Norm estimates for generalized translation operators associated with a singular differential operator

Abstract

Delsarte's approach via the solution of a Cauchy problem is used to estimate generalized translation operators associated with a singular Sturm-Liouville differential operator of the form D q, x α ,=( d 2/ dx 2) + ((2 α + 1)/ x)( d/ dx)- q( x), a≥ 1 2 , 0<×<∞. Such estimates are fundamental for establishing a convolution structure. As a general purpose, the relation between translations belonging to different potential functions q are studied. This leads to norm estimates for classes of “related” translation operators mainly under hypotheses on the Riemann function associated with an explicitly given member of the class. Specifically, our aim is to investigate perturbations of the Hankel translation as well as of the Laguerre translation, the underlying Sturm-Liouville equations of which are prototypes of singular equations on the positive half-axis with continuous and discrete spectra, respectively.