Extension of the Drasin-Shea-Jordan theorem

Abstract

Passing from regular variation of a function $f$ to regular variation of its integral transform $k*f$ of Mellin-convolution form with kernel $k$ is an Abelian problem; its converse, under suitable Tauberian conditions, is a Tauberian one. In either case, one has a comparison statement that the ratio of $f$ and $k*f$ tends to a constant at infinity. Passing from a comparison statement to a regular-variation statement is a Mercerian problem. The prototype results here are the Drasin-Shea theorem (for non-negative $k$) and Jordan's theorem (for $k$ which may change sign). We free Jordan's theorem from its non-essential technical conditions which reduce its applicability. Our proof is simpler than the counter-parts of the previous results and does not even use the Pólya Peak Theorem which has been so essential before. The usefulness of the extension is highlighted by an application to Hankel transforms.