DOI QR코드

DOI QR Code

RELATIONSHIPS BETWEEN INTEGRAL TRANSFORMS AND CONVOLUTIONS ON AN ANALOGUE OF WIENER SPACE

Cho, Dong Hyun

  • Received : 2013.01.13
  • Accepted : 2013.01.28
  • Published : 2013.03.25

Abstract

In the present paper, we evaluate the analytic conditional Fourier-Feynman transforms and convolution products of unbounded function which is the product of the cylinder function and the function in a Banach algebra which is defined on an analogue o Wiener space and useful in the Feynman integration theories and quantum mechanics. We then investigate the inverse transforms of the function with their relationships and finally prove that th analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions, can be expressed in terms of the product of the conditional Fourier-Feynman transforms of each function.

Keywords

analogue of Wiener space;analytic conditional Feynman integral;analytic conditional Fourier-Feynman transform;conditional convolution product;conditional Wiener integral;Wiener space

References

  1. Cameron R.H., Storvick D.A., Some Banach algebras of analytic Feynman integrable functionals, Lecture Notes in Mathematics, 798, Springer, Berlin-New York, 1980
  2. Chang S.J., Skoug D., The eect of drift on conditional Fourier-Feynman trans-forms and conditional convolution products, Int. J. Appl. Math., 2000, 2(4), 505-527
  3. Cho D.H., Conditional Fourier-Feynman transforms and convolutions of unbounded functions on a generalized Wiener space, preprint
  4. Cho D.H., A time-independent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space, Bull. Korean Math. Soc., (in Press, in Korea)
  5. Cho D.H., A time-dependent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space, preprint
  6. Cho D.H., Conditional integral transforms and convolutions of bounded functions on an analogue of Wiener space, preprint
  7. Cho D.H., Conditional integral transforms and conditional convolution products on a function space, Integral Transforms Spec. Funct., 2012, 23(6), 405-420 https://doi.org/10.1080/10652469.2011.596482
  8. Cho D.H., A simple formula for an analogue of conditional Wiener integrals and its applications II, Czechoslovak Math. J., 2009, 59(2), 431-452 https://doi.org/10.1007/s10587-009-0030-6
  9. Cho D.H., A simple formula for an analogue of conditional Wiener integrals and its applications, Trans. Amer. Math. Soc., 2008, 360(7), 3795-3811 https://doi.org/10.1090/S0002-9947-08-04380-8
  10. Cho D.H., Kim B.J., Yoo I., Analogues of conditional Wiener integrals and their change of scale transformations on a function space, J. Math. Anal. Appl., 2009, 359(2), 421-438 https://doi.org/10.1016/j.jmaa.2009.05.023
  11. Im M.K., Ryu K.S., An analogue of Wiener measure and its applications, J. Korean Math. Soc., 2002, 39(5), 801-819 https://doi.org/10.4134/JKMS.2002.39.5.801
  12. Johnson G.W., Skoug D.L., The Cameron-Storvick function space integral: an ${\mathcal{L}}({{\mathit{L}}_p},{{{\mathit{L}}_p{\prime}}})$ theory, Nagoya Math. J., 1976, 60, 93-137 https://doi.org/10.1017/S0027763000017189
  13. Kim M.J., Conditional Fourier-Feynman transform and convolution product on a function space, Int. J. Math. Anal., 2009, 3(10), 457-471
  14. Park S.B., Cho D.H., Choi Y.H., Conditional Fourier-Feynman transforms and convolutions over continuous paths, Int. Math. Forum. 2013, 8(9), 443-456
  15. Ryu K.S., Im M.K., A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula, Trans. Amer. Math. Soc., 2002, 354(12), 4921-4951 https://doi.org/10.1090/S0002-9947-02-03077-5
  16. Ryu K.S., Im M.K., Choi K.S., Survey of the theories for analogue of Wiener measure space, Interdiscip. Inform. Sci., 2009, 15(3), 319-337
  17. Stein E.M., Weiss G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.

Acknowledgement

Supported by : National Research Foundation(NRF)