Analytic regularisation of inversion transform integrals by Cesàro summability

Abstract

The conformal inversion operator is represented in Hilbert space by Hankel transforms. These transforms occur in a wide variety of problems,e.g. high-energy diffraction scattering, conformal invariant quantum field theory, the improvements of Borel summability of QCD asymptotic expansions, etc. The convergence of these transforms is generally not guaranteed everywhere, hence the need for their regularisation and analytic continuation. It is proposed in this paper that Cesaro summability is not only very simple but also most effective as a method of regularisation and analytic continuation of these integrals. Its power and utility are greatly enhanced by the availability of fast computing facilities. The hierarchy of Cesaro means of the transform integrals is generated, and their convergence investigated, numerically. In all cases we obtain results in agreement with analytic continuation. Cesaro summability can therefore be applied directly to effect the analytic continuation involved in QCD sum rules and not only to the Borel improvements of the relevant integrals. This would provide an additional check of the results obtained via Borel summability.