A mixed Parseval equation and the generalized Hankel transformations

Abstract:

Let ${T_1}$ and ${T_2}$ be two classical integral transforms whose inverse formulas coincide with themselves, satisfying the mixed Parseval equation \[ \int _0^\infty {f(x)g(x)dx = \int _0^\infty {{F_1}(y){G_2}(y)} \;dy,} \] where ${F_1}(y) = ({T_1}f)(y)$ and ${G_2}(y) = ({T_2}g)(y)$. We propose to define the generalized transformation ${T’_1}$ as the adjoint operator of ${T_2}$, and conversely. This procedure provides a new approach to extend the Hankel transform to certain spaces of distributions.