Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED CONDITIONAL INTEGRAL TRANSFORMS, CONDITIONAL CONVOLUTIONS AND FIRST VARIATIONS

  • Kim, Bong Jin (Department of Mathematics Daejin University) ;
  • Kim, Byoung Soo (School of Liberal Arts Seoul National University of Science and Technology)
  • Received : 2011.09.20
  • Accepted : 2012.01.10
  • Published : 2012.03.30
Download PDF
⟨ Previous Next ⟩

Abstract

We study various relationships that exist among generalized conditional integral transform, generalized conditional convolution and generalized first variation for a class of functionals defined on K[0, T], the space of complex-valued continuous functions on [0, T] which vanish at zero.

Keywords

Acknowledgement

Supported by : Daejin University

References

  1. R.H. Cameron, The first variation of an indefinite Wiener integral, Proc. Amer. Math. Soc. 2 (1951), 914-924. https://doi.org/10.1090/S0002-9939-1951-0045937-X
  2. R.H. Cameron and W.T. Martin, Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 489-507. https://doi.org/10.1215/S0012-7094-45-01244-0
  3. R.H. Cameron and D.A. Storvick, An $L_{2}$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), 1-30. https://doi.org/10.1307/mmj/1029001617
  4. 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.
  5. K.S. Chang, B.S. Kim and I. Yoo, Integral transform and convolution of analytic functionals on abstract Wiener spaces, Numer. Funct. Anal. Optim. 21 (2000), 97-105. https://doi.org/10.1080/01630560008816942
  6. S.J. Chang and D. Skoug, Parts formulas involving conditional Feynman integrals, Bull. Aust. Math. Soc. 65 (2002), 353-369. https://doi.org/10.1017/S0004972700020402
  7. D.M. Chung ,C. Park and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J. 40 (1993), 377-391. https://doi.org/10.1307/mmj/1029004758
  8. D.M. Chung and D. Skoug, Conditional analytic Feynman integrals and a related Schrodinger integral equation, SIAM J. Math. Anal. 20 (1989), 950-965. https://doi.org/10.1137/0520064
  9. B.A. Fuks, Theory of analytic functions of several complex variables, Amer. Math. Soc., Providence, Rhode Island, 1963.
  10. T. Huffman, C. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), 661-673. https://doi.org/10.1090/S0002-9947-1995-1242088-7
  11. B.J. Kim, B.S. Kim and D. Skoug, Integral transforms, convolution products and first variations, Int. J. Math. Math. Sci. 11 (2004), 579-598.
  12. B.J. Kim, B.S. Kim and D. Skoug, Conditional integral transforms, conditional convolution products and first variations, Panamer. Math. J. 14 (2004), 27-47.
  13. B.S. Kim and D. Skoug, Integral transforms of functionals in $L_{2}(C_{0}[0,T])$, Rocky Mountain J. Math. 33 (2003), 1379 - 1393. https://doi.org/10.1216/rmjm/1181075469
  14. Y.J. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), 153-164. https://doi.org/10.1016/0022-1236(82)90103-3
  15. C. Park and D. Skoug, Conditional Fourier-Feynman transforms and conditional convolution products, J. Korean Math. Soc. 38 (2001), 61-76.
  16. C. Park and D. Skoug, A simple formula for conditional Wiener integrals with applications, Pacific J. Math. 135 (1988), 381-394. https://doi.org/10.2140/pjm.1988.135.381
  17. C. Park and D. Skoug, A Kac-Feynman integral equation for conditional Wiener integrals, J. Integral Equations Appl. 3 (1991), 411-427. https://doi.org/10.1216/jiea/1181075633
  18. 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. https://doi.org/10.1216/rmjm/1181071725
  19. D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math. 34 (2004), 1147-1175. https://doi.org/10.1216/rmjm/1181069848
  20. J. Yeh, Convolution in Fourier-Wiener transform, Pacific J. Math. 15 (1965), 731-738. https://doi.org/10.2140/pjm.1965.15.731
  21. J. Yeh, Inversion of conditional Wiener integral, Pacific J. Math. 59 (1975), 623-638. https://doi.org/10.2140/pjm.1975.59.623
  22. I. Yoo, Convolution and the Fourier-Wiener transform on abstract Wiener space, Rocky Mountain J. Math. 25 (1995), 1577-1587. https://doi.org/10.1216/rmjm/1181072163

Cited by

  1. CONDITIONAL INTEGRAL TRANSFORMS AND CONVOLUTIONS FOR A GENERAL VECTOR-VALUED CONDITIONING FUNCTIONS vol.24, pp.3, 2012, https://doi.org/10.11568/kjm.2016.24.3.573
  2. Movie Box-office Prediction using Deep Learning and Feature Selection : Focusing on Multivariate Time Series vol.25, pp.6, 2020, https://doi.org/10.9708/jksci.2020.25.06.035