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On Hankel transform. (English) Zbl 0711.46034

It is proved that for \(\alpha >-1\), the operator \[ {\mathcal H}_{\alpha}(f)(x)=\int^{\infty}_{0}f(t)(tx)^{-\alpha /2} t^{\alpha}J_{\alpha}(\sqrt{xt})dt, \] is an isomorphism of \(S^+\) onto itself and \({\mathcal H}^ 2_{\alpha}=Id\), where \(S^+=\{f\in C^{\infty}(0,+\infty):\;\forall k,n\geq 0| \quad t^ kf^ n(t)<C_{k,n}\},\) which reduces to \(H_{\mu}\) [A. H. Zemanian, SIAM J. Appl. Math. 14, 561-576 (1986; Zbl 0154.138)] by a simple transformation \((t^{-1}D)^ nf(t)=2^ ng^ n(x),\) \(x=t^ 2\). Finally a characterization of the functions in \(S^+\) which satisfy \(\int^{\infty}_{0}f(t)t^{\alpha +n}dt=0\), for all \(n\geq 0\), \(\alpha >-1\) is given.
Reviewer: B.M.Agrawal

MSC:

46F12 Integral transforms in distribution spaces
44A15 Special integral transforms (Legendre, Hilbert, etc.)
44A05 General integral transforms

Citations:

Zbl 0154.138
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References:

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