On E-Pólya frequency functions

Let E be a subset of R. We say that f :R→R is an E-Pólya frequency function if the kernel f (u − x) is totally positive on E × R and f is integrable on R. In this paper, we show that under some conditions on E, an E-Pólya frequency function is a Pólya frequency function. This strengthens Schoenberg’s result on Pólya frequency functions.