Cylinder functions in the Fresnel class of functions on abstract Wiener spaces

Chung, Dong Myung; Hwang, Hong Taek

doi:10.1090/S0002-9939-1992-1097340-8

Cylinder functions in the Fresnel class of functions on abstract Wiener spaces
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by Dong Myung Chung and Hong Taek Hwang PDF

Proc. Amer. Math. Soc. 115 (1992), 381-388 Request permission

Abstract:

In this paper we consider the class of cylinder functions on abstract Wiener space $B$ and give necessary and sufficient conditions of cylinder functions on $B$ to be in the Banach algebra $\mathfrak {F}\left ( B \right )$ (resp. ${\mathfrak {F}^*}\left ( B \right )$) of analytic (resp. sequential) Feynman integrable functions on $B$. The results here subsume similar known results obtained by Chang, Johnson, and Skoug in the setting of Hilbert and Wiener spaces.

References

Sergio A. Albeverio and Raphael J. Høegh-Krohn, Mathematical theory of Feynman path integrals, Lecture Notes in Mathematics, Vol. 523, Springer-Verlag, Berlin-New York, 1976. MR 0495901
Robert Horton Cameron and David Arne Storvick, Some Banach algebras of analytic Feynman integrable functionals, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), Lecture Notes in Math., vol. 798, Springer, Berlin-New York, 1980, pp. 18–67. MR 577446
R. H. Cameron and D. A. Storvick, A simple definition of the Feynman integral, with applications, Mem. Amer. Math. Soc. 46 (1983), no. 288, iv+46. MR 719157, DOI 10.1090/memo/0288
Kun Soo Chang, G. W. Johnson, and D. L. Skoug, Necessary and sufficient conditions for the Fresnel integrability of certain classes of functions, J. Korean Math. Soc. 21 (1984), no. 1, 21–29. MR 760389
K. S. Chang, G. W. Johnson, and D. L. Skoug, Necessary and sufficient conditions for membership in the Banach algebra $S$ for certain classes of functions, Rend. Circ. Mat. Palermo (2) Suppl. 17 (1987), 153–171 (1988). Functional integration with emphasis on the Feynman integral (Sherbrooke, PQ, 1986). MR 950414
K. S. Chang, G. W. Johnson, and D. L. Skoug, Functions in the Fresnel class, Proc. Amer. Math. Soc. 100 (1987), no. 2, 309–318. MR 884471, DOI 10.1090/S0002-9939-1987-0884471-6

Functions in the Banach algebra

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Dong Myung Chung, Scale-invariant measurability in abstract Wiener spaces, Pacific J. Math. 130 (1987), no. 1, 27–40. MR 910652
Dong Myung Chung and Soon Ja Kang, Evaluation formulas for conditional abstract Wiener integrals, Stochastic Anal. Appl. 7 (1989), no. 2, 125–144. MR 997275, DOI 10.1080/07362998908809173
G. W. Johnson, The equivalence of two approaches to the Feynman integral, J. Math. Phys. 23 (1982), no. 11, 2090–2096. MR 680005, DOI 10.1063/1.525250
G. W. Johnson and D. L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), no. 1, 157–176. MR 555044
G. W. Johnson and D. L. Skoug, Notes on the Feynman integral. I, Pacific J. Math. 93 (1981), no. 2, 313–324. MR 623567
G. W. Johnson and D. L. Skoug, Notes on the Feynman integral. I, Pacific J. Math. 93 (1981), no. 2, 313–324. MR 623567
Gopinath Kallianpur and C. Bromley, Generalized Feynman integrals using analytic continuation in several complex variables, Stochastic analysis and applications, Adv. Probab. Related Topics, vol. 7, Dekker, New York, 1984, pp. 217–267. MR 776983
G. Kallianpur, D. Kannan, and R. L. Karandikar, Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces, and a Cameron-Martin formula, Ann. Inst. H. Poincaré Probab. Statist. 21 (1985), no. 4, 323–361 (English, with French summary). MR 823080
Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643
Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528