Hankel transformations and spaces of type \(S\) (Q1174832)

It is well nown, that the \(L_ 1\)-function \(\varphi\) belongs to the Schwartz space \(S\) if and only if for all \(k,\ell\in\mathbb{N}_ 0\) \[ \sup_{x\in\mathbb{R}}| x^ k\varphi(x)|<\infty\text{ and } \sup_{x\in\mathbb{R}}| x^ \ell(F\varphi)(x)|<\infty, \] where \(F\varphi\) denotes the Fourier transform of \(\varphi\). Similar results are obtained for even functions \(\varphi\) belonging to Gelfand-Shilov space \(S^ \beta_ \alpha\), \(\alpha>0\), \(\beta>0\), \(\alpha+\beta\geq 1\), and for their Hankel transform \(H_ \nu\varphi\). A straightforward corollary is \(H_ \nu(S^ \beta_{\alpha,\hbox{even}})=S^ \alpha_{\beta,\hbox{even}}\), and the operators \(H_ \mu H_ \nu\) yield Weyl's fractional calculus.