Article ID: MTJPAM-D-23-00004

Title: Truncation error upper bounds in derivative Whittaker–type plane sampling reconstruction


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-23-00004; Volume 6 / Issue 3 / Year 2024, Pages 1-10

Document Type: Research Paper

Author(s): ‪Tibor K. Pogány a

aFaculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia and Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary

Received: 17 March 2023, Accepted: 16 May 2023, Published: 5 June 2023

Corresponding Author: Tibor K. Pogány (Email address: poganj@uniri.hr; pogany.tibor@uni-obuda.hu)

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Abstract

A survey is presented on the author’s mean square and almost sure Whittaker–type derivative sampling theorems obtained for the class L^\alpha( \Omega , {\mathfrak F}, {\mathsf P});\; 0 \leq \alpha \le 2 of stochastic processes having spectral representation, with the aid of the Weierstraß \sigma function. Processes of this class are represented by interpolation series. The results are valid for harmonizable and weakly stationary (or in the Hinčin sense stationary) processes (\alpha=2) as well. The formulæ are interpreted in the \alpha–mean and also almost sure sense when the input function and its derivatives are sampled at the points of the integer lattice {\mathbb{Z}}^2 . The circular truncation error is introduced and used in the truncation error analysis and related sampling sum convergence rates are shown.

Keywords: Circular truncation error, derivative sampling, Leont’ev spaces of entire functions, Piranashvili–type stochastic processes, truncation error upper bounds, Weierstraß sigma–function, Whittaker–type sampling, (p, q)–order weighted differential operator

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