A unified approach to the integrals of Mellin--Barnes--Hecke type

In this paper we provide a unified approach to a family of integrals of Mellin--Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributions via smooth submersive maps defined by H\"ormander. We derive by this method the integrals of Hecke and Sonine relating to various types of Bessel functions which have found applications in analytic and algebraic number theory.