Korean J. Math.  Vol 29, No 2 (2021)  pp.321-332
DOI: https://doi.org/10.11568/kjm.2021.29.2.321

Generalized sequential convolution product for the generalized sequential Fourier-Feynman transform

Byoung Soo Kim, Il Yoo

Abstract


This paper is a further development of the recent results by the authors on the generalized sequential Fourier-Feynman transform for functionals in a Banach  algebra $\hat{\mathcal S}$ and some related functionals. We investigate various relationships between the generalized sequential Fourier-Feynman transform and the generalized sequential convolution product of functionals. Parseval's relation for the generalized sequential Fourier-Feynman transform is also given.


Keywords


generalized sequential Feynman integral, generalized sequential Fourier-Feynman transform, generalized sequential convolution product, generalized first variation, Parseval's relation

Subject classification

28C20, 46G12

Sponsor(s)

Seoul National University of Science and Technology

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References


R.H. Cameron and D.A. Storvick, A simple definition of the Feynman integral, with applications, Mem. Amer. Math. Soc. No. 288, Amer. Math. Soc., 1983. (Google Scholar)

R.H. Cameron and D.A. Storvick, Sequential Fourier-Feynman transforms, Annales Acad. Sci- ent. Fenn. 10 (1985), 107–111. (Google Scholar)

R.H. Cameron and D.A. Storvick, New existence theorems and evaluation formulas for sequential Feynman integrals, Proc. London Math. Soc. 52 (1986), 557–581. (Google Scholar)

(Google Scholar)

R.H. Cameron and D.A. Storvick, New existence theorems and evaluation formulas for analytic Feynman integrals, Deformations Math. Struct., Complex Analy. Phys. Appl., Kluwer Acad. Publ., Dordrecht (1989), 297–308. (Google Scholar)

(Google Scholar)

K.S. Chang, D.H. Cho, B.S. Kim, T.S. Song and I. Yoo, Relationships involving generalized Fourier-Feynman transform, convolution and first variation, Integral Transform. Spec. Funct. 16 (2005), 391–405. (Google Scholar)

(Google Scholar)

K.S. Chang, D.H. Cho, B.S. Kim, T.S. Song and I. Yoo, Sequential Fourier-Feynman transform, convolution and first variation, Trans. Amer. Math. Soc. 360 (2008), 1819–1838. (Google Scholar)

(Google Scholar)

K.S. Chang, B.S. Kim and I. Yoo, Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space, Integral Transform. Spec. Funct. 10 (2000), 179–200. (Google Scholar)

(Google Scholar)

S.J. Chang and J.G. Choi, Rotation of Gaussian paths on Wiener space with applications, Ba- nach J. Math. Anal. 12 (2018), 651–672. (Google Scholar)

(Google Scholar)

D.M. Chung, C. Park and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J. 40 (1993), 377–391. (Google Scholar)

(Google Scholar)

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

(Google Scholar)

T. Huffman, C. Park and D. Skoug, Convolutions and Fourier-Feynman transforms of function- als involving multiple integrals, Michigan Math. J. 43 (1996), 247–261. (Google Scholar)

(Google Scholar)

T. Huffman, C. Park and D. Skoug, Generalized transforms and convolutions, Int. J. Math. Math. Sci. 20 (1997), 19–32. (Google Scholar)

(Google Scholar)

C. Park and D. Skoug, A Kac-Feynman integral equation for conditional Wiener integrals, J. Integral Equations Appl. 3 (1991), 411–427. (Google Scholar)

(Google Scholar)

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)

(Google Scholar)

D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math. 34 (2004), 1147–1176. (Google Scholar)

(Google Scholar)

I. Yoo and B.S. Kim, Generalized sequential Feynman integral and Fourier-Feynman transform, Rocky Mountain J. Math. accepted. (2021) . (Google Scholar)

(Google Scholar)


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