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## Studia Mathematica

1996 | 121 | 1 | 15-33
Tytuł artykułu

### Stochastic continuity and approximation

Autorzy
Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
This work is concerned with the study of stochastic processes which are continuous in probability, over various parameter spaces, from the point of view of approximation and extension. A stochastic version of the classical theorem of Mergelyan on polynomial approximation is shown to be valid for subsets of the plane whose boundaries are sets of rational approximation. In a similar vein, one can obtain a version in the context of continuity in probability of the theorem of Arakelyan on the uniform approximation of continuous functions on a closed set by entire functions. Locally bounded processes continuous in probability are characterized via operators from $L^1$-spaces to spaces of continuous functions. This characterization is utilized in a discussion of the problem of extension of the parameter space.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
15-33
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-07-06
poprawiono
1996-04-12
Twórcy
autor
• Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A., lbrown@math.wayne.edu
autor
• Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A., berts@math.wayne.edu
Bibliografia
• [1] Andrus, G. F., and Nishiura, T., Stochastic approximation of random functions, Rend. Mat. (6) 13 (1980), 593-615.
• [2] Arakelyan, N. V., Uniform approximation on closed sets by entire functions, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1187-1206 (in Russian).
• [3] Arens, R., Extension of functions on fully normal spaces, Pacific J. Math. 2 (1952), 11-22.
• [4] Blanc-Lapierre, A., and Fortet (transl. by J. Gani), R., Theory of Random Functions, Vol. 1, Gordon and Breach, New York, 1965.
• [5] Brown, L., and Schreiber, B. M., Approximation and extension of random functions, Monatsh. Math. 107 (1989), 111-123.
• [6] Diestel, J., and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., 1977.
• [7] Doob, J. L., Stochastic Processes, Wiley, New York, 1953.
• [8] Dugué, D., Traité de statistique théorique et appliquée: analyse aléatoire, algèbre aléatoire, Masson, Paris, 1958.
• [9] Dugundji, J., An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353-367.
• [10] Dugundji, Topology, Allyn and Bacon, Boston, 1966.
• [11] Fan, K., Sur l'approximation et l'intégration des fonctions aléatoires, Bull. Soc. Math. France 72 (1944), 97-117.
• [12] Fedorchuk, V. V., The fundamentals of dimension theory, in: General Topology I, A. V. Arkhangel'skiĭ and L. S. Pontryagin (eds.), Encyclopaedia Math. Sci. 17, Springer, Berlin, 1990, 91-192.
• [13] Fernique, X., Les fonctions aléatoires cadlag, la compacité de leurs lois, Liet. Mat. Rink. 34 (1994), 288-306.
• [14] Gamelin, T. W., Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969.
• [15] Getoor, R. K., The Brownian escape process, Ann. Probab. 7 (1979), 864-867.
• [16] Gikhman, I. I., and Skorokhod, A. V., Introduction to the Theory of Random Processes, Saunders, Philadelphia, Penn., 1969.
• [17] Himmelberg, C. J., Measurable relations, Fund. Math. 87 (1975), 53-72.
• [18] Istrătescu, V. I., and Onicescu, O., Approximation theorems for random functions, Rend. Mat. (6) 8 (1975), 65-81.
• [19] Kakutani, S., Simultaneous extension of continuous functions considered as a positive linear operation, Japan. J. Math. 17 (1940), 1-4.
• [20] Kelley, J. L., General Topology, Van Nostrand, New York, 1955.
• [21] Mergelyan, S. N., Uniform approximations of functions of a complex variable, Uspekhi Mat. Nauk 7 (2 (48)) (1952), 31-122 (in Russian); English transl.: Amer. Math. Soc. Transl. 101 (1954).
• [22] Michael, E., Some extension theorems for continuous functions, Pacific J. Math. 3 (1953), 789-806.
• [23] Pełczyński, A., On simultaneous extension of continuous functions, Studia Math. 24 (1964), 285-304; Supplement: Studia Math. 25 (1964), 157-161.
• [24] Pełczyński, Linear extensions, linear averagings, and their application to linear topological classification of spaces of continuous functions, Dissertationes Math. (Rozprawy Mat.) 58 (1968).
• [25] Rudin, W., Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
• [26] Runge, C., Zur Theorie der eindeutigen analytischen Funktionen, Acta Math. 6 (1885), 229-244.
• [27] Semadeni, Z., Simultaneous Extensions and Projections in Spaces of Continuous Functions, Lecture Notes, Aarhus Univ., May 1965.
• [28] Syski, R., Stochastic processes, in: Encyclopedia Statist. Sci. 8, S. Kotz and N. L. Johnson (eds.), Wiley-Interscience, New York, 1988, 836-851.
• [29] Vitushkin, A. G., Conditions on a set which are necessary and sufficient in order that any continuous function, analytic at its interior points, admit uniform approximation by rational fractions, Soviet Math. Dokl. 7 (1966), 1622-1625.
• [30] Vitushkin, Analytic capacity of sets in problems of approximation theory, Russian Math. Surveys 22 (1967), 139-200.
• [31] Zalcman, L., Analytic Capacity and Rational Approximation, Lecture Notes in Math. 50, Springer, Berlin, 1968.
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Bibliografia
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