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

Olenko, A.; Pogány, T.

doi:10.1090/S0094-9000-05-00655-1

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
DOI: https://doi.org/10.1090/S0094-9000-05-00655-1
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.

References

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 https://doi.org/10.1109/tit.1969.1054335
John L. Brown Jr., Truncation error for band-limited random processes, Information Sci. 1 (1968/1969), 261–271. MR 0249170, DOI https://doi.org/10.1016/s0020-0255%2869%2980012-5
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 https://doi.org/10.1109/TIT.1982.1056527
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 https://doi.org/10.1080/07362998308809003

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 https://doi.org/10.1006/acha.1999.0273

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 https://doi.org/10.1006/jath.1993.1027

D. Jagerman, Bounds for truncation error of the sampling expansion, SIAM J. Appl. Math. 14 (1966), 714–723. MR 213816, DOI https://doi.org/10.1137/0114060

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 https://doi.org/10.1109/tit.1975.1055404

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 https://doi.org/10.1109/tit.1975.1055453

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 https://doi.org/10.1111/j.1467-842x.1989.tb00987.x

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 https://doi.org/10.1023/B%3AUKMA.0000027044.99266.31

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 https://doi.org/10.1016/S0169-7161%2885%2905012-X

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 https://doi.org/10.1137/1104027

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 https://doi.org/10.1137/0147073

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 https://doi.org/10.1016/S0019-9958%2881%2990343-0

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.