Calderón's reproducing formula associated with the Bessel operator
The ``classical'' Calderón reproducing formula on the real line can be written in the form \[ f= \int^\infty_0 (f*\mu_a)da/a, \] where \(f*\mu_a\) denotes a convolution with a dilated version of the ``wavelet measure'' \(\mu\). The authors establish an analogue of this formula for the case when ``\(*\)'' is replaced by the generalized convolution involving the generalized translation operator related to the Bessel differential operator \[ L_\alpha= {d^2\over dx^2}+ {2\alpha+ 1\over x} {d\over dx} \] on the half-line \(\mathbb{R}_+\). The consideration is carried out in the relevant weighted \(L^p\)-spaces on \(\mathbb{R}_+\).
scientific article; zbMATH DE number 1218658