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Characterization of generalized Hankel integral type transformable generalized functions. (English) Zbl 1030.46052
The first author, in 2000, has discussed a generalized Hankel-type transformation defined by \[ F(y)= \biggl(F_{1, \mu,\alpha,\beta, \nu} \bigl\{f(x) \bigr\}\biggr) (y)=\nu\beta y^{-1-2\alpha+2 \nu}\int^\infty_0 (xy)^\alpha J_\mu \bigl[\beta (xy)\bigr]^\nu f(x)dx \] where \(\alpha,\beta,\nu\) are real and \(J_\mu(z)\) is the Bessel function of first kind and of order \(\mu\). He also extended the results to a class of generalized functions.
Similar to the spaces of \(H_\mu\) [A. H. Zehmanian, Generalized integral transformations, Chapter V, New York: Wiley (1968; Zbl 0181.12701)] the spaces \(H_{1,\mu, 1/2,1}\), \(H_{2,\mu,1/2,1}\) and their duals are defined in a paper by the first author and Bandewar (to appear).
In the paper under review, the authors proved a characterization theorem for generalized Hankel-type transformable generalized functions in \(H_{2,\mu, \alpha,\beta, \nu}'\). They also maintain that similar results can be deduced for elements of the space \(H_{1,\mu,\alpha, \beta,\mu}'\).
46F12 Integral transforms in distribution spaces
46F10 Operations with distributions and generalized functions
44A20 Integral transforms of special functions
44A05 General integral transforms