Sampling theorems for nonstationary random processes

Lee, Alan J.

doi:10.1090/S0002-9947-1978-0482995-6

Sampling theorems for nonstationary random processes
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by Alan J. Lee

Trans. Amer. Math. Soc. 242 (1978), 225-241

DOI: https://doi.org/10.1090/S0002-9947-1978-0482995-6

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Abstract:

Consider a second order stochastic process $\{ X(t),t \in \textbf {R}\}$, and let $H(X)$ be the Hilbert space generated by the random variables of the process. The process is said to be linearly determined by its samples $\{ X(nh),n \in \textbf {Z}\}$ if the random variables $X(nh)$ generate $H(X)$. In this paper we give a sufficient condition for a wide class of nonstationary processes to be determined by their samples, and present sampling theorems for such processes. We also consider similar problems for harmonizable processes indexed by LCA groups having suitable subgroups.

References

Moshe Zakai, Band-limited functions and the sampling theorem, Information and Control 8 (1965), 143–158. MR 174403, DOI 10.1016/S0019-9958(65)90038-0

On the problem of interpolation of stochastic processes

12

Alan J. Lee, On band limited stochastic processes, SIAM J. Appl. Math. 30 (1976), no. 2, 269–277. MR 401319, DOI 10.1137/0130028
Alan J. Lee, Characterization of bandlimited functions and processes, Information and Control 31 (1976), no. 3, 258–271. MR 405573, DOI 10.1016/S0019-9958(76)90409-5
S. P. Lloyd, A sampling theorem for stationary (wide sense) stochastic processes, Trans. Amer. Math. Soc. 92 (1959), 1–12. MR 107301, DOI 10.1090/S0002-9947-1959-0107301-6
M. M. Rao, Inference in stochastic processes. III. Nonlinear prediction, filtering, and samlping theorems, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 8 (1967), 49–72. MR 216713, DOI 10.1007/BF00533944
Stamatis Cambanis and Elias Masry, On the representation of weakly continuous stochastic processes, Information Sci. 3 (1971), 277–290; erratum, ibid. 4 (1972), 289–290. MR 0305477, DOI 10.1016/s0020-0255(71)80011-7
Emanuel Parzen, An appproach to time series analysis, Ann. Math. Statist. 32 (1961), 951–989. MR 143315, DOI 10.1214/aoms/1177704840
A. H. Zemanian, Distribution theory and transform analysis. An introduction to generalized functions, with applications, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0177293
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0152834
Stamatis Cambanis and Bede Liu, On harmonizable stochastic processes, Information and Control 17 (1970), 183–202. MR 282401, DOI 10.1016/S0019-9958(70)90541-3

Abstract harmonic analysis

28

R. A. Mayer, Summation of Fourier series on compact groups, Amer. J. Math. 89 (1967), 661–692. MR 218492, DOI 10.2307/2373239