Some properties of the matrix Wiener transform with related topics on Hilbert space

Main purpose of this paper is to obtain fundamental relationships for the integrals and the matrix Wiener transforms on Hilbert space. Using some technics and properties of matrices of real numbers, we state some algebraic structure of matrices. We then establish evaluation formulas with examples. Furthermore, we define the matrix Wiener transform, and investigate some properties of the matrix Wiener transform. Finally, we establish relationships for the matrix Wiener transform.

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[1] Chang KS, Kim BS, Yoo I. Integral transforms and convolution of analytic functionals on abstract Wiener space. Numerical Functional Analysis and Optimization 2000; 21 (1-2): 97-105.
[2] Chang SJ, Chung HS. Generalized Cameron-Storvick type theorem via the bounded linear operators. Journal of the Korean Mathematical Society 2020; 57 (3): 655-668.
[3] Chang SJ, Skoug D, Chung HS. Relationships for modified generalized integral transforms and modified convolution products and first variations on function space. Integral Transforms and Special Functions 2014; 25 (10): 790-804.
[4] Chung DM, Ji UC. Transforms on white noise functionals with their applications to Cauchy problems. Nagoya Mathematical Journal 1997; 147: 1-23.
[5] Chung HS. Fundamental formulas for modified generalized integral transforms. Banach Journal of Mathematical Analysis 2020; 14 (3): 970-986.
[6] Chung HS. Generalized integral transforms via the series expressions. Mathematics 2020; 8: 569.
[7] Chung HS. A matrix transform on function space with related topics. Filomat 2021; 35 (13): 4459-4468.
[8] Gross L. Potential theory on Hilbert space. Journal Functional Analysis 1967; 1 (2): 123-181.
[9] Kim BJ, Kim BS, Skoug D. Integral transforms, convolution products and first variations. International Journal of Mathematics and Mathematical Sciences 2004; 2004: 579-598.
[10] Lee YJ. Integral transforms of analytic functions on abstract Wiener spaces. Journal Functional Analysis 1982; 47 (2): 153-164.
[11] Chang KS, Cho DH, Kim BS, Song TS, Yoo I. Relationships involving generalized Fourier–Feynman transform, convolution and first variation. Integral Transforms and Special Functions 2005; 16 (5-6): 391-405.
[12] Hayek N, Gonzalez BJ, Negrin ER. The second quantization and its general integral finite-dimensional representation. Integral Transforms and Special Functions 2002; 13 (4): 373-378.
[13] Hayek N, Gonzalez BJ, Negrin ER. Matrix Wiener transform. Applied Mathematics and Computation 2011; 218 (3): 773-776.
[14] Hayek N, Srivastava HM, Gonzalez BJ, Negrin ER. A family of Wiener transforms associated with a pair of operators on Hilbert space. Integral Transforms and Special Functions 2012; 24 (1): 1-8.
[15] Hida T. Stationary stochastic process: Series on Mathematical Notes. Princeton University Press, Tokyo, Japan: NJ/University of Tokyo Press, 1970.
[16] Hida T. Brownian motion. Series on Applications of Mathematics, Springer, New York, NJ,USA: Springer-Verlag New York 1980.
[17] Negrin ER. Integral representation of the second quantization via Segal duality transform. Journal Functional Analysis 1996; 141 (1): 37-44.
[18] Segal IE. Tensor algebra over Hilbert spaces I. Transactions of the American Mathematical Society 1956; 81 (1): 106-134.
[19] Segal IE. Tensor algebra over Hilbert spaces II. Annals of Mathematics 1956; 63 (1): 160-175.
[20] Segal IE. Distributions in Hilbert space and canonical systems of operators. Transactions of the American Mathematical Society 1958; 88 (1): 12-41.