Korean J. Math. Vol. 22 No. 1 (2014) pp.57-69
DOI: https://doi.org/10.11568/kjm.2014.22.1.57

Parts formulas involving conditional integral transforms on function space

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Published: Mar 27, 2014
Mathematics Subject Classification:
28C20

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Bong Jin Kim
Daejin University
Byoung Soo Kim
Seoul National University of Science and Technology

Abstract

We obtain a formula for the conditional Wiener integral of the first variation of functionals and establish several integration by parts formulas of conditional Wiener integrals of functionals on a function space. We then apply these results to obtain various integration by parts formulas involving conditional integral transforms and conditional convolution products on the function space.


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References

[1] R.H.Cameron, The first variation of an indefinite Wiener integral, Proc. Amer. Math. Soc. 2 (1951), 914-924. Google Scholar

[2] R.H.Cameron and W.T.Martin, Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 489-507. Google Scholar

[3] R.H.Cameron and D.A.Storvick, A translation theorem for analytic Feynman integrals, Trans. Amer. Math. Soc. 125 (1966), 1-6. Google Scholar

[4] R.H.Cameron and D.A.Storvick, An $L_2$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), 1-30. Google Scholar

[5] R.H.Cameron and D.A.Storvick, Feynman integral of variations of functionals, Gaussian Random Fields (Nagoya, 1990), Ser. Probab. Statist. 1, World Sci. Publ. 1991, 144-157. Google Scholar

[6] S.J.Chang and D.Skoug, Parts formulas involving conditional Feynman integrals, Bull. Austral. Math. Soc. 65 (2002), 353-369. Google Scholar

[7] T.Huffman, C.Park and D.Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), 661-673. Google Scholar

[8] B.J.Kim and B.S.Kim, Parts formulas involving integral transforms on function space, Commun. Korean Math. Soc. 22 (2007), 553-564. Google Scholar

[9] B.J.Kim, B.S.Kim and D.Skoug, Integral transforms, convolution products and first variations, Internat. J. Math. Math. Sci. 2004 (2004), 579-598 Google Scholar

[10] B.J.Kim, B.S.Kim and D.Skoug, Conditional integral transforms, conditional convolution products and first variations, PanAmerican Math. J. 14 (2004), 27-47. Google Scholar

[11] Y.J.Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), 153-164. Google Scholar

[12] C.Park, D.Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), 381-394. Google Scholar

[13] C.Park, D.Skoug and D.Storvick, Relationships among the first variation, the convolution product, and the Fourier-Feynman transform, Rocky Mountain J. Math. 28 (1998), 1447-1468. Google Scholar

[14] D.Skoug and D.Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math. 34 (2004), 1147-1175. Google Scholar

[15] J.Yeh, Convolution in Fourier-Wiener transform, Pacific J. Math. 15 (1965), 731-738. Google Scholar

[16] J.Yeh, Inversion of conditional Wiener integral, Google Scholar

[17] Pacific J. Math. 59 (1975), 623-638. Google Scholar