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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1164142-5

MathSciNet review:
1164142

Full-text PDF Free Access

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.

**[1]**A. V. Balakrishnan,*Parameter estimation in stochastic differential systems: theory and application*, Developments in statistics, Vol. 1, Academic Press, New York, 1978, pp. 1–32. MR**505445****[2]**G. Kallianpur and R. L. Karandikar,*White noise calculus and nonlinear filtering theory*, Ann. Probab.**13**(1985), no. 4, 1033–1107. MR**806211****[3]**D. H. Griffel,*Applied functional analysis*, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1981. Ellis Horwood Series in Mathematics and its Applications. MR**637334****[4]**A. Germani and Prodip Sen,*White noise solutions for a class of distributed feedback systems with multiplicative noise*, Ricerche Automat.**10**(1979), no. 1, 38–65 (1980). MR**614562****[5]**Hui Hsiung Kuo,*Gaussian measures in Banach spaces*, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR**0461643****[6]**Leonard Gross,*Integration and nonlinear transformations in Hilbert space*, Trans. Amer. Math. Soc.**94**(1960), 404–440. MR**112025**, https://doi.org/10.1090/S0002-9947-1960-0112025-3**[7]**Leonard Gross,*Harmonic analysis on Hilbert space*, Mem. Amer. Math. Soc. No.**46**(1963), ii+62. MR**0161095****[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), no. 6, 435–444. MR**727391**, https://doi.org/10.1016/0016-0032(83)90090-X**[9]**K. R. Parthasarathy,*Probability measures on metric spaces*, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR**0226684****[10]**P. M. Prenter,*A Weierstrass theorem for normed linear spaces*, Bull. Amer. Math. Soc.**75**(1969), 860–862. MR**244685**, https://doi.org/10.1090/S0002-9904-1969-12329-3**[11]**Patricia M. Prenter,*On polynomial operators and equations*, Nonlinear Functional Anal. and Appl. (Proc. Advanced Sem., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970) Academic Press, New York, 1971, pp. 361–398. MR**0290208****[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.

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

DOI:
https://doi.org/10.1090/S0002-9939-1994-1164142-5

Article copyright:
© Copyright 1994
American Mathematical Society