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A mixed Parseval’s equation and a generalized Hankel transformation of distributions. (English) Zbl 0666.46046

We introduce a new integral transform, that is a generalization of the Hankel transformation, depending on three parameters and denoted by \(F_{\alpha_ 0,\alpha_ 1,\alpha_ 2}\). It is extended to a space of distributions. This transforms satisfy the mixed Parseval equation \[ \int^{\infty}_{0}F_{\alpha_ 0,\alpha_ 1,\alpha_ 2}\{f\}(x)g(x)dx=\int^{\infty}_{0}f(x)F_{\alpha_ 2,\alpha_ 1,\alpha_ 0}\{g\}(x)dx. \] This equality suggests that the generalized transform \(F'_{\alpha_ 0,\alpha_ 1,\alpha_ 2}\) be defined as the adjoint operator of \(F_{\alpha_ 0,\alpha_ 1,\alpha_ 2}\). Well- known results due to A. H. Zemanian about the Hankel transformation of distributions can be seen as special cases of the ones obtained here.
Reviewer: J.J.Betancor

MSC:

46F12 Integral transforms in distribution spaces
44A15 Special integral transforms (Legendre, Hilbert, etc.)
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