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On Generalized Derivative Sampling Series Expansion

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Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

Abstract

Master generalized sampling series expansion is presented for entire functions (signals) coming from a class whose members satisfy an extended exponential boundedness condition. Firstly, estimates are given for the remainder of Maclaurin series of those functions and consequent derivative sampling results are obtained and discussed.

The established results are employed in evaluating the related remainder term of signals which occur in sampling series expansion of stochastic processes and random fields (not necessarily stationary or homogeneous) whose spectral kernel satisfies the relaxed exponential boundedness. The derived truncation error upper bounds enable to obtain mean-square master generalized derivative sampling series expansion formulae either for harmonizable Piranashvili-type stochastic processes or for random fields.

Finally, being the sampling series convergence rate exponential, almost sure P sampling series expansion formulae are presented.

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Notes

  1. 1.

    The Bernstein class \(B_\sigma ^p\) consists of entire functions (in the complex plane) of exponential type at most σ, whose restriction to \(\mathbb R\) belongs to \(L^p(\mathbb R)\). We are interested here in \(B_\sigma ^2\)-functions since our study belongs to the L 2-correlation theory area.

  2. 2.

    The existence of such representation is ensured by the Karhunen–Cramèr theorem.

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Author information

Authors and Affiliations

  1. Vladimir Chavchanidze Institute of Cybernetics, Georgian Technical University, Tbilisi, Georgia

    Zurab A. Piranashvili

  2. Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia

    Tibor K. Pogány

  3. Institute of Applied Mathematics, Óbuda University, Budapest, Hungary

    Tibor K. Pogány

Authors
  1. Zurab A. Piranashvili
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  2. Tibor K. Pogány
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Correspondence to Tibor K. Pogány .

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Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  2. University of Nis, Faculty of Sciences and Mathematics, Aleksandrovac, Serbia

    Ljubiša D. R. Kočinac

  3. Dept. Mathematics and Statistics, University of Victoria, Victoria, BC, Canada

    Hari M. Srivastava

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Piranashvili, Z.A., Pogány, T.K. (2019). On Generalized Derivative Sampling Series Expansion. In: Dutta, H., Kočinac, L.D.R., Srivastava, H.M. (eds) Current Trends in Mathematical Analysis and Its Interdisciplinary Applications. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-15242-0_14

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