Korean Journal of Mathematics

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

DOI QR Code

CONDITIONAL INTEGRAL TRANSFORMS OF FUNCTIONALS ON A FUNCTION SPACE OF TWO VARIABLES

  • Received : 2022.09.01
  • Accepted : 2022.10.01
  • Published : 2022.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let C(Q) denote Yeh-Wiener space, the space of all real-valued continuous functions x(s, t) on Q ≡ [0, S] × [0, T] with x(s, 0) = x(0, t) = 0 for every (s, t) ∈ Q. For each partition τ = τm,n = {(si, tj)|i = 1, . . . , m, j = 1, . . . , n} of Q with 0 = s0 < s1 < . . . < sm = S and 0 = t0 < t1 < . . . < tn = T, define a random vector Xτ : C(Q) → ℝmn by Xτ (x) = (x(s1, t1), . . . , x(sm, tn)). In this paper we study the conditional integral transform and the conditional convolution product for a class of cylinder type functionals defined on K(Q) with a given conditioning function Xτ above, where K(Q)is the space of all complex valued continuous functions of two variables on Q which satify x(s, 0) = x(0, t) = 0 for every (s, t) ∈ Q. In particular we derive a useful equation which allows to calculate the conditional integral transform of the conditional convolution product without ever actually calculating convolution product or conditional convolution product.

Keywords

References

  1. K.S.Chang, B.S.Kim and I.Yoo, Integral transform and convolution of analytic functionals on abstract Wiener space, Numer. Fuct. Anal. and Optim., 21 (2000), 97-105.  https://doi.org/10.1080/01630560008816942
  2. B.A.Fuks, Theory of analytic functions of several complex variables, Amer. Math. Soc. Providence, Rhodo Island, 1963. 
  3. B.J.Kim Conditional Fourier-Feynman transform and convolution product for a vector valued conditoning function, Korean J. Math. 30 (2022), 239-247. 
  4. B.J.Kim, B.S. Kim and D.Skoug Conditional integral transforms, conditional convolution product products and first variations, PanAmerican Math. J. 14 (2004), 27-47. 
  5. B.J.Kim, B.S.Kim and I.Yoo, Integral transforms of functionals on a function space of two variables, J. Chungcheong Math. Soc. 23 (2010), 349-362. 
  6. B.S.Kim, Integral transforms of square integrable functionals on Yeh-Wiener space, Kyungpook Math. J. 49 (2009), 155-166.  https://doi.org/10.5666/KMJ.2009.49.1.155
  7. R.G.Laha and V.K.Rohatgi, Probability theory, John Wiley & Sons, New York-Chichester-Brisbane, 1979. 
  8. Y.J.Lee Integrsl transfoms of analytic functions on abstract Wiener space, J. Funct. Anal. 47 (1982), 153-164.  https://doi.org/10.1016/0022-1236(82)90103-3
  9. C.Park and D.Skoug, Conditional Yeh-Wiener integrals with vector-valued conditioning functions, Proc. Amer. Math. Soc. 105 (1989), 450-461.  https://doi.org/10.1090/S0002-9939-1989-0960650-6
  10. D.Skoug, Feynman integrals involving quadratic potentials, stochastic integration formulas, and bounded variation for functionals of several variables, Supplemento al Rendiconti del Circolo Mathematico di Palermo, Series II, No.17 (1987), 331-347. 
  11. 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
  12. J.Yeh, Wiener measure in a space of functions of two variables, Trans. Amer. Math. Soc. 95 (1960), 433-450. https://doi.org/10.1090/S0002-9947-1960-0125433-1