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Let C[0, t] denote the function space of real-valued continuous paths on [0, t]. Define Xn : C[0, t] → Rn+1 and Xn+1 C[0, t] → Rn+2 by Xn(x) = (x(t0), x(t1), ..., x(tn)) and Xn+1(x) = (x(t0), x(t1), ...,x(tn), x(tn+1)), respectively, where 0 = t0 < t1 < … < tn<tn+1 = t. In the present paper, using simple formulas for the conditional expectations with the conditioning functions Xn and Xn+1, we evaluate the Lp(1 ≤ p ≤ ∞)-analytic conditional Fourier-Feynman transforms and the conditional convolution products of the functions, which have the form fr((v1,x),...,(vr,x)) ∫L2[0,t] exp{i(v,x)} dσ(v) for x ∈ C[0,t], where {v1, ..., vr} is an orthonormal subset of L2[0,t], fr ∈ Lp(Rr), and σ is the complex Borel measure of bounded variation on L2[0,t]. We then investigate the inverse conditional Fourier-Feynman transforms of the function and prove that the analytic conditional Fourier-Feynman transforms of the conditional convolution products for the functions can be expressed by the products of the analytic conditional Fourier-Feynman transform of each function.
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참고문헌 (25)
참고문헌 신청
B. J. Kim / 2004 / Conditional integral transforms, conditional con- volution products and first variations / Panamer. Math. J 14 (3) : 27 ~ 47
C. Park / 1988 / A simple formula for conditional Wiener integrals with applications / Pacific J. Math 135 (2) : 381 ~ 394
C. Park / 2001 / Conditional Fourier-Feynman transforms and conditional con- volution products / J. Korean Math. Soc 38 (1) : 61 ~ 76
D. H. Cho / 2008 / A simple formula for an analogue of conditional Wiener integrals and its ap- plications / Trans. Amer. Math. Soc 360 (7) : 3795 ~ 3811
D. H. Cho / 2009 / A simple formula for an analogue of conditional Wiener integrals and its ap- plications II / Czechoslovak Math. J 59 (2) : 431 ~ 452
C. Park / 1988 / A simple formula for conditional Wiener integrals with applications / Pacific J. Math 135 (2) : 381 ~ 394
C. Park / 2001 / Conditional Fourier-Feynman transforms and conditional con- volution products / J. Korean Math. Soc 38 (1) : 61 ~ 76
D. H. Cho / 2008 / A simple formula for an analogue of conditional Wiener integrals and its ap- plications / Trans. Amer. Math. Soc 360 (7) : 3795 ~ 3811
D. H. Cho / 2009 / A simple formula for an analogue of conditional Wiener integrals and its ap- plications II / Czechoslovak Math. J 59 (2) : 431 ~ 452
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