Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RELATIONS AMONG THE FIRST VARIATION, THE CONVOLUTIONS AND THE GENERALIZED FOURIER-GAUSS TRANSFORMS

  • Im, Man-Kyu (Department of Mathematics Hannam University) ;
  • Ji, Un-Cig (Department of Mathematics Research institute of Mathematical Finance Chungbuk National University) ;
  • Park, Yoon-Jung (Department of Mathematics Chungbuk National University)
  • Received : 2009.06.26
  • Published : 2011.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

We first study the generalized Fourier-Gauss transforms of functionals defined on the complexification $\cal{B}_C$ of an abstract Wiener space ($\cal{H}$, $\cal{B}$, ${\nu}$). Secondly, we introduce a new class of convolution products of functionals defined on $\cal{B}_C$ and study several properties of the convolutions. Then we study various relations among the first variation the convolutions, and the generalized Fourier-Gauss transforms.

Keywords

References

  1. R. H. Cameron, Some examples of Fourier-Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), 485-488. https://doi.org/10.1215/S0012-7094-45-01243-9
  2. 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
  3. 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
  4. R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1-30. https://doi.org/10.1307/mmj/1029001617
  5. D. M. Chung and U. C. Ji, Transforms on white noise functionals with their applications to Cauchy problems, Nagoya Math. J. 147 (1997), 1-23. https://doi.org/10.1017/S0027763000006292
  6. D. M. Chung, Transformation groups on white noise functionals and their applications, Appl. Math. Optim. 37 (1998), no. 2, 205-223. https://doi.org/10.1007/s002459900074
  7. L. Gross, Potential theory on Hilbert space, J. Funct. Anal. 1 (1967), 123-181. https://doi.org/10.1016/0022-1236(67)90030-4
  8. T. Hida, Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes, No. 13. Carleton Univ., Ottawa, Ont., 1975.
  9. T. Hida, H. H. Kuo, and N. Obata, Transformations for white noise functionals, J. Funct. Anal. 111 (1993), no. 2, 259-277. https://doi.org/10.1006/jfan.1993.1012
  10. T. Huffman, C. Park, and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), no. 2, 661-673. https://doi.org/10.2307/2154908
  11. U. C. Ji and N. Obata, Quantum white noise calculus, Non-commutativity, infinite-dimensionality and probability at the crossroads, 143-191, QP-PQ: Quantum Probab. White Noise Anal., 16, World Sci. Publ., River Edge, NJ, 2002.
  12. U. C. Ji and N. Obata, Bogoliubov transformations in terms of generalized Fourier-Gauss transformations, preprint, 2008.
  13. H. H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Mathematics, Vol. 463. Springer-Verlag, Berlin-New York, 1975.
  14. H. H. Kuo, White Noise Distribution Theory, CRC Press, Boca Raton, FL, 1996.
  15. Y. J. Lee, Integral transforms of analytic functions on abstract Wiener spaces, J. Funct. Anal. 47 (1982), no. 2, 153-164. https://doi.org/10.1016/0022-1236(82)90103-3
  16. N. Obata, White Noise Calculus and Fock Space, Lecture Notes in Mathematics, 1577. Springer-Verlag, Berlin, 1994.
  17. J. Yeh, Convolution in Fourier-Wiener transform, Pacific J. Math. 15 (1965), 731-738. https://doi.org/10.2140/pjm.1965.15.731
  18. I. Yoo, Convolution and the Fourier-Wiener transform on abstract Wiener space, Rocky Mountain J. Math. 25 (1995), no. 4, 1577-1587. https://doi.org/10.1216/rmjm/1181072163

Cited by

  1. YEH CONVOLUTION OF WHITE NOISE FUNCTIONALS vol.31, pp.5_6, 2013, https://doi.org/10.14317/jami.2013.825
  2. SOME RELATIONSHIPS BETWEEN THE INTEGRAL TRANSFORM AND THE CONVOLUTION PRODUCT ON ABSTRACT WIENER SPACE vol.31, pp.1, 2015, https://doi.org/10.7858/eamj.2015.014
  3. Generalized Cameron–Storvick theorem and its applications 2017, https://doi.org/10.1080/10652469.2017.1346636
  4. STOCHASTIC DIFFERENTIAL EQUATION FOR WHITE NOISE FUNCTIONALS vol.29, pp.2, 2016, https://doi.org/10.14403/jcms.2016.29.2.337
  5. SOME EXPRESSIONS FOR THE INVERSE INTEGRAL TRANSFORM VIA THE TRANSLATION THEOREM ON FUNCTION SPACE vol.53, pp.6, 2016, https://doi.org/10.4134/JKMS.j150485
  6. Relationships for modified generalized integral transforms, modified convolution products and first variations on function space vol.25, pp.10, 2014, https://doi.org/10.1080/10652469.2014.918614
  7. Double integral transforms and double convolution products of functionals on abstract Wiener space vol.24, pp.11, 2013, https://doi.org/10.1080/10652469.2013.783577
  8. Factorization property of convolutions of white noise operators vol.46, pp.4, 2015, https://doi.org/10.1007/s13226-015-0146-3
  9. Generalized convolution product for an integral transform on a Wiener space vol.41, pp.13036149, 2017, https://doi.org/10.3906/mat-1601-79
  10. New expressions of the modified generalized integral transform via the translation theorem with applications vol.29, pp.2, 2018, https://doi.org/10.1080/10652469.2017.1414214