, A. & Skopina, M. Differential and Falsified Sampling Expansions. J Fourier Anal Appl 24, 1276–1305 (2018). https://doi.org/10.1007/s00041-017-9559-1

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  • DOI: https://doi.org/10.1007/s00041-017-9559-1

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