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Generalized integral transforms and convolution products on function space. (English) Zbl 1229.60096

Integral Transforms Spec. Funct. 22, No. 8, 573-586 (2011); erratum ibid. 24, No. 6, 509 (2013).
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
\[ \mathcal F_{\alpha,\beta,h}(F)(y):= \int_{C_0[0,T]}F(\alpha Z_h(x,\cdot),\beta y)\mu(\d x),\;\;y\in K_0[0,T] \]
is called the generalized integral transform (GIT) of \(F\) if it exists. Moreover, the generalized convolution function of \(F\) and \(G\) is defined by
\[ (F\* G)_\alpha(y)= \int_{C_0[0,T]}F\Big(\frac{y+\alpha Z_h(x,\cdot)}{2^{1/2}}\Big)G\Big(\frac{y-\alpha Z_h(x,\cdot)}{2^{1/2}}\Big)\mu(\d x) ,\;\;y\in K_0[0,T]. \]

MSC:

60J65 Brownian motion
28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
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