$L_1$ analytic fourier-feynman transform on the fresnel class of abstract wiener space

  • Ahn, Jae-Moon (Department of Mathematics Education, College of Education, Kon-Kuk University, Seoul 143-701)
  • Published : 1998.02.01


Let $(B, H, p_1)$ be an abstract Wiener space and $F(B)$ the Fresnel class on $(B, H, p_1)$ which consists of functionals F of the form : $$ F(x) = \int_{H} exp{i(h,x)^\sim} df(h), x \in B, $$ where $(\cdot, \cdot)^\sim$ is a stochastic inner product between H and B, and f is in $M(H)$, the space of complex Borel measures on H. We introduce an $L_1$ analytic Fourier-Feynman transforms for functionls in $F(B)$. Furthermore, we introduce a convolution on $F(B)$, and then verify the existence of the $L_1$ analytic Fourier-Feynman transform for the convolution product of two functionals in $F(B)$, and we establish the relationships between the $L_1$ analytic Fourier-Feynman tranform of the convolution product for two functionals in $F(B)$ and the $L_1$ analytic Fourier-Feynman transforms for each functional. Finally, we show that most results in [7] follows from our results in Section 3.


  1. A Functional Transform for Feynman Integrals similar to theFourier Transform M. D. Brue
  2. Michigan Math. J. v.23 An L₂ analytic FourierFeynman Transform R. H. Cameron;D. A. Storvick
  3. Analytic functions (Kozubnik, 1979), Lecture Notes in Math. v.798 Some Banach algebras of analytic Feynman integrable functionals R. H. Cameron;D. A. Storvick
  4. Pacific J. Math. v.130 ScaleInvariant Measurability in Abstract Wiener Space D. M. Chung
  5. Measure Theory D. L. Cohn
  6. Trans. of the Amer. Math. Soc. v.347 Analytic FourierFeynman Transforms and Convolution T. Huffman;C. Park;D. L. Skoug
  7. Michigan Math. J. v.43 Convolutions and FourierFeynman Transforms of Functionals involving Multiple Integrals T. Huffman;C. Park;D. L. Skoug
  8. Michigan Math. J. v.26 An Lp Analytic FourierFeynman Transform G. W. Johnson;D. L. Skoug
  9. Stochastic Analysis and Application Generalized Feynman Integrals using Analytic Continuation in several complex variables G. Kallianpur;C. Bromley;M.H.Pinsky(ed.)
  10. Pacific J. Math. v.31 Abstract Wiener Spaces and Applications to Analysis J. Kuelbs
  11. Lecture Notes in Math. v.463 Gaussian Measures in Banach Spaces H. H. Kuo
  12. J. Funct. Anal. v.47 Integarl Transforms of Analytic Functions on Abstract Wiener Spaces Y. J. Lee
  13. Pacific J. Math. v.15 Convolution in FourierWiener Transform J. Yeh