Hankel convolution operators on entire functions and distributions
The Hankel transformation \(\varphi\to h_\mu (\varphi)\) of order \(\mu>-1/2\) is defined here as: \[ h_\mu (\varphi)(x)= \int^\infty_0 (xy)^{-\mu} J_\mu(xy) \varphi(y)y^{2\mu+1} dy\;(x<0), \] where \(J_\mu\) is the Bessel function of the first kind and order \(\mu\). The Hankel convolution \(f,g\to f*g\) is defined such that \(h_\mu (f*g)= h_\mu(f) h_\mu(g)\). The authors study Hankel convolution operators on the space of even and entire functions and on distribution spaces and prove that they are hypercyclic and chaotic. The Hankel convolution operators are characterized as those operators commuting with the Hankel translations or, equivalently, with the Bessel operator \(\Delta_\mu= z^{-2\mu-1} Dz^{2\mu+1}D\).
scientific article; zbMATH DE number 1866730
scientific article