An estimate of the mean square error of interpolation of stochastic processes
Author:
A. Ya. Olenko
Translated by:
V. Semenov
Journal:
Theor. Probability and Math. Statist. 72 (2006), 113-123
MSC (2000):
Primary 94A20, 60G12, 26D15; Secondary 30D15, 41A05
DOI:
https://doi.org/10.1090/S0094-9000-06-00669-7
Published electronically:
September 5, 2006
MathSciNet review:
2168141
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: For functions with an unbounded support of the spectrum, we obtain precise estimates of the error of interpolation by Whittaker–Kotelnikov–Shannon sums. We study the uniform and mean square errors of interpolation. Examples of extreme functions are given for which the estimate is precise. We obtain the rate of the mean square convergence of the error of interpolation for stochastic processes of the weak Cramér class and for processes generated by an orthogonal stochastic measure.
References
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- J. R. Higgins, Sampling in Fourier and Signal Analysis: Foundations, Clarendon Press, Oxford, 1996.
- 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 bandlimited functions, Proc. IEEE 65/5 (1977), 1005–1028.
- A. Ya. Olenko and T. K. Poganī, The least upper bound for error in the interpolation of random processes, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 133–144 (Ukrainian, with Russian summary); English transl., Theory Probab. Math. Statist. 71 (2005), 151–163. MR 2144328, DOI https://doi.org/10.1090/S0094-9000-05-00655-1
- A. Olenko, Aliasing error upper bounds for truncated cardinal series, Theory Probab. Math. Stat. 73 (2005), 120–133. (Ukrainian)
- T. Pogány, Almost sure sampling restoration of bandlimited stochastic signals, Sampling Theory in Fourier and Signal Analysis: Advanced Topics (J. R. Higgins and R. L. Stens, eds.), Oxford University Press, 1999, 203–232.
- C. J. Standish, Two remarks on the reconstruction of sampled non-bandlimited functions, IBM J. Res. Develop. 11 (1967), 648–649.
References
- Y. Bresler, Bounds on the aliasing error in multidimensional Shannon sampling, IEEE Trans. Inform. Theory 42/6 (1996), 2238–2241.
- J. L. Brown, Jr., Estimation of energy aliasing error for nonbandlimited signals, Multdimens. Syst. Signal Process. 15 (2004), 51–56. MR 2042163 (2005b:94020)
- P. L. Butzer, W. Splettstößer, and R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein. 90 (1988), 1–70. MR 0928745 (89b:94006)
- I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. I, “Nauka”, Moscow, 1971; English transl., Springer-Verlag, New York–Heidelberg, 1974. MR 0341539 (49:6287); MR 0346882 (49:11603)
- J. R. Higgins, Sampling in Fourier and Signal Analysis: Foundations, Clarendon Press, Oxford, 1996.
- Y. Kakihara, Multidimensional Second Order Stochastic Processes, World Scientific, Singapore, 1997. MR 1625379 (2000g:60061)
- Yu. I. Khurgin and V. P. Yakovlev, Progress in the Soviet Union on the theory and applications of bandlimited functions, Proc. IEEE 65/5 (1977), 1005–1028.
- A. Olenko and T. Pogány, The least upper bound for error in the interpolation of random processes, Teor. Ĭmovir. Mat. Stat. 71 (2004), 133–144; English transl. in Theory Probab. Math. Statist. 71 (2005), 151–163. MR 2144328 (2006f:60037)
- A. Olenko, Aliasing error upper bounds for truncated cardinal series, Theory Probab. Math. Stat. 73 (2005), 120–133. (Ukrainian)
- T. Pogány, Almost sure sampling restoration of bandlimited stochastic signals, Sampling Theory in Fourier and Signal Analysis: Advanced Topics (J. R. Higgins and R. L. Stens, eds.), Oxford University Press, 1999, 203–232.
- C. J. Standish, Two remarks on the reconstruction of sampled non-bandlimited functions, IBM J. Res. Develop. 11 (1967), 648–649.
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Additional Information
A. Ya. Olenko
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
olenk@univ.kiev.ua
Keywords:
Errors of approximation/interpolation,
extreme functions,
Fréchet semivariation,
convergence of random variables,
unbounded spectrum,
aliasing,
Paley–Wiener function classes,
Kotelnikov–Shannon theorem,
precise estimate,
upper estimate,
interpolation,
truncation error,
stochastic processes,
weak Cramér class
Received by editor(s):
July 3, 2004
Published electronically:
September 5, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
