## A Parseval equation and a generalized finite Hankel transformation.(English)Zbl 0763.46028

For $$\mu\geq-{1\over 2}$$ a space $$S_ \mu$$ of certain $$C^ \infty$$- functions on $$]0,1]$$ and a sequence space $$L_ \mu$$ are introduced. If $$J_ \mu$$ denotes the Bessel function of first kind and order $$\mu$$ and if $$(\lambda_ n)_{n\in\mathbb{N}_ 0}$$ denotes the positive roots of $$J_ \mu$$ (arranged increasingly) then the finite Hankel transform $(h_ \mu^* f)(n):=2J_{\mu+1}^{-2}(\lambda_ n) \int_ 0^ 1 J_ \mu(\lambda_ n x)f(x)dx, \quad f\in S_ \mu, \quad \mu\in\mathbb{N}_ 0,$ is shown to be an isomorphism between $$S_ \mu$$ and $$L_ \mu$$. Then the generalized finite Hankel transform $$h_ \mu': S_ \mu'\to L_ \mu'$$ is defined as the adjoint of $$(h_ \mu^*)^{-1}$$ and it is shown that it extends a previous definition given in the literature.

### MSC:

 46F12 Integral transforms in distribution spaces 44A15 Special integral transforms (Legendre, Hilbert, etc.)
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