# MULTIPLE Lp ANALYTIC GENERALIZED FOURIER-FEYNMAN TRANSFORM ON THE BANACH ALGEBRA

• Published : 2004.01.01
• 51 4

#### Abstract

In this paper, we use a generalized Brownian motion process to define a generalized Feynman integral and a generalized Fourier-Feynman transform. We also define the concepts of the multiple Lp analytic generalized Fourier-Feynman transform and the generalized convolution product of functional on function space $C_{a,\;b}[0,\;T]$. We then verify the existence of the multiple $L_{p}$ analytic generalized Fourier-Feynman transform for functional on function space that belong to a Banach algebra $S({L_{a,\;b}}^{2}[0, T])$. Finally we establish some relationships between the multiple $L_{p}$ analytic generalized Fourier-Feynman transform and the generalized convolution product for functionals in $S({L_{a,\;b}}^{2}[0, T])$.ﾖ⨀

#### Keywords

generalized Brownian motion process;generalized analytic Feynman integral;generalized analytic Fourier-Feynman transform;generalized convolution product;multiple $L_{p}$ analytic generalized fourier-Feynman transform

#### References

1. Bull. Korean Math. Soc. v.35 no.1 L₁analytic Fourier-Feynman transform on the Fresenel class of abstract Wiener space J.M.Ahn
2. Lectrure Notes in Math. v.798 Some Banach algebras of analytic Feyuman interable functionals, Analytic Funtions(Kozubnik,1979), R.H.Cameron https://doi.org/10.1007/BFb0097256
3. Math. Note Dynamical theries of the Browian motion(2nd ed.) E.Nelson
4. J. Korean Math. Soc. v.38 no.1 Conditional Fourier-Feynman transforms and conditonal convolution products C.Park;D.Skoug
5. Marcel Dekker, Inc. Stochastic Processes and the Wiener Integral J.Yeh
6. Trans. Amer. Math. Soc. v.355 no.7 Integration by parts formulas involving generalized Fourier-Feynman transforms on funtion space S.J.Chang;J.G.Choi;D.Skoug https://doi.org/10.1090/S0002-9947-03-03256-2
7. Rocky Moutain J. of Math. v.27 Convoluton and Fourier-Feynman Transforms T.Huffman https://doi.org/10.1216/rmjm/1181071896
8. Michigan Math. J. v.26 An Lρ Analytic Fourier-Feynman transform G.W.Johnson;D.L.Skoug https://doi.org/10.1307/mmj/1029002166
9. thesis, University of Minnesota M.D.Brue;A.Functional Transform for Feynman Integrals Similar to the Fourier Transform
10. Trans. Amer. Math. Soc. v.347 Analytic Fourier-Feynman transforms and convoution T.Huffman;C.Park;D.Skoug https://doi.org/10.2307/2154908
11. Oxford Mathematical Monographs The Feynman Integral and Feynman's Operational Calculus G.W.Johnson;M.L.Lapidus
12. J. Korean Math. Soc. v.38 no.2 Analytic Fourier-Feynman transform and first variation on abstract Wiener space K.S.Chang;T.S.Song;I.Yoo
13. Rocky Mountain J. of Math. v.26 no.1 Conditional function space integrals with applications S.J.Chang;D.M.Chung https://doi.org/10.1216/rmjm/1181072102
14. Integral Transforms and Special Functions Generalized Fourier-Feynman transforms and the first variation on function space S.J.Chang;D.Skoug https://doi.org/10.1080/1065246031000074425
15. Michigan Math. J. v.43 Convolutions and Fourier-Feynman transforms of funtionals involving multiple integrals T.Huffman https://doi.org/10.1307/mmj/1029005461
16. Michigan Math. J. v.23 An L₂analytic Fourier-Feynman transform R.H.Cameron;D.A.Storvick https://doi.org/10.1307/mmj/1029001617
17. Pacific J. Math v.83 Scale-invariant measurability in Wiener space G.W.Johnson https://doi.org/10.2140/pjm.1979.83.157
18. Rocky Mountain J. of Math. v.28 no.4 Relationship among the first variation the convolution product, and the Fourier-Feyman transform C.Park;D.Skoug;D.Storvick https://doi.org/10.1216/rmjm/1181071725

#### Cited by

1. CONDITIONAL GENERALIZED FOURIER-FEYNMAN TRANSFORM OF FUNCTIONALS IN A FRESNEL TYPE CLASS vol.26, pp.2, 2011, https://doi.org/10.4134/CKMS.2011.26.2.273
2. GENERALIZED ANALYTIC FOURIER-FEYNMAN TRANSFORMS AND CONVOLUTIONS ON A FRESNEL TYPE CLASS vol.48, pp.2, 2011, https://doi.org/10.4134/BKMS.2011.48.2.223
3. CHANGE OF SCALE FORMULAS FOR FUNCTION SPACE INTEGRALS RELATED WITH FOURIER-FEYNMAN TRANSFORM AND CONVOLUTION ON Ca,b[0, T] vol.23, pp.1, 2015, https://doi.org/10.11568/kjm.2015.23.1.47