Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

• Autorzy: Byoung Kim , Il Yoo , Dong Cho
• Języki publikacji: EN

Abstrakty

EN

Huffman, Park and Skoug established several results involving Fourier-Feynman transform and convolution for functionals in a Banach algebra S on the classical Wiener space. Chang, Kim and Yoo extended these results to abstract Wiener space for a more generalized Fresnel class $$ \mathcal{F}_{\mathcal{A}_1 ,\mathcal{A}_2 } $$ A1,A2 than the Fresnel class $$ \mathcal{F} $$(B)which corresponds to the Banach algebra S. In this paper we study Fourier-Feynman transform, convolution and first variation of unbounded functionals on abstract Wiener space having the form $$ F\left( x \right) = G\left( x \right)\psi \left( {\left( {\vec e,x} \right)^ \sim } \right) $$, where G∈$$ \mathcal{F} $$(B)and Ψ = ψ + ϕ with ψ ∈ L 1(ℝn) and ϕ is the Fourier transform of a complex Borel measure of bounded variation on ℝn. We also prove a translation theorem for the analytic Feynman integral of the above functionals.

Słowa kluczowe

EN

Abstract Wiener space; Fresnel class; Analytic Feynman integral; Fourier-Feynman transform; Convolution; First variation; Translation theorem

Kategorie tematyczne

• 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 3
• Strony: 616-632

Daty

wydano

2010-06-01

online

2010-05-30

Twórcy

autor

Byoung Kim

• Seoul National University of Technology

autor

Il Yoo

• Yonsei University

autor

Dong Cho

• Kyonggi University

Bibliografia

• [1] Ahn J.M., Chang K.S., Kim B.S., Yoo I., Fourier-Feynman transform, convolution and first variation, Acta Math. Hungar., 2003, 100, 215–235 http://dx.doi.org/10.1023/A:1025041525913
• [2] Cameron R.H., Storvick D.A., An L 2 analytic Fourier-Feynman transform, Michigan Math. J., 1976, 23, 1–30 http://dx.doi.org/10.1307/mmj/1029001617
• [3] Cameron R.H., Storvick D.A., Some Banach algebras of analytic Feynman integrable functionals, Analytic Functions (Kozubnik, 1979), Lecture Notes in Mathematics 798, Springer-Verlag, Berlin-New York, 1980, 18–67
• [4] Cameron R.H., Storvick D.A., A new translation theorem for the analytic Feynman integral, Rev. Roumaine Math. Pures Appl., 1982, 27(9), 937–944
• [5] Cameron R.H., Storvick D.A., Feynman integral of variations of functionals, Gaussian random fields (Nagoya, 1990), Ser. Probab. Statist. 1, World Sci. Publ., River Edge, NJ, 1991, 1, 144–157
• [6] Chang K.S., Cho D.H., Kim B.S., Song T.S., Yoo I., Relationships involving generalized Fourier-Feynman transform, convolution and first variation, Integral Transform. Spec. Funct., 2005, 16, 391–405 http://dx.doi.org/10.1080/10652460412331320359
• [7] Chang K.S., Cho D.H., Kim B.S., Song T.S., Yoo I., Sequential Fourier-Feynman transform, convolution and first variation, Trans. Amer. Math. Soc., 2008, 360, 1819–1838 http://dx.doi.org/10.1090/S0002-9947-07-04383-8
• [8] Chang K.S., Kim B.S., Yoo I., Analytic Fourier-Feynman transform and convolution of functionals on abstract Wiener space, Rocky Mountain J. Math., 2000, 30, 823–842 http://dx.doi.org/10.1216/rmjm/1021477245
• [9] Chang K.S., Kim B.S., Yoo I., Fourier-Feynman transform, convolution and first variation of functionals on abstract Wiener space, Integral Transform. Spec. Funct., 2000, 10, 179–200 http://dx.doi.org/10.1080/10652460008819285
• [10] Huffman T., Park C., Skoug D., Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc., 1995, 347, 661–673 http://dx.doi.org/10.2307/2154908
• [11] Huffman T., Park C., Skoug D., Convolutions and Fourier-Feynman transforms of functionals involving multiple integrals, Michigan Math. J., 1996, 43, 247–261 http://dx.doi.org/10.1307/mmj/1029005461
• [12] Johnson G.W., Skoug D., An L p analytic Fourier-Feynman transform, Michigan Math. J., 1979, 26, 103–127 http://dx.doi.org/10.1307/mmj/1029002166
• [13] Kallianpur G., Bromley C., Generalized Feynman integrals using analytic continuation in several complex variables, In: Stochastic Analysis and Applications, Dekker, New York, 1984, 433–450
• [14] Kallianpur G., Kannan D., Karandikar R.L., Analytic and sequential Feynman integrals on abstract Wiener and Hilbert spaces and a Cameron-Martin formula, Ann. Inst. Henri Poincare, 1985, 21, 323–361
• [15] Kuo H.H., Gaussian measures in Banach spaces, Lecture Notes in Mathematics 463, Springer-Verlag, Berlin, 1975
• [16] Park C., Skoug D., Storvick D., Relationships among the first variation, the convolution product, and the Fourier-Feynman transform, Rocky Mountain J. Math., 1998, 28, 1447–1468 http://dx.doi.org/10.1216/rmjm/1181071725
• [17] Skoug D., Storvick D., A survey of results involving transforms and convolutions in function space, Rocky Mountain J. Math., 2004, 34, 1147–1176 http://dx.doi.org/10.1216/rmjm/1181069848
• [18] Yeh J., Convolution in Fourier-Wiener transform, Pacific J. Math., 1965, 15, 731–738
• [19] Yoo I., Convolution and the Fourier-Wiener transform on abstract Wiener space, Rocky Mountain J. Math., 1995, 25, 1577–1587 http://dx.doi.org/10.1216/rmjm/1181072163
• [20] Yoo I., Song T.S., Kim B.S., A change of scale formula for Wiener integrals of unbounded functions II, Commun. Korean Math. Soc., 2006, 21, 117–133 http://dx.doi.org/10.4134/CKMS.2006.21.1.117
• [21] Yoo I., Song T.S., Kim B.S., Chang K.S., A change of scale formula for Wiener integrals of unbounded functions, Rocky Mountain J. Math., 2004, 34, 371–389 http://dx.doi.org/10.1216/rmjm/1181069911

Identyfikatory

DOI

10.2478/s11533-010-0019-2