Fourier-Feynman transforms of unbounded functionals on abstract Wiener space

Abstract

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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [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

OPEN ACCESS

Journal + Issues

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.

Search