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January  2020, 19(1): 371-389. doi: 10.3934/cpaa.2020019

Generalized transforms and generalized convolution products associated with Gaussian paths on function space

Department of Mathematics, Dankook University, Cheonan 330-714, Republic of Korea

* Corresponding author

Received  December 2018 Revised  April 2019 Published  July 2019

In this paper we define a more general convolution product (associated with Gaussian processes) of functionals on the function space $ C_{a, b}[0, T] $. The function space $ C_{a, b}[0, T] $ is induced by a generalized Brownian motion process. Thus the Gaussian processes used in this paper are non-centered processes. We then develop the fundamental relationships between the generalized Fourier–Feynman transform associated with the Gaussian process and the convolution product.

Citation: Seung Jun Chang, Jae Gil Choi. Generalized transforms and generalized convolution products associated with Gaussian paths on function space. Communications on Pure and Applied Analysis, 2020, 19 (1) : 371-389. doi: 10.3934/cpaa.2020019
References:
[1]

M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D. Thesis, University of Minnesota, Minneapolis, 1972.

[2]

R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier–Feynman transform, Michigan Math. J., 23 (1976), 1-30. 

[3]

R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, in Analytic Functions, Kozubnik 1979 (eds. A. Dold and B. Eckmann), Lecture Notes in Math. 798, Springer, Berlin, (1980), 18–67.

[4]

S. J. Chang and J. G. Choi, Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral, Arch. Math., 106 (2016), 591-600.  doi: 10.1007/s00013-016-0899-x.

[5]

S. J. ChangJ. G. Choi and A. Y. Ko, Multiple generalized analytic Fourier–Feynman transform via rotation of Gaussian paths on function space, Banach J. Math. Anal., 9 (2015), 58-80.  doi: 10.15352/bjma/09-4-4.

[6]

S. J. ChangJ. G. Choi and D. Skoug, Integration by parts formulas involving generalized Fourier–Feynman transforms on function space, Trans. Amer. Math. Soc., 355 (2003), 2925-2948.  doi: 10.1090/S0002-9947-03-03256-2.

[7]

S. J. ChangJ. G. Choi and D. Skoug, Generalized Fourier–Feynman transforms, convolution products, and first variations on function space, Rocky Mount. J. Math., 40 (2010), 761-788.  doi: 10.1216/RMJ-2010-40-3-761.

[8]

S. J. Chang and D. Skoug, Generalized Fourier–Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct., 14 (2003), 375-393.  doi: 10.1080/1065246031000074425.

[9]

J. G. Choi, H. S. Chung and S. J. Chang, Sequential generalized transforms on function space, Abstr. Appl. Anal., 2013 (2013), Article ID: 565832. doi: 10.1155/2013/565832.

[10]

J. G. Choi, D. Skoug and S. J. Chang, A multiple generalized Fourier–Feynman transform via a rotation on Wiener space, Int. J. Math., 23 (2012), Article ID: 1250068. doi: 10.1142/S0129167X12500681.

[11]

D. M. ChungC. Park and and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.  doi: 10.1307/mmj/1029004758.

[12]

T. HuffmanC. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347 (1995), 661-673.  doi: 10.2307/2154908.

[13]

T. HuffmanC. Park and D. Skoug, Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals, Michigan Math. J., 43 (1996), 247-261.  doi: 10.1307/mmj/1029005461.

[14]

T. HuffmanC. Park and D. Skoug, Convolution and Fourier-Feynman transforms, Rocky Mountain J. Math., 27 (1997), 827-841.  doi: 10.1216/rmjm/1181071896.

[15]

T. HuffmanC. Park and D. Skoug, Generalized transforms and convolutions, Int. J. Math. Math. Sci., 20 (1997), 19-32.  doi: 10.1155/S0161171297000045.

[16]

G. W. Johnson and D. L. Skoug, An $L_p$ analytic Fourier–Feynman transform, Michigan Math. J., 26 (1979), 103-127. 

[17]

E. Nelson, Dynamical Theories of Brownian Motion, 2nd edition, Math. Notes, Princeton University Press, Princeton, 1967.

[18]

R. E. A. C. PaleyN. Wiener and A. Zygmund, Notes on random functions, Math. Z., 37 (1933), 647-668.  doi: 10.1007/BF01474606.

[19]

C. Park and D. Skoug, Generalized Feynman integrals: the $\mathcal L(L_2, L_2)$ theory, Rocky Mountain J. Math., 25 (1995), 739-756.  doi: 10.1216/rmjm/1181072247.

[20]

C. ParkD. Skoug and D. Storvick, Relationships among the first variation, the convolution product, and the Fourier–Feynman transform, Rocky Mountain J. Math., 28 (1998), 1447-1468.  doi: 10.1216/rmjm/1181071725.

[21]

D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math., 34 (2004), 1147-1175.  doi: 10.1216/rmjm/1181069848.

[22]

J. Yeh, Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments, Illinois J. Math., 15 (1971), 37-46. 

[23]

J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.

show all references

References:
[1]

M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D. Thesis, University of Minnesota, Minneapolis, 1972.

[2]

R. H. Cameron and D. A. Storvick, An $L_2$ analytic Fourier–Feynman transform, Michigan Math. J., 23 (1976), 1-30. 

[3]

R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, in Analytic Functions, Kozubnik 1979 (eds. A. Dold and B. Eckmann), Lecture Notes in Math. 798, Springer, Berlin, (1980), 18–67.

[4]

S. J. Chang and J. G. Choi, Effect of drift of the generalized Brownian motion process: an example for the analytic Feynman integral, Arch. Math., 106 (2016), 591-600.  doi: 10.1007/s00013-016-0899-x.

[5]

S. J. ChangJ. G. Choi and A. Y. Ko, Multiple generalized analytic Fourier–Feynman transform via rotation of Gaussian paths on function space, Banach J. Math. Anal., 9 (2015), 58-80.  doi: 10.15352/bjma/09-4-4.

[6]

S. J. ChangJ. G. Choi and D. Skoug, Integration by parts formulas involving generalized Fourier–Feynman transforms on function space, Trans. Amer. Math. Soc., 355 (2003), 2925-2948.  doi: 10.1090/S0002-9947-03-03256-2.

[7]

S. J. ChangJ. G. Choi and D. Skoug, Generalized Fourier–Feynman transforms, convolution products, and first variations on function space, Rocky Mount. J. Math., 40 (2010), 761-788.  doi: 10.1216/RMJ-2010-40-3-761.

[8]

S. J. Chang and D. Skoug, Generalized Fourier–Feynman transforms and a first variation on function space, Integral Transforms Spec. Funct., 14 (2003), 375-393.  doi: 10.1080/1065246031000074425.

[9]

J. G. Choi, H. S. Chung and S. J. Chang, Sequential generalized transforms on function space, Abstr. Appl. Anal., 2013 (2013), Article ID: 565832. doi: 10.1155/2013/565832.

[10]

J. G. Choi, D. Skoug and S. J. Chang, A multiple generalized Fourier–Feynman transform via a rotation on Wiener space, Int. J. Math., 23 (2012), Article ID: 1250068. doi: 10.1142/S0129167X12500681.

[11]

D. M. ChungC. Park and and D. Skoug, Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.  doi: 10.1307/mmj/1029004758.

[12]

T. HuffmanC. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347 (1995), 661-673.  doi: 10.2307/2154908.

[13]

T. HuffmanC. Park and D. Skoug, Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals, Michigan Math. J., 43 (1996), 247-261.  doi: 10.1307/mmj/1029005461.

[14]

T. HuffmanC. Park and D. Skoug, Convolution and Fourier-Feynman transforms, Rocky Mountain J. Math., 27 (1997), 827-841.  doi: 10.1216/rmjm/1181071896.

[15]

T. HuffmanC. Park and D. Skoug, Generalized transforms and convolutions, Int. J. Math. Math. Sci., 20 (1997), 19-32.  doi: 10.1155/S0161171297000045.

[16]

G. W. Johnson and D. L. Skoug, An $L_p$ analytic Fourier–Feynman transform, Michigan Math. J., 26 (1979), 103-127. 

[17]

E. Nelson, Dynamical Theories of Brownian Motion, 2nd edition, Math. Notes, Princeton University Press, Princeton, 1967.

[18]

R. E. A. C. PaleyN. Wiener and A. Zygmund, Notes on random functions, Math. Z., 37 (1933), 647-668.  doi: 10.1007/BF01474606.

[19]

C. Park and D. Skoug, Generalized Feynman integrals: the $\mathcal L(L_2, L_2)$ theory, Rocky Mountain J. Math., 25 (1995), 739-756.  doi: 10.1216/rmjm/1181072247.

[20]

C. ParkD. Skoug and D. Storvick, Relationships among the first variation, the convolution product, and the Fourier–Feynman transform, Rocky Mountain J. Math., 28 (1998), 1447-1468.  doi: 10.1216/rmjm/1181071725.

[21]

D. Skoug and D. Storvick, A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math., 34 (2004), 1147-1175.  doi: 10.1216/rmjm/1181069848.

[22]

J. Yeh, Singularity of Gaussian measures on function spaces induced by Brownian motion processes with non-stationary increments, Illinois J. Math., 15 (1971), 37-46. 

[23]

J. Yeh, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973.

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