## 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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