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Generalized Fourier–Feynman transforms and generalized convolution products on Wiener space II

  • Sang Kil Shim
  • Jae Gil ChoiEmail author
Original Paper

Abstract

The purpose of this article is to present the second type fundamental relationship between the generalized Fourier–Feynman transform and the generalized convolution product on Wiener space. The relationships in this article are also natural extensions (to the case on an infinite dimensional Banach space) of the structure which exists between the Fourier transform and the convolution of functions on Euclidean spaces.

Keywords

Wiener space Gaussian process Generalized Fourier–Feynman transform Generalized convolution product 

Mathematics Subject Classification

46G12 28C20 60G15 60J65 

Notes

Acknowledgements

The authors would like to express their gratitude to the editor and the referees for their valuable comments and suggestions which have improved the original paper. The present research was supported by the research fund of Dankook University in 2019.

References

  1. 1.
    Cameron, R.H., Storvick, D.A.: Some Banach Algebras of Analytic Feynman Integrable Functionals. Analytic Functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), Lecture Notes in Math, vol. 798, pp. 18–67. Springer, Berlin (1980)Google Scholar
  2. 2.
    Chang, K.S., Cho, D.H., Kim, B.S., Song, T.S., Yoo, I.: Relationships involving generalized Fourier–Feynman transform, convolution and first variation. Integral Transforms Spec. Funct. 16, 391–405 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chang, S.J., Chung, H.S., Choi, J.G.: Generalized Fourier–Feynman transforms and generalized convolution products on Wiener space. Indag. Math. 28, 566–579 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chung, D.M., Park, C., Skoug, D.: Generalized Feynman integrals via conditional Feynman integrals. Mich. Math. J. 40, 377–391 (1993)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Huffman, T., Park, C., Skoug, D.: Analytic Fourier–Feynman transforms and convolution. Trans. Am. Math. Soc. 347, 661–673 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Huffman, T., Park, C., Skoug, D.: Convolutions and Fourier–Feynman transforms of functionals involving multiple integrals. Mich. Math. J. 43, 247–261 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Huffman, T., Park, C., Skoug, D.: Convolution and Fourier–Feynman transforms. Rocky Mt. J. Math. 27, 827–841 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Huffman, T., Park, C., Skoug, D.: Generalized transforms and convolutions. Int. J. Math. Math. Sci. 20, 19–32 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Paley, R.E.A.C., Wiener, N., Zygmund, A.: Notes on random functions. Math. Z. 37, 647–668 (1933)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Park, C.: A generalized Paley–Wiener–Zygmund integral and its applications. Proc. Am. Math. Soc. 23, 388–400 (1969)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Park, C., Skoug, D.: A note on Paley–Wiener–Zygmund stochastic integrals. Proc. Am. Math. Soc. 103, 591–601 (1988)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Park, C., Skoug, D.: A Kac–Feynman integral equation for conditional Wiener integrals. J. Integral Equations Appl. 3, 411–427 (1991)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Park, C., Skoug, D., Storvick, D.: Relationships among the first variation, the convolution product, and the Fourier–Feynman transform. Rocky Mt. J. Math. 28, 1447–1468 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Skoug, D., Storvick, D.: A survey of results involving transforms and convolutions in function space. Rocky Mt. J. Math. 34, 1147–1175 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Department of MathematicsDankook UniversityCheonanRepublic of Korea
  2. 2.School of General EducationDankook UniversityCheonanRepublic of Korea

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