Conditional Fourier-Feynman transforms and convolutions of unbounded functions on a generalized Wiener space
1
조회
0
다운로드
2
인용
0
참고문헌
키워드
초록
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.
Abstract
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.
참고문헌 (25)
- M. D. Brue, "A functional transform for Feynman integrals similar to the Fourier trans- form", pp.1972
- R. H. Cameron, "Some Banach algebras of analytic Feynman in- tegrable functionals", Lecture Notes in Mathematics, 798 (1980), Springer
- K. S. Chang, "Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space", Integral Transforms Spec. Funct, 14(3), pp.217~235 (2003)
- S. J. Chang, "A class of conditional Wiener integrals", J. Korean Math. Soc, 30(1), pp.161~172 (1993)
- S. J. Chang, "The effect of drift on conditional Fourier-Feynman trans- forms and conditional convolution products", Int. J. Appl. Math, 2(4), pp.505~527 (2000)
- S. J. Chang, "The effect of drift on the Fourier-Feynman transform, the convolution product and the first variation", Panamer. Math. J, 10(2), pp.25~38 (2000)
- 조동현, "A TIME-INDEPENDENT CONDITIONAL FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION PRODUCT ON AN ANALOGUE OF WIENER SPACE", 호남수학학술지, 35(2), pp.179~200 (2013), 호남수학회[DOI]
- D. H. Cho, "A time-dependent conditional Fourier-Feynman transform and convolution product on an analogue of Wiener space", Houston J. Math (2012)
- D. H. Cho, "Conditional integral transforms and convolutions of bounded functions on an analogue of Wiener space", J. Chungcheong Math. Soc. (2012)
- D. H. Cho, "Conditional integral transforms and conditional convolution products on a func- tion space", Integral Transforms Spec. Funct, 23(6), pp.405~420 (2012)
- D. H. Cho, "A simple formula for an analogue of conditional Wiener integrals and its ap- plications II", Czechoslovak Math. J, 59(2), pp.431~452 (2009)
- D. H. Cho, "A simple formula for an analogue of conditional Wiener integrals and its ap- plications", Trans. Amer. Math. Soc, 360(7), pp.3795~3811 (2008)
- Dong Hyun Cho, "Conditional Fourier-Feynman transform and convolution product over Wiener paths in abstract Wiener space : an{L}$_{\text{p}}$ theory", 대한수학회지, 41(2), pp.265~294 (2004), 대한수학회
- D. H. Cho, "Analogues of conditional Wiener integrals and their change of scale transformations on a function space", J. Math. Anal. Appl, 359(2), pp.421~438 (2009)
- T. Huffman, "Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals", Michigan Math. J, 43(2), pp.247~261 (1996)
- 임만규, "An analogue of Wiener measure and its applications", 대한수학회지, 39(5), pp.801~819 (2002), 대한수학회
- G. W. Johnson, "The Cameron-Storvick function space integral: an L(Lp,Lp′ ) theory", Nagoya Math. J, 60, pp.93~137 (1976)
- M. J. Kim, "Conditional Fourier-Feynman transform and convolution product on a func- tion space", Int. J. Math. Anal, 3(10), pp.457~471 (2009)
- B. J. Kim, "Conditional integral transforms, conditional con- volution products and first variations", Panamer. Math. J, 14(3), pp.27~47 (2004)
- C. Park, "Conditional Fourier-Feynman transforms and conditional con- volution products", J. Korean Math. Soc, 38(1), pp.61~76 (2001)
- C. Park, "A simple formula for conditional Wiener integrals with applications", Pacific J. Math, 135(2), pp.381~394 (1988)
- K. S. Ryu, "A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula", Trans. Amer. Math. Soc, 354(12), pp.4921~4951 (2002)
- K. S. Ryu, "Survey of the theories for analogue of Wiener measure space", Interdiscip. Inform. Sci, 15(3), pp.319~337 (2009)
- E. M. Stein, "Introduction to Fourier analysis on Euclidean spaces" (1971), Prince- ton Univ. Press
- J. Yeh, "Inversions of conditional Wiener integrals", Pacific J. Math, 59(2), pp.623~638 (1975)