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Zbl 1229.60096
Chung, Hyun Soo; Tuan, Vu Kim
Generalized integral transforms and convolution products on function space.
(English)
[J] Integral Transforms Spec. Funct. 22, No. 8, 573-586 (2011). ISSN 1065-2469; ISSN 1476-8291/e

Existence and properties of the generalized integral transform (GIT) and the generalized convolution product function (GCP) on Wiener space are studied. Let $C_0[0,T]=\{x\in C([0,T];\mathbb R):\ x_0=0\}$ and $K_0[0,T]=\{y\in C([0,T];\mathbb C):\ y_0=0\} .$ Let $\mu$ be the Wiener measure on $C_0[0,T].$ For $h\in L^2[0,T], t\in [0,T]$, let $Z_h(x,t)$ be the Paley-Wiener-Zygmund integral of $h$ w.r.t. $x$ over $[0,t],$ which is $\mu$-a.s. defined. For any $\alpha,\beta\in\mathbb C$, and a measurable function $F$ on $K_0[0,T]$, the function $$\mathcal F_{\alpha,\beta,h}(F)(y):= \int_{C_0[0,T]}F(\alpha Z_h(x,\cdot),\beta y)\mu(x),\ \ y\in K_0[0,T]$$ is called the generalized integral transform (GIT) of $F$ if it exists. Moreover, the generalized convolution function of $F$ and $G$ is defined by $$(F\* G)_\alpha(y)= \int_{C_0[0,T]}F\Big(\frac{y+\alpha Z_h(x,\cdot)}{2^{1/2}}\Big)G\Big(\frac{y-\alpha Z_h(x,\cdot)}{2^{1/2}}\Big)\mu(x) ,\ \ y\in K_0[0,T].$$
[Feng-Yu Wang (Swansea)]
MSC 2000:
*60J65 Brownian motion
28C20 Set functions and measures and integrals in infinite-dim. spaces

Keywords: generalized integral transform; generalized convolution product; inverse integral transform Wiener space

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