Differential and Falsified Sampling Expansions

Article

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.

Keywords

Differential expansion Falsified sampling expansion Approximation order Matrix dilation Strang–Fix condition 

Mathematics Subject Classification

41A58 41A25 41A63 

Notes

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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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Mechanics of NAS of UkraineSlov’yans’kUkraine
  2. 2.Institut für MathematikUniversität zu LübeckLübeckGermany
  3. 3.St. Petersburg State UniversitySaint PetersburgRussia

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