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
Full-text PDF Free Access
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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
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References
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- J. R. Brown, Jr., Mean square truncation error in series expansion of random functions, J. Soc. Industr. Appl. Math. 8 (1960), no. 1, 28–32. MR 0151999 (27:1980)
- J. R. Brown, Jr., Bounds for truncation error in sampling expansion of band-limited signals, IEEE Trans. Inform. Theory IT-15 (1969), no. 4, 440–444. MR 0249169 (40:2416)
- J. R. Brown, Jr., Truncation error for band-limited random processes, Inform. Sci. 1 (1969), 261–271. MR 0249170 (40:2417)
- 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)
- S. Cambanis and E. Masry, Truncation error bounds for the cardinal sampling expansion of band-limited signals, IEEE Trans. Inform. Theory IT-28 (1982), no. 4, 605–612. MR 0674210 (83k:94004)
- D. K. Chang and M. M. Rao, Bimeasures and sampling theorems for weakly harmonizable processes, Stoch. Anal. Appl. 1 (1983), no. 1, 21–55. MR 0700356 (85f:60055)
- 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.
- K. M. Flornes, Yu. Lyubarskiĭ, and K. Seip, A direct interpolation method for irregular sampling, Appl. Comput. Harmon. Anal. 7 (1999), no. 3, 305–314. MR 1721809 (2000i:41001)
- 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 (26:5707)
- J. R. Higgins, Sampling in Fourier and Signal Analysis: Foundations, Clarendon Press, Oxford, 1996.
- G. Hinsen, Irregular sampling of band-limited $L^p$-functions, J. Approximation Theory 72 (1993), 346–364. MR 1209973 (94e:41006)
- D. L. Jagerman, Bounds for truncation error of the sampling expansion, SIAM J. Appl. Math. 14 (1966), 714–723. MR 0213816 (35:4673)
- A. J. Jerri, The Shannon sampling theorem — its various extensions and applications: A tutorial review, Proc. IEEE 65 (1977), no. 11, 1565–1596.
- 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 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.
- H. S. Piper, Jr., Bounds for truncation error in sampling expansion of finite energy band-limited signals, IEEE Trans. Inform. Theory IT-21 (1975), 482–485. MR 0378988 (51:15154)
- H. S. Piper, Jr., Best asymptotic bounds for truncation error in sampling expansion of band-limited signals, IEEE Trans. Inform. Theory IT-21 (1975), 687–690. MR 0396045 (52:16835)
- Z. A. Piranashvili, On the problem of interpolation of random processes, Teor. Veroyatnost. i Primenen. XII (1967), no. 4, 708–717; English transl. in Theory Probab. Appl. XII (1968), no. 4, 647–657. MR 0219125 (36:2208)
- Z. A. Piranashvili, Stability in the theory of the transfer of continuous information on the basis of Kotel’nikov’s formula, Tekhnicheskaya Kibernetika, Institut Kibernetiki, Akademiya Nauk Gruzinskoj SSR, “Metsniereba”, Tbilisi, 1986, pp. 60–74. (Russian) MR 0901035 (88j:94010)
- T. Pogány, An approach to the sampling theorem for continuous time processes, Austral. J. Statist. 31(3) (1989), 427–432. MR 1054510 (91f:62155)
- 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 interpolation via Kotel’nikov–Shannon sampling formulas, Ukr. Math. J. 50 (2003), no. 11, 1810–1827. MR 2075705 (2005h:41006)
- M. M. Rao, Harmonizable, Cramér, and Karhunen classes of processes, Handbook of Statistics: Time Series in Time Domain (E. J. Hanna et al., eds.), vol. 5, Elsevier, Amsterdam, 1985, pp. 279–310. MR 0831752
- A. G. Robatashvili, Z. A. Piranashvili, and N. G. Kharatishvili, On time discretization of signals, Radiotekhn. i Elektron. XVIII (1973), no. 7, 1384–1388.
- Yu. A. Rozanov, Spectral analysis of abstract functions, Teor. Veroyatnost. Primenen. IV (1959), no. 3, 291–310; English transl. in Theory Probab. Appl. IV (1960), no. 3, 271–287. MR 0123357 (23:A685)
- K. Seip, An irregular sampling theorem for functions band-limited in a general sense, SIAM J. Appl. Math. 47 (1987), no. 5, 1112–1116. MR 0908468 (88k:94004)
- K. Seip, A note on sampling of band-limited stochastic processes, IEEE Trans. Inform. Theory IT-36 (1990), no. 5, 1186.
- W. Splettstößer, Sampling series approximation of continuous weak sense stationary processes, Inform. and Control 50 (1981), 228–241. MR 0691590 (84d:94006)
- W. Splettstößer, L. R. Stens, and G. Wilmes, On approximation by the interpolating series of G. Valiron, Funct. Approx. Comment. Meth. XI (1981), 39–56. MR 0692712 (84f:41004)
- 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.
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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
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