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A mixed Parseval equation and the generalized Hankel transformations

Author: J. M. Méndez
Journal: Proc. Amer. Math. Soc. 102 (1988), 619-624
MSC: Primary 46F12; Secondary 44A05, 47B35
MathSciNet review: 928991
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Abstract: Let $ {T_1}$ and $ {T_2}$ be two classical integral transforms whose inverse formulas coincide with themselves, satisfying the mixed Parseval equation

$\displaystyle \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.

References [Enhancements On Off] (What's this?)

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Keywords: Hankel transformation, Parseval equation, generalized functions, Bessel operator
Article copyright: © Copyright 1988 American Mathematical Society