Abstract
A jacobi field is understood to be a family (Ã(ϕ))ϕ∈Φ of commuting selfadjoint operatorsÃ(ϕ) acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions ϕ. In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. M. Berezansky,Expansion in Eigenfunctions of Selfadjoint Operators, AMS, Providence, R. I., 1968 (Russian edition: Naukova Dumka, Kiev, 1965).
Yu. M. Berezansky,Selfadjoint Operators in Spaces of Functions of Infinitely Many Variables, AMS, Providence, R. I., 1986 (Russian edition: Naukova Dumka, Kiev, 1978).
Yu. M. Berezansky,Spectral approach to white noise analysis, Proceedings of Symposium “Dynamics of Complex and Irregular Systems”, 16–20 December 1991, Germany. In: Bielefeld Encounters in Mathematics and Physics VIII, 1993, World Scientific, Singapore-New Jersey-London-Hong Kong, 131–140.
Yu. M. Berezansky,Infinite-dimensional non-Gaussian analysis and generalized translation operators, Funkts. Anal. Prilozhen.30, no. 4 (1996), 61–65.
Yu. M. Berezansky,Infinite dimensional analysis connected with generalized translation operators, Ukr. Mat. Zh.49, no. 3 (1997), 364–409.
Yu. M. Berezansky,Construction of the operators of generalized translation by Appel characters, Proceedings of works, devoted to S. G. Krein, World Scientific, 1997 (in print).
Yu. M. Berezansky and Yu. G. Kondratiev,Spectral Methods in Infinite Dimensional Analysis, v. 1, 2, Kluwer, Dordrecht-Boston-London, 1995 (Russian edition: Naukova Dumka, Kiev, 1988).
Yu. M. Berezansky and V. D. Koshmanenko,An axiomatic field theory in terms of operator Jacobian matrices, Soviet Physics. Doklady14 (1968/70), 1064–1066.
Yu. M. Berezansky, V. O. Livinsky, and E. W. Lytvynov,Spectral approach to white noise analysis, Ukr. Mat. Zh.46, no. 3 (1994), 177–197.
Yu. M. Berezansky, V. O. Livinsky, and E. W. Lytvynov,A generalization of Gaussian white noise analysis, Methods Funct. Anal. Topology1, no. 1 (1995), 28–55.
E. Brüning,On Jacobi-fields, Colloquia Mathematica Societatic Janos Bolyai, Random Fields (1979), 171–196.
T. Hida, H.-H. Kuo, J. Potthoff, and L. Streit,White Noise. An Infinite Dimensional Calculus, Kluwer, Dordrecht-Boston-London, 1993.
M. G. Krein and M. A. Krasnoselskii,Basic theorems on extensions of Hermitian operators and certain their applications to the theory of orthogonal polynomials and moment problem, Uspekhi Mat. Nauk2, no. 3 (1947), 60–106.
E. W. Lytvynov,Spectral approach to multiple Wiener integrals, Doklady Akademii Nauk Ukrainy (1994), no. 11, 29–32.
E. W. Lytvynov,Multiple Wiener integrals and non-Gaussian white noise: a Jacobi field approach, Methods Funct. Anal. Topology1, no. 1 (1995), 61–85.
L. Nachbin,Topology on Spaces of Holomorphic Mappings, Springer, Berlin-Heidelberg-New York, 1969.
G. F. Us,Dual Appel systems in Poisson analysis, Methods Funct. Anal. Topology1, no. 1 (1995), 93–108.
Author information
Authors and Affiliations
Additional information
This article is dedicated to the memory of my dear teacher Mark G. Krein
The work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.