Sequential Fourier-Feynman transform, convolution and first variation

Chang, K.; Cho, D.; Kim, B.; Song, T.; Yoo, I.

doi:10.1090/S0002-9947-07-04383-8

Sequential Fourier-Feynman transform, convolution and first variation
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by K. S. Chang, D. H. Cho, B. S. Kim, T. S. Song and I. Yoo PDF

Trans. Amer. Math. Soc. 360 (2008), 1819-1838 Request permission

Abstract:

Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebra $\hat {\mathcal S}$ of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous. Also we study the relationships involving this transform and the first variation.

References

M.D.Brue, A functional transform for Feynman integrals similar to the Fourier transform, Thesis, Univ. Minnesota, Minneapolis, 1972.
R. H. Cameron and D. A. Storvick, An $L_{2}$ analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), no. 1, 1–30. MR 404571, DOI 10.1307/mmj/1029001617
Robert Horton Cameron and David Arne Storvick, Some Banach algebras of analytic Feynman integrable functionals, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), Lecture Notes in Math., vol. 798, Springer, Berlin-New York, 1980, pp. 18–67. MR 577446
R. H. Cameron and D. A. Storvick, A simple definition of the Feynman integral, with applications, Mem. Amer. Math. Soc. 46 (1983), no. 288, iv+46. MR 719157, DOI 10.1090/memo/0288
R. H. Cameron and D. A. Storvick, Sequential Fourier-Feynman transforms, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 107–111. MR 802472, DOI 10.5186/aasfm.1985.1013
R. H. Cameron and D. A. Storvick, New existence theorems and evaluation formulas for sequential Feynman integrals, Proc. London Math. Soc. (3) 52 (1986), no. 3, 557–581. MR 833650, DOI 10.1112/plms/s3-52.3.557
Robert Horton Cameron and David Arne Storvick, New existence theorems and evaluation formulas for analytic Feynman integrals, Deformations of mathematical structures (Łódź/Lublin, 1985/87) Kluwer Acad. Publ., Dordrecht, 1989, pp. 297–308. MR 987746
K. S. Chang, B. S. Kim, T. S. Song, and I. Yoo, Convolution and analytic Fourier-Feynman transforms over paths in abstract Wiener space, Integral Transforms Spec. Funct. 13 (2002), no. 4, 345–362. MR 1918957, DOI 10.1080/10652460213523
Kun Soo Chang, Byoung Soo Kim, and Il Yoo, Analytic Fourier-Feynman transform and convolution of functionals on abstract Wiener space, Rocky Mountain J. Math. 30 (2000), no. 3, 823–842. MR 1797816, DOI 10.1216/rmjm/1021477245
K. S. Chang, B. S. Kim, and I. Yoo, Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space, Integral Transform. Spec. Funct. 10 (2000), no. 3-4, 179–200. Analytical methods of analysis and differential equations (Minsk, 1999). MR 1811008, DOI 10.1080/10652460008819285
Kun Soo Chang, Teuk Seob Song, and Il Yoo, Analytic Fourier-Feynman transform and first variation on abstract Wiener space, J. Korean Math. Soc. 38 (2001), no. 2, 485–501. MR 1817632
Timothy Huffman, Chull Park, and David Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), no. 2, 661–673. MR 1242088, DOI 10.1090/S0002-9947-1995-1242088-7
Timothy Huffman, Chull Park, and David Skoug, Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals, Michigan Math. J. 43 (1996), no. 2, 247–261. MR 1398153, DOI 10.1307/mmj/1029005461
Timothy Huffman, Chull Park, and David Skoug, Convolution and Fourier-Feynman transforms, Rocky Mountain J. Math. 27 (1997), no. 3, 827–841. MR 1490278, DOI 10.1216/rmjm/1181071896
G. W. Johnson and D. L. Skoug, An $L_{p}$ analytic Fourier-Feynman transform, Michigan Math. J. 26 (1979), no. 1, 103–127. MR 514964, DOI 10.1307/mmj/1029002166
Chull Park, David Skoug, and David Storvick, Relationships among the first variation, the convolution product and the Fourier-Feynman transform, Rocky Mountain J. Math. 28 (1998), no. 4, 1447–1468. MR 1681677, DOI 10.1216/rmjm/1181071725
David Skoug and David Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math. 34 (2004), no. 3, 1147–1175. MR 2087452, DOI 10.1216/rmjm/1181069848