Asymptotic expansions of Mellin convolution integrals: An oscillatory case

In a recent paper [J.L. López, Asymptotic expansions of Mellin convolution integrals, SIAM Rev. 50 (2) (2008) 275–293], we have presented a new, very general and simple method for deriving asymptotic expansions of ∫ 0 ∞ f ( t ) h ( x t ) d t for small x . It contains Watson’s Lemma and other classical methods, Mellin transform techniques, McClure and Wong’s distributional approach and the method of analytic continuation used in this approach as particular cases. In this paper we generalize that idea to the case of oscillatory kernels, that is, to integrals of the form ∫ 0 ∞ e i c t f ( t ) h ( x t ) d t , with c ∈ R , and we give a method as simple as the one given in the above cited reference for the case c = 0 . We show that McClure and Wong’s distributional approach for oscillatory kernels and the summability method for oscillatory integrals are particular cases of this method. Some examples are given as illustration.