Geodesic
A Parseval equation and a generalized finite Hankel transformation
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 4, pp. 627-638 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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In this paper, we study the finite Hankel transformation on spaces of ge\-ne\-ra\-lized 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 $$ \langle (h'_\mu f), ((h_\mu ^{\ast } \varphi )(n))_{n=0}^{\infty }\rangle =\langle f,\varphi \rangle, \quad \text{for } \varphi \in S_\mu . $$
In this paper, we study the finite Hankel transformation on spaces of ge\-ne\-ra\-lized 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 $$ \langle (h'_\mu f), ((h_\mu ^{\ast } \varphi )(n))_{n=0}^{\infty }\rangle =\langle f,\varphi \rangle, \quad \text{for } \varphi \in S_\mu . $$
Classification : 44A15, 46F12
Keywords: finite Hankel transformation; distribution; Parseval equation 
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Betancor, Jorge J.; Flores, Manuel T. A Parseval equation and a generalized finite Hankel transformation. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 4, pp. 627-638. http://geodesic.mathdoc.fr/item/CMUC_1991_32_4_a4/

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