Distributions of positive type with an application in quantum field theory

Distributions of positive type arise naturally in quantum physics as integral kerels of inner products on the space of test functions. When performing constructions with cut-off functions, however, we sometimes run into the obstruction that f(x,y) cannot be of positive type for properly supported functions f wich are identically 1 on a neighbourhood of the diagonal x=y. In this talk I will show how this obstruction can be overcome in R^n by using test-functions of positive type and estimating lower bounds on their Fourier transforms. If time permits I will show an application of this result in quantum field theory. (Based on J. Phys. A 56, 505201 (2023) and CMP 405, 132 (2024)).