Korean J. Math.  Vol 23, No 1 (2015)  pp.47-64
DOI: https://doi.org/10.11568/kjm.2015.23.1.47

Change of scale formulas for function space integrals related with Fourier-Feynman transform and convolution on $C_{a,b}[0,T]$

Bong Jin Kim, Byoung Soo Kim, Il Yoo

Abstract


We express generalized Fourier-Feynman transform and convolution product of  functionals in a Banach algebra ${\mathcal S}(L_{a,b}^2[0,T])$ as limits of  function space integrals on $C_{a,b}[0,T]$. Moreover we obtain change of scale formulas for function space integrals related with generalized Fourier-Feynman transform and convolution product of these functionals.

Keywords


change of scale formula, function space integral, generalized analytic Feynman integral, generalized Fourier-Feynman transform, convolution

Subject classification

28C20, 60J25, 60J65

Sponsor(s)

This study was supported by the Research Program funded by the Seoul National University of Science and Technology.

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References


R.H. Cameron, The translation pathology of Wiener space, Duke Math. J. 21 (1954), 623–628. (Google Scholar)

R.H. Cameron and W.T. Martin, The behavior of measure and measurability under change of scale in Wiener space, Bull. Amer. Math. Soc. 53 (1947), 130– 137. (Google Scholar)

R.H. Cameron and D.A. Storvick, An L2 analytic Fourier-Feynman transform, Michigan Math. J. 23 (1976), 1–30. (Google Scholar)

R.H. Cameron and D.A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, in Analytic Functions (Kozubnik, 1979), Lecture Notes in Math. 798, Springer-Verlag, (1980), 18–67. (Google Scholar)

R.H. Cameron and D.A. Storvick, Change of scale formulas for Wiener integral, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II-Numero 17 (1987), 105–115. (Google Scholar)

R.H. Cameron and D.A. Storvick, Relationships between the Wiener integral and the analytic Feynman integral, Supplemento ai Rendiconti del Circolo Matem- atico di Palermo, Serie II-Numero 17 (1987), 117–133. (Google Scholar)

S.J. Chang and J.G. Choi, Multiple Lp analytic generalized Fourier-Feynman transform on the Banach algebra, Commun. Korean Math. Soc. 19 (2004), 93– 111. (Google Scholar)

S.J. Chang, J.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. (Google Scholar)

S.J. Chang and D. Skoug, Generalized Fourier-Feynman transforms and a first variation on function space, Integral Transform. Spec. Funct. 14 (2003), 375– 393. (Google Scholar)

T. Huffman, C. Park and D. Skoug, Analytic Fourier-Feynman transforms and convolution, Trans. Amer. Math. Soc. 347 (1995), 661–673. (Google Scholar)

T. Huffman, C. Park and D. Skoug, Convolutions and Fourier-Feynman trans- forms of functionals involving multiple integrals, Michigan Math. J. 43 (1996), 247–261. (Google Scholar)

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

G.W. Johnson and D.L. Skoug, Scale-invariant measurability in Wiener space, Pacific J. Math. 83 (1979), 157–176. (Google Scholar)

B.J. Kim, B.S. Kim and I. Yoo, Change of scale formulas for Wiener integrals related with Fourier-Feynman transform and convolution, J. Function Spaces 2014 (2014), 1–7. (Google Scholar)

J. Yeh, Convolution in Fourier-Wiener transform, Pacific J. Math. 15 (1965), 731–738. (Google Scholar)

J. Yeh, Stochastic processes and the Wiener integral, Marcel Dekker, Inc., New York, 1973. (Google Scholar)

I. Yoo, B.J. Kim and B.S. Kim, A change of scale formula for a function space integrals on Ca,b[0, T ], Proc. Amer. Math. Soc. 141 (2013), 2729–2739. (Google Scholar)

I. Yoo and D. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces, Internat. J. Math. Math. Sci. 17 (1994), 239–248. (Google Scholar)

I. Yoo and D. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces II, J. Korean Math. Soc. 31 (1994), 115–129. (Google Scholar)

I. Yoo, T.S. Song and B.S. Kim, A change of scale formula for Wiener integrals of unbounded functions II, Commun. Korean Math. Soc. 21 (2006), 117–133. (Google Scholar)

I. Yoo, T.S. Song, B.S. Kim and K.S. Chang, A change of scale formula for Wiener integrals of unbounded functions, Rocky Mountain J. Math. 34 (2004), 371–389. (Google Scholar)


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