Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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), 261-266
MSC: Primary 60B12; Secondary 28C20, 46G12, 47N30
MathSciNet review: 1164142
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In white noise theory on Hilbert spaces, it is known that maps which are uniformly continuous around the origin in the S-topology 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 S-continuous around the origin as the composition of a continuous map with a Hilbert-Schmidt operator.

References [Enhancements On Off] (What's this?)

  • [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 non-linear filtering theory, Ann. Probab. 13 (1985), 1033-1107. 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), 38-65. 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 non-linear transformations in Hilbert space, Trans. Amer. Math. Soc. 94 (1960), 404-440. MR 0112025 (22:2883)
  • [7] -, Harmonic analysis on Hilbert spaces, Mem. Amer. Math. Soc., no. 46, Amer. Math. Soc., Providence, RI, 1963, pp. 1-62. MR 0161095 (28:4304)
  • [8] 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), 435-444. MR 727391 (85e:93045)
  • [9] 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), 860-862. 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 Radon-Nikodym derivatives in the white-noise theory, Proc. Internat. Conf. on Mathematical Theory of Control (Bombay, India, December 10-15, 1990), Marcel Dekker, New York, 1993, pp. 109-123.
  • [13] R. R. Mazumdar and A. Bagchi, A representation result for nonlinear filters, Proc. COMCON 3 (Victoria, Canada, October 15-18, 1991), Vol. 2, UNLV Publications, Las Vegas, 1992, pp. 794-805.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60B12, 28C20, 46G12, 47N30

Retrieve articles in all journals with MSC: 60B12, 28C20, 46G12, 47N30

Additional Information

Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society