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Theory of Probability and Mathematical Statistics

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A precise upper bound for the error of interpolation of stochastic processes


Authors: A. Ya. Olenko and T. K. Pogány
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 71 (2004).
Journal: Theor. Probability and Math. Statist. 71 (2005), 151-163
MSC (2000): Primary 94A20, 60G12, 26D15; Secondary 30D15, 41A05
Published electronically: December 30, 2005
MathSciNet review: 2144328
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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain a precise upper bound for the truncation error of interpolation of functions of the Paley-Wiener class with the help of finite Whittaker-Kotelnikov-Shannon sums. We construct an example of an extremal function for which the upper bound is achieved. We study the error of interpolation and the rate of the mean square convergence for stochastic processes of the weak Cramér class. The paper contains an extensive list of references concerning the upper bounds for errors of interpolation for both deterministic and stochastic cases. The final part of the paper contains a discussion of new directions in this field.


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Additional Information

A. Ya. Olenko
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: olenk@univ.kiev.ua

T. K. Pogány
Affiliation: Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Studentska 2, Croatia
Email: poganj@brod.pfri.hr

DOI: http://dx.doi.org/10.1090/S0094-9000-05-00655-1
Keywords: Errors of approximation/interpolation, extremal functions, Fr\'echet (semi-) variation, mean square convergence, Paley--Wiener classes of functions, Kotelnikov--Shannon theorem, precise bounds, upper bound of error of interpolation, truncation error, stochastic processes, Cram\'{e}r class
Received by editor(s): April 29, 2004
Published electronically: December 30, 2005
Article copyright: © Copyright 2005 American Mathematical Society