Pseudodifferential calculus in Schwinger--DeWitt formalism: UV and IR parts

We consider expansions for the kernels of operator functions of second-order minimal operators on a curved background. We show that the terms of these expansions originate in the ultraviolet or infrared regions. We propose a systematic approach to obtaining ultraviolet terms using term-by-term integration of the DeWitt expansion of the heat kernel. We discuss two methods for regularizing infrared divergences arising at intermediate computational steps – using analytic continuation and introducing a mass term – and the relationship between them.