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

Betancor, J. J.

Can. J. Math. 41, No. 2, 274-284 (1989).

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.)

Keywords:

integral transform; generalization of the Hankel transformation, depending on three parameters; space of distributions; mixed Parseval equation