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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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