Abstract
One stohastic process after A. Kolmogorov can be considered as a curve in Hilbert space. The stationary random curves, i. e. such curves, the correlation functions (CF) of which depend on the difference of the arguments, have been studied by S. Bohner [1] and A. Hinchin [2]. These authors have obtained spectral representation of the stohastic stationary processes in the form
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Bohner, S., Monotone functionen, Ctiltjesche integrale and harmonishe Analyse, Math. Ann., 108, 1933 378–410.
Hinchin, A. Y., Correlation Theory of stationary random processes, Uspekhi Mat. Nauk 5, 1938, 42–51.
Livshic, M. S., On a certain class of linear operators in Hillert space, Mat. sb. 19 (61), 1946, 236–260 (Russian).
Livshic, M. S. and Jancevich, A. A., Theory of Operator colligations in Hilbert space, Engl. Transl., J. Wiley, N. Y., 1979.
Kirchev, K. P., On a certain class of non-stationary random random processes, Teoria funci, func. analisis (Kharkov 1971) 14 150–169 (Russian).
Kirchev, K. P., Linear representable random processes, Annuaire de l'Universite de Sofia, 66, 1974, 288–306 (Russian).
Kirchev, K. P., Dissipative random processes, Serdika, Bulg. Mat. j., 1, 1975 199–217 (Russian).
Brodskij, M. S. and Livshic, M. S., Spectral analysis of nonselfadjoint operators and intermediate systems, Uspekhi Mat. Nauk 13, 1958, 3–85, Engl. Transl. Amer. Math. Soc. Transl (2), 13, 1960, 265–346.
Brodskij, M. S., Triangular and Jordan Representations of Operators, Amer. Math. Soc. Transl. Math. Mon. v. 13, 1974.
Brodskij, M. S., On integral representation of the bounded nonselfadjoint operators with real spectrum, Dokl. Akad. Nauk SSSR, 126, 6, 1959, 1166–1169 (Russian).
Handbook of Mathematical Functions with Formulas, Graphs and Math. Tables, ed. Abranowitz M. and Stegun I. A. appl. math. ser. 55, 1964.
Zolotarev V. A., Universal models of linear operators with determinate growth of resolvents, Teoria funci, func. analisis (Kharkov 1986), 45, 40–45 (Russian).
Djrbashian M. M., Integral transformations and representations of functions in complex plane (Russian), M., Nauka, 1966.
Livshic M. S., Waksman L. L., Commuting Nonselfadjoint Operators in Hilbert Space, Lect. Notes in Math, 1272, 1987.
Zolotarev V. A., On open systems and characteristic functions of commuting operator systems, VINITI 857-79, 1–37, (Russian).
Zolotarev V. A., Triangle models and Cauchy problem for characteristic functions of commuting operator systems, VINITI 3978-80, 1–65. (Russian).
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Partially supported by Grant MM-22/1991 of MESC.
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Kirchev, K., Zolotarev, V. Nonstationary curves in Hilbert spaces and their correlation functions I. Integr equ oper theory 19, 270–289 (1994). https://doi.org/10.1007/BF01203666
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DOI: https://doi.org/10.1007/BF01203666