## Banach Journal of Mathematical Analysis

### Analytic Fourier–Feynman transforms and convolution products associated with Gaussian processes on Wiener space

#### Abstract

Using Gaussian processes, we define a very general convolution product of functionals on Wiener space and we investigate fundamental relationships between the generalized Fourier–Feynman transforms and the generalized convolution products. Using two rotation theorems of Gaussian processes, we establish that both of the generalized Fourier–Feynman transform of the generalized convolution product and the generalized convolution product of the generalized Fourier–Feynman transforms of functionals on Wiener space are represented as products of the generalized Fourier–Feynman transforms of each functional, with examples.

#### Article information

Source
Banach J. Math. Anal., Volume 11, Number 4 (2017), 785-807.

Dates
Received: 4 July 2016
Accepted: 1 November 2016
First available in Project Euclid: 30 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1498809807

Digital Object Identifier
doi:10.1215/17358787-2017-0017

Mathematical Reviews number (MathSciNet)
MR3708529

Zentralblatt MATH identifier
1382.28010

#### Citation

Chang, Seung Jun; Choi, Jae Gil. Analytic Fourier–Feynman transforms and convolution products associated with Gaussian processes on Wiener space. Banach J. Math. Anal. 11 (2017), no. 4, 785--807. doi:10.1215/17358787-2017-0017. https://projecteuclid.org/euclid.bjma/1498809807

#### References

• [1] J. E. Bearman, Rotations in the product of two Wiener spaces, Proc. Amer. Math. Soc. 3 (1952), 129–137.
• [2] M. D. Brue, A functional transform for Feynman integrals similar to the Fourier transform, Ph.D. dissertation, University of Minnesota, Minneapolis, Minnesota, 1972.
• [3] R. H. Cameron and D. A. Storvick, An $L_{2}$ analytic Fourier–Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1–30.
• [4] R. H. Cameron and D. A. Storvick, An operator valued Yeh–Wiener integral, and a Wiener integral equation, Indiana Univ. Math. J. 25 (1976), no. 3, 235–258.
• [5] R. H. Cameron and D. A. Storvick, “Some Banach algebras of analytic Feynman integrable functionals” in Analytic Functions (Kozubnik, 1979), Lecture Notes in Math. 798, Springer, Berlin, 1980, 18–67.
• [6] M. Carter and B. van Brunt, The Lebesgue–Stieltjes Integral, Undergrad. Texts Math., Springer, Berlin, 2000.
• [7] K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song, and I. Yoo, Relationships involving generalized Fourier–Feynman transform, convolution and first variation, Integral Transforms Spec. Funct. 16 (2005), no. 5–6, 391–405.
• [8] J. G. Choi, D. Skoug, and S. J. Chang, A multiple generalized Fourier–Feynman transform via a rotation on Wiener space, Internat. J. Math. 23 (2012), no. 7, article ID 1250068.
• [9] D. M. Chung, C. Park, and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J. 40 (1993), no. 2, 377–391.
• [10] T. Huffman, C. Park, and D. Skoug, Analytic Fourier–Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), no. 2, 661–673.
• [11] T. Huffman, C. Park, and D. Skoug, Convolutions and Fourier–Feynman transforms of functionals involving multiple integrals, Michigan Math. J. 43 (1996), no. 2, 247–261.
• [12] T. Huffman, C. Park, and D. Skoug, Convolution and Fourier–Feynman transforms, Rocky Mountain J. Math. 27 (1997), no. 3, 827–841.
• [13] T. Huffman, C. Park, and D. Skoug, Generalized transforms and convolutions, Int. J. Math. Math. Sci. 20 (1997), no. 1, 19–32.
• [14] G. W. Johnson and D. Skoug, An $L_{p}$ analytic Fourier–Feynman transform, Michigan Math. J. 26 (1979), no. 1, 103–127.
• [15] G. W. Johnson and D. Skoug, Notes on the Feynman integral, II, J. Funct. Anal. 41 (1981), no. 3, 277–289.
• [16] R. E. A. C. Paley, N. Wiener, and A. Zygmund, Notes on random functions, Math. Z. 37 (1933), 647–668.
• [17] C. Park and D. Skoug, A note on Paley–Wiener–Zygmund stochastic integrals, Proc. Amer. Math. Soc. 103 (1988), no. 2, 591–601.
• [18] C. Park and D. Skoug, A Kac–Feynman integral equation for conditional Wiener integrals, J. Integral Equations Appl. 3 (1991), no. 3, 411–427.
• [19] C. Park and D. Skoug, Integration by parts formulas involving analytic Feynman integrals, PanAmer. Math. J. 8 (1998), no. 4, 1–11.
• [20] C. Park and D. Skoug, Conditional Fourier–Feynman transforms and conditional convolution products, J. Korean Math. Soc. 38 (2001), no. 1, 61–76.
• [21] D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math. 34 (2004), no. 3, 1147–1175.
• [22] J. Yeh, Convolution in Fourier–Wiener transform, Pacific J. Math. 15 (1965), 731–738.