Summability experiments with a class of divergent inverse Kontorovich-Lebedev Transforms

The computation of inverse Kontorovich-Lebedev transforms is examined with the help of the extrapolative mW-transformation first presented by Sidi in user-friendly form in Sidi (1988). Recently published improved values for the Macdonald function of purely imaginary order have enabled this strategy to give 10 digit accuracy in the inversion, including an example with competing oscillations having different phase functions. Moreover the method is extended, with similar accuracy, to cases where the integral diverges but is summable in the sense of Abel. Further probing with divergent integrals that are not Abel summable but instead summable in the sense defined by Jones (1980) is also yielding encouraging results and some analysis, culminating in a formal theorem of convergence in that sense, is presented to support the interpretation and computation of these.