Two variants of fractional powers of Hankel integral transforms of arbitrary order
The paper introduces two variants of fractional Hankel transforms and extend them to spaces of generalized functions. Let \(0< \theta <\pi\), \(\nu\geq -1/2\) and set \[ \left({\mathcal H}^\theta_{1, \nu,\mu}\, f \right)(y) =\int_0^\infty K^\theta_1(x, y) f(x)dx, \] where \[ K^\theta_1(x,y)=\begin{cases} C_{\nu, \mu, \theta}\, e^{i(x^2+y^2) \cot \theta/2} \left(xy\csc \theta \right)^{-\mu} J_\nu (xy \csc \theta) y^{1+2\mu}, &\theta \neq n\pi, \\ (xy)^{-\mu}J_\nu (xy)y^{1+2\mu}, &\theta =\pi/2, \\ \delta(x-y,) &\theta = n\pi, \end{cases} \] and \[ \left({\mathcal H}^\theta_{2, \nu,\mu}\, g \right)(y) =\int_0^\infty K^\theta_2(x, y) g(x)dx, \] \[ K^\theta_2(x,y)=\begin{cases} C_{\nu, \mu, \theta}\, e^{i(x^2+y^2) \cot \theta/2 }\left(xy\csc \theta \right)^{-\mu} J_\nu (xy \csc \theta) x^{1+2\mu}, &\theta \neq n\pi, \\ (xy)^{-\mu}J_\nu (xy)x^{1+2\mu}, &\theta =\pi/2, \\ \delta(x-y), &\theta = n\pi, \end{cases} \] where \(C_{\nu, \mu, \theta}\) is a constant that depends on \(\mu, \nu, \theta.\) The above first and second fractional Hankel transforms reduce to the standard Hankel transform pair when \(\theta = \pi/2.\) The fractional Hankel transforms are then extended to \({\mathcal H}^\theta_{1, \nu,\mu, m}, {\mathcal H}^\theta_{2, \nu,\mu, m} ,\) where \(m\) is a natural number such that \(\nu+\mu+m \geq -1 .\) The authors then introduce two countable families of multinorms by which they construct two testing-function spaces \(\mathbf{H}^\theta_{1,\nu,\mu}(I), \mathbf{H}^\theta_{2,\nu,\mu}(I)\) as in \textit{A. H. Zemanian}'s book [Generalized integral transformations. New York etc.: Interscience Publishers, a division of John Wiley \& Sons, Inc. (1968; Zbl 0181.12701)], where \(I=(0, \infty).\) It is shown that \( {\mathcal H}^\theta_{1, \nu,\mu, m}\) is a continuous linear mapping on \(\mathbf{H}^\theta_{1,\nu,\mu}(I)\) into itself. The fractional Hankel transform \({\mathcal H}^\theta_{1, \nu,\mu, m}\) is then extended as a continuous linear mapping on a space of generalized functions, namely, \({(\mathbf{H})'}^\theta_{2,\nu,\mu}(I),\) which is the dual of the space \(\mathbf{H}^\theta_{2,\nu,\mu}(I) ,\) and it also works as the adjoint of \({\mathcal H}^\theta_{2, \nu,\mu, m}\) acting on \(\mathbf{H}^\theta_{2,\nu,\mu}(I) ,\) according to the relation \[ \langle {\mathcal H}^\theta_{1, \nu,\mu, m} f, \Phi \rangle =\langle f, {\mathcal H}^\theta_{2, \nu,\mu, m} \Phi \rangle \quad \forall f \in {\mathbf{H}'}^\theta_{2, \nu,\mu,}(I) \quad \text{ and } \Phi \in \mathbf{H}^\theta_{2, \nu,\mu,}(I). \] The paper is concluded with an application to solving a partial differential equation involving Hankel differential operator.
scientific article; zbMATH DE number 6516562
scientific article