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Algebraic structure of the $ L_2 $ analytic Fourier–Feynman transform associated with Gaussian paths on Wiener space
1. | School of General Education, Dankook University, Cheonan 31116, Republic of Korea |
2. | Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588-0130, USA |
In this paper we study algebraic structures of the classes of the $ L_2 $ analytic Fourier–Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian paths. We then proceed to analyze the $ L_2 $ analytic Fourier–Feynman transforms associated with Gaussian paths. Our results show that these $ L_2 $ analytic Fourier–Feynman transforms are actually linear operator isomorphisms from a Hilbert space into itself. We finally investigate the algebraic structures of these classes of the transforms on Wiener space, and show that they indeed are group isomorphic.
References:
[1] |
J. E. Bearman,
Rotations in the product of two Wiener spaces, Proc. Amer. Math. Soc., 3 (1952), 129-137.
doi: 10.2307/2032469. |
[2] |
M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D Thesis, University of Minnesota, 1972. |
[3] |
R. H. Cameron and D. A. Storvick,
An $L_2$ analytic Fourier–Feynman transform, Michigan Math. J., 23 (1976), 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), 235-258.
doi: 10.1512/iumj.1976.25.25020. |
[5] |
S. J. Chang and J. G. Choi,
Rotation of Gaussian paths on Wiener space with applications, Banach J. Math. Anal., 12 (2018), 651-672.
doi: 10.1215/17358787-2017-0057. |
[6] |
S. J. Chang, H. S. Chung and J. G. Choi,
Generalized Fourier–Feynman transforms and generalized convolution products on Wiener space, Indag. Math., 28 (2017), 566-579.
doi: 10.1016/j.indag.2017.01.004. |
[7] |
J. G. Choi and S. J. Chang,
Note on generalized Wiener integrals, Arch. Math., 101 (2013), 569-579.
doi: 10.1007/s00013-013-0595-z. |
[8] |
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), Art. 1250068.
doi: 10.1142/S0129167X12500681. |
[9] |
D. M. Chung, C. Park and D. Skoug,
Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.
doi: 10.1307/mmj/1029004758. |
[10] |
T. Huffman, C. Park and D. Skoug,
Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347 (1995), 661-673.
doi: 10.2307/2154908. |
[11] |
T. Huffman, C. Park and D. Skoug,
Generalized transforms and convolutions, Int. J. Math. Math. Sci., 20 (1997), 19-32.
doi: 10.1155/S0161171297000045. |
[12] |
T. Huffman, D. Skoug and D. Storvick,
Integration formulas involving Fourier-Feynman transforms via a Fubini theorem, J. Korean Math., 38 (2001), 421-435.
|
[13] |
G. W. Johnson and D. L. Skoug,
An $L_p$ analytic Fourier-Feynman transform, Michigan Math. J., 26 (1979), 103-127.
|
[14] |
G. W. Johnson and D. L. Skoug,
Scale-invariant measurability in Wiener space, Pac. J. Math., 83 (1979), 157-176.
|
[15] |
G. W. Johnson and D. L. Skoug,
Notes on the Feynman integral, II, J. Funct. Anal., 41 (1981), 277-289.
doi: 10.1016/0022-1236(81)90075-6. |
[16] |
R. E. A. C. Paley, N. Wiener and A. Zygmund,
Notes on random functions, Math. Z., 37 (1933), 647-668.
doi: 10.1007/BF01474606. |
[17] |
C. Park and D. Skoug,
A note on Paley–Wiener–Zygmund stochastic integrals, Proc. Amer. Math. Soc., 103 (1988), 591-601.
doi: 10.2307/2047184. |
[18] |
C. Park and D. Skoug,
A Kac-Feynman integral equation for conditional Wiener integrals, J. Integral Equ. Appl., 3 (1991), 411-427.
doi: 10.1216/jiea/1181075633. |
[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] |
D. Robinson, A course in the Theory of Groups, 2$^{nd}$ edition, Graduate texts in mathematics, Vol. 80, Springer, New York, 1996.
doi: 10.1007/978-1-4419-8594-1. |
[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, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973. |
show all references
References:
[1] |
J. E. Bearman,
Rotations in the product of two Wiener spaces, Proc. Amer. Math. Soc., 3 (1952), 129-137.
doi: 10.2307/2032469. |
[2] |
M. D. Brue, A Functional Transform for Feynman Integrals Similar to the Fourier Transform, Ph.D Thesis, University of Minnesota, 1972. |
[3] |
R. H. Cameron and D. A. Storvick,
An $L_2$ analytic Fourier–Feynman transform, Michigan Math. J., 23 (1976), 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), 235-258.
doi: 10.1512/iumj.1976.25.25020. |
[5] |
S. J. Chang and J. G. Choi,
Rotation of Gaussian paths on Wiener space with applications, Banach J. Math. Anal., 12 (2018), 651-672.
doi: 10.1215/17358787-2017-0057. |
[6] |
S. J. Chang, H. S. Chung and J. G. Choi,
Generalized Fourier–Feynman transforms and generalized convolution products on Wiener space, Indag. Math., 28 (2017), 566-579.
doi: 10.1016/j.indag.2017.01.004. |
[7] |
J. G. Choi and S. J. Chang,
Note on generalized Wiener integrals, Arch. Math., 101 (2013), 569-579.
doi: 10.1007/s00013-013-0595-z. |
[8] |
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), Art. 1250068.
doi: 10.1142/S0129167X12500681. |
[9] |
D. M. Chung, C. Park and D. Skoug,
Generalized Feynman integrals via conditional Feynman integrals, Michigan Math. J., 40 (1993), 377-391.
doi: 10.1307/mmj/1029004758. |
[10] |
T. Huffman, C. Park and D. Skoug,
Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 347 (1995), 661-673.
doi: 10.2307/2154908. |
[11] |
T. Huffman, C. Park and D. Skoug,
Generalized transforms and convolutions, Int. J. Math. Math. Sci., 20 (1997), 19-32.
doi: 10.1155/S0161171297000045. |
[12] |
T. Huffman, D. Skoug and D. Storvick,
Integration formulas involving Fourier-Feynman transforms via a Fubini theorem, J. Korean Math., 38 (2001), 421-435.
|
[13] |
G. W. Johnson and D. L. Skoug,
An $L_p$ analytic Fourier-Feynman transform, Michigan Math. J., 26 (1979), 103-127.
|
[14] |
G. W. Johnson and D. L. Skoug,
Scale-invariant measurability in Wiener space, Pac. J. Math., 83 (1979), 157-176.
|
[15] |
G. W. Johnson and D. L. Skoug,
Notes on the Feynman integral, II, J. Funct. Anal., 41 (1981), 277-289.
doi: 10.1016/0022-1236(81)90075-6. |
[16] |
R. E. A. C. Paley, N. Wiener and A. Zygmund,
Notes on random functions, Math. Z., 37 (1933), 647-668.
doi: 10.1007/BF01474606. |
[17] |
C. Park and D. Skoug,
A note on Paley–Wiener–Zygmund stochastic integrals, Proc. Amer. Math. Soc., 103 (1988), 591-601.
doi: 10.2307/2047184. |
[18] |
C. Park and D. Skoug,
A Kac-Feynman integral equation for conditional Wiener integrals, J. Integral Equ. Appl., 3 (1991), 411-427.
doi: 10.1216/jiea/1181075633. |
[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] |
D. Robinson, A course in the Theory of Groups, 2$^{nd}$ edition, Graduate texts in mathematics, Vol. 80, Springer, New York, 1996.
doi: 10.1007/978-1-4419-8594-1. |
[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, Stochastic Processes and the Wiener Integral, Marcel Dekker, Inc., New York, 1973. |
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