Polynomial approximation for a class of physical random variables
Authors:
A. De Santis, A. Gandolfi, A. Germani and P. Tardelli
Journal:
Proc. Amer. Math. Soc. 120 (1994), 261266
MSC:
Primary 60B12; Secondary 28C20, 46G12, 47N30
MathSciNet review:
1164142
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Abstract 
References 
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Abstract: In white noise theory on Hilbert spaces, it is known that maps which are uniformly continuous around the origin in the Stopology constitute an important class of "physical" random variables. We prove that random variables having such a continuity property can be approximated in the gaussian measure by polynomial random variables. The proof relies on representing functions which are uniformly Scontinuous around the origin as the composition of a continuous map with a HilbertSchmidt operator.
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 [1]
 A. V. Balakrishnan, Parameter estimation in stochastic differential systems: theory and application, Developments in Statistics, vol. 1, Academic Press, New York, 1978. MR 505445 (80a:60059)
 [2]
 G. Kallianpur and R. Karandikar, White noise calculus and nonlinear filtering theory, Ann. Probab. 13 (1985), 10331107. MR 806211 (87b:60067)
 [3]
 A. V. Balakrishnan, Applied functional analysis, Springer, New York, 1981. MR 637334 (83h:00004)
 [4]
 A. Germani and P. Sen, White noise solution for a class of distributed feedback systems with multiplicative noise, Ricerche Automat. 10 (1979), 3865. MR 614562 (82j:60119)
 [5]
 H. H. Kuo, Gaussian measures in Banach Spaces, Lecture Notes in Math., vol. 463, Springer, New York, 1975. MR 0461643 (57:1628)
 [6]
 L. Gross, Integration and nonlinear transformations in Hilbert space, Trans. Amer. Math. Soc. 94 (1960), 404440. MR 0112025 (22:2883)
 [7]
 , Harmonic analysis on Hilbert spaces, Mem. Amer. Math. Soc., no. 46, Amer. Math. Soc., Providence, RI, 1963, pp. 162. MR 0161095 (28:4304)
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 A. Gandolfi and A. Germani, On the definition of a topology in Hilbert spaces with applications to the White Noise Theory, J. Franklin Inst. 316 (1983), 435444. MR 727391 (85e:93045)
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 K. R. Parthasarathy, Probability measures on metric spaces, Academic Press, New York, 1967. MR 0226684 (37:2271)
 [10]
 P. M. Prenter, A Weierstrass theorem for normed linear spaces, Bull. Amer. Math. Soc. 75 (1969), 860862. MR 0244685 (39:5999)
 [11]
 , On polynomial operators and equations, Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971. MR 0290208 (44:7392)
 [12]
 A. DeSantis, A. Gandolfi, A. Germani, and P. Tardelli, A representation theorem for RadonNikodym derivatives in the whitenoise theory, Proc. Internat. Conf. on Mathematical Theory of Control (Bombay, India, December 1015, 1990), Marcel Dekker, New York, 1993, pp. 109123.
 [13]
 R. R. Mazumdar and A. Bagchi, A representation result for nonlinear filters, Proc. COMCON 3 (Victoria, Canada, October 1518, 1991), Vol. 2, UNLV Publications, Las Vegas, 1992, pp. 794805.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199411641425
PII:
S 00029939(1994)11641425
Article copyright:
© Copyright 1994
American Mathematical Society
