Skip to Main Content
Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



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

Authors: A. Ya. Olenko and T. K. Pogány
Translated by: Oleg Klesov
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
Full-text PDF Free Access

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.

References [Enhancements On Off] (What's this?)

  • Yu. K. Belyaev, Analytic random processes, Teor. Veroyatnost. i Primenen 4 (1959), 437–444 (Russian, with English summary). MR 0112174
  • J. L. Brown Jr., Mean square truncation error in series expansions of random functions, J. Soc. Indust. Appl. Math. 8 (1960), 28–32. MR 151999
  • John L. Brown Jr., Bounds for truncation error in sampling expansions of band-limited signals, IEEE Trans. Inform. Theory IT-15 (1969), 440–444. MR 249169, DOI
  • John L. Brown Jr., Truncation error for band-limited random processes, Information Sci. 1 (1968/1969), 261–271. MR 0249170, DOI
  • P. L. Butzer, W. Splettstösser, and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein. 90 (1988), no. 1, 70. MR 928745
  • Stamatis Cambanis and Elias Masry, Truncation error bounds for the cardinal sampling expansions of band-limited signals, IEEE Trans. Inform. Theory 28 (1982), no. 4, 605–612. MR 674210, DOI
  • Derek K. Chang and M. M. Rao, Bimeasures and sampling theorems for weakly harmonizable processes, Stochastic Anal. Appl. 1 (1983), no. 1, 21–55. MR 700356, DOI
  • M. M. Dodson and M. G. Beaty, Abstract harmonic analysis and the sampling theorem, Sampling Theory in Fourier and Signal Analysis: Advanced Topics (J. R. Higgins and R. L. Stens, eds.), Oxford University Press, 1999, pp. 233–265.
  • Kristin M. Flornes, Yurii Lyubarskii, and Kristian Seip, A direct interpolation method for irregular sampling, Appl. Comput. Harmon. Anal. 7 (1999), no. 3, 305–314. MR 1721809, DOI
  • O. Gulyás, On the truncation of the series in the sampling theorem, Proc. Colloquium on Microwave Communication, Budapest, 1970, vol. 1, Communication System Theory (G. Bognár, ed.), Akadémiai Kiadó, Budapest, 1970, pp. 13/1–13/5. (Russian)
  • H. D. Helms and J. B. Thomas, Truncation error of sampling-theorem expansions, Proc. IRE 50 (1962), 179–184. MR 0148199
  • J. R. Higgins, Sampling in Fourier and Signal Analysis: Foundations, Clarendon Press, Oxford, 1996.
  • G. Hinsen, Irregular sampling of bandlimited $L^p$-functions, J. Approx. Theory 72 (1993), no. 3, 346–364. MR 1209973, DOI
  • D. Jagerman, Bounds for truncation error of the sampling expansion, SIAM J. Appl. Math. 14 (1966), 714–723. MR 213816, DOI
  • A. J. Jerri, The Shannon sampling theorem — its various extensions and applications: A tutorial review, Proc. IEEE 65 (1977), no. 11, 1565–1596.
  • Yûichirô Kakihara, Multidimensional second order stochastic processes, Series on Multivariate Analysis, vol. 2, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. MR 1625379
  • Yu. I. Khurgin and V. P. Yakovlev, Progress in the Soviet Union on the theory and applications of band-limited functions, Proc. IEEE 65 (1977), no. 5, 1005–1028.
  • A. Olenko and T. Pogány, Direct Lagrange–Yen type interpolation of random fields, Theory Stoch. Process. 9(25) (2003), no. 3–4, 242–254.
  • A. Papoulis, Limits on band-limited signals, Proc. IEEE 55 (1967), no. 10, 1677–1686.
  • Harvey S. Piper Jr., Bounds for truncation error in sampling expansions of finite energy band-limited signals, IEEE Trans. Inform. Theory IT-21 (1975), 482–485. MR 378988, DOI
  • Harvey S. Piper Jr., Best asymptotic bounds for truncation error in sampling expansions of band-limited signals, IEEE Trans. Inform. Theory IT-21 (1975), no. 6, 687–690. MR 396045, DOI
  • Z. A. Piranašvili, The problem of interpolation of random processes, Teor. Verojatnost. i Primenen 12 (1967), 708–717 (Russian, with English summary). MR 0219125
  • Z. A. Piranashvili, Stability in the theory of the transfer of continuous information on the basis of Kotel′nikov’s formula, Engineering cybernetics, “Metsniereba”, Tbilisi, 1986, pp. 60–74 (Russian). MR 901035
  • Tibor Pogány, An approach to the sampling theorem for continuous time processes, Austral. J. Statist. 31 (1989), no. 3, 427–432. MR 1054510, DOI
  • T. Pogány, Almost sure sampling restoration of band-limited stochastic signals, Sampling Theory in Fourier and Signal Analysis: Advanced Topics, (J. R. Higgins and R. L. Stens, eds.), Oxford University Press, 1999, pp. 203–232.
  • T. K. Pogány, Multidimensional Lagrange-Yen type interpolation via Kotel′nikov-Shannon sampling formulae, Ukraïn. Mat. Zh. 55 (2003), no. 11, 1503–1519 (English, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 55 (2003), no. 11, 1810–1827. MR 2075705, DOI
  • M. M. Rao, Harmonizable, Cramér, and Karhunen classes of processes, Time series in the time domain, Handbook of Statist., vol. 5, North-Holland, Amsterdam, 1985, pp. 279–310. MR 831752, DOI
  • A. G. Robatashvili, Z. A. Piranashvili, and N. G. Kharatishvili, On time discretization of signals, Radiotekhn. i Elektron. XVIII (1973), no. 7, 1384–1388.
  • Ju. A. Rozanov, Spectral analysis of abstract functions, Theor. Probability Appl. 4 (1959), 271–287. MR 123357, DOI
  • Kristian Seip, An irregular sampling theorem for functions bandlimited in a generalized sense, SIAM J. Appl. Math. 47 (1987), no. 5, 1112–1116. MR 908468, DOI
  • K. Seip, A note on sampling of band-limited stochastic processes, IEEE Trans. Inform. Theory IT-36 (1990), no. 5, 1186.
  • Wolfgang Splettstösser, Sampling series approximation of continuous weak sense stationary processes, Inform. and Control 50 (1981), no. 3, 228–241. MR 691590, DOI
  • W. Splettstösser, R. L. Stens, and G. Wilmes, On approximation by the interpolating series of G. Valiron, Funct. Approx. Comment. Math. 11 (1981), 39–56. MR 692712
  • B. S. Tsybakov and V. P. Yakovlev, On the reconstruction error level in the approximation by the partial sum of the Kotel’nikov series, Radiotekhn. i Elektron. IV (1959), 542.
  • Ya. S. Zatuliveter, B. I. Olejnikov, and V. N. Filinov, On the approximation error level in reconstructing signals with Kotel’nikov series partial sums, Radiotekhnika i Elektronika XVII (1972), no. 4, 881–882.
  • K. Yao and J. B. Thomas, On truncation error bounds for sampling representations of band-limited signals, IEEE Trans. Aerospace Electron. Systems AES-2 (1966), no. 6, 640–647.
  • J. L. Yen, On nonuniform sampling of bandwidth-limited signals, IRE Trans. Circuit Theory CS-3 (1956), 251–257.

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 94A20, 60G12, 26D15, 30D15, 41A05

Retrieve articles in all journals with MSC (2000): 94A20, 60G12, 26D15, 30D15, 41A05

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

T. K. Pogány
Affiliation: Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Studentska 2, Croatia

Keywords: Errors of approximation/interpolation, extremal functions, Fréchet (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, Cramér class
Received by editor(s): April 29, 2004
Published electronically: December 30, 2005
Article copyright: © Copyright 2005 American Mathematical Society