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An analytic Yeh-Feynman-Fourier transform and convolution
Abstract
Let $C\sb0\lbrack0,T\rbrack$ denote Wiener space. Brue introduced the idea of an $L\sp1$ analytic Feynman-Fourier transform of functionals on $C\sb0\lbrack0,T\rbrack$ in 1971. Since then many people including Cameron, Johnson, Martin, Skoug and Storvick have extended this theory to $L\sp{p}$ with 1 $\le$ p $\le$ 2 for many classes of functionals. Recently, there has also been interest in convolution of functionals on $C\sb0\lbrack0,T\rbrack$ and its relationship to the analytic Feynman-Fourier Transform. Let Q = (0,b) $\times \lbrack 0,\beta\rbrack$ and let $C\sb2\lbrack Q\rbrack = \{$x(s,t): x is real valued, continuous on Q and x(0,t) = x(s,0) = 0$\}$. Yeh developed a measure m on this space and hence we will call $C\sb2\lbrack Q\rbrack$ together with m, Yeh-Wiener Space. In this dissertation we will create an $L\sp{p}$ analytic Yeh-Feynman-Fourier transform of functionals on $C\sb2\lbrack Q\rbrack$. Also a convolution product will be introduced for functionals on $C\sb2\lbrack Q\rbrack$. We then show that this transform and convolution product have many of the same properties as the Fourier transform of functions on $\Re\sp{n}$. That is, we show the inverse transform of the transform of a functional is the original functional. Also we show that the transform of the convolution equals the product of the transforms. Finally, we consider an identity similar to the Plancherel identity. In chapter one we give the basic definitions of Yeh-Wiener space, the analytic Yeh-Feynman-Fourier transform and convolution product. We then consider three classes of functionals on $C\sb2\lbrack Q\rbrack$. In chapter two we consider functionals of the form F(x) = $f(x(s\sb1,t\sb1),x(s\sb1,t\sb2),\...,x(s\sb{m},t\sb{n})).$ Next, in chapter three we consider functionals of the form F(x) = $\int\sb{Q}\int$ f(s,t,x(s,t)ds dt. Finally we consider functionals of the form F(x) = $f(\int\int\sb{Q}\alpha\sb1dx(s,t),\...,\int \int \alpha\sb{n}dx(s,t))$ in chapter four. We then close our work by giving a number of specific examples of the transform and its properties in chapter five.
Subject Area
Mathematics
Recommended Citation
Huffman, Timothy John, "An analytic Yeh-Feynman-Fourier transform and convolution" (1994). ETD collection for University of Nebraska-Lincoln. AAI9507814.
https://digitalcommons.unl.edu/dissertations/AAI9507814