Jorge J. Betancor, Manuel T. Flores
A Parseval equation and a generalized finite Hankel transformation

Comment.Math.Univ.Carolinae 32,4 (1991) 627-638.

Abstract:In this paper, we study the finite Hankel transformation on spaces of generalized functions by developing a new procedure. We consider two Hankel type integral transformations $h_\mu $ and $h_\mu ^{\ast }$ connected by the Parseval equation $$ \sum _{n=0}^{\infty }(h_\mu f)(n)(h_\mu ^{\ast } \varphi )(n)= \int _{0}^{1}f(x)\varphi (x) dx. $$ A space $S_\mu $ of functions and a space $L_\mu $ of complex sequences are introduced. $h_\mu ^{\ast }$ is an isomorphism from $S_\mu $ onto $L_\mu $ when $\mu \geq -\frac {1}{2}$. We propose to define the generalized finite Hankel transform $h'_\mu f$ of $f\in S'_\mu $ by $$ \langle (h'_\mu f), ((h_\mu ^{\ast } \varphi )(n))_{n=0}^{\infty }\rangle =\langle f,\varphi \rangle , \quad \text {for } \varphi \in S_\mu. $$

Keywords: finite Hankel transformation, distribution, Parseval equation
AMS Subject Classification: 46F12

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