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Introduction to the theory of \((\nu _ 1,\dots ,\nu _{r-1})\)-transforms. (Russian) Zbl 0637.44007

Let r be an integer, \(r\geq 2\) and let \(\nu =(\nu_ 1,...,\nu_{r-1})\in R^{r-1}\). Then the \(\nu\)-transform resp. the inverse \(\nu\)-transform are defined by means of \[ \phi_{\nu}(x)=P_{\nu}f(x)=\int^{\infty}_{0}... \int^{\infty}_{0}f(x\prod t_ i)\exp [-\sum t^ r_ i]\Pi (t_ i^{r\nu_ i+r-1}dt_ i)\text{ resp.} \]
\[ f(x)=P_{\nu}^{- 1}\phi_{\nu}(x)=(r/2\pi i)^{r-1}\int^{(0+)}_{-\infty}... \int^{(0+)}_{-\infty}\phi_{\nu}(x\Pi t_ i^{-1/r})\exp (\sum t_ i)\Pi (t_ i^{-\nu_ i-1}dt_ i). \] Here the sums and the products have to be taken from \(i=1\) to \(i=r-1\). Theorems on the existence and properties of the transforms in consideration, an uniqueness theorem, a convolution theorem and differentiation rules are proved. Applications to the solution of differential equations are given. The Mellin transforms \(\phi_{\nu}=M[\phi_{\nu}]\) and \(F=M[f]\) are connected by \(\phi_{\nu}(s)=r^{-r-1}\pi \Gamma (\nu_ i+1-s/r)F(s).\)
Reviewer: H.-J.Glaeske

MSC:

44A30 Multiple integral transforms
44A45 Classical operational calculus
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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