Extension of the Drasin-Shea-Jordan theorem

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 Abe lian problem; its converse, under suitable Tauberian conditions, is a Taube rian 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 stat ement to a regular-variation statement is a Mercerian problem. The prototyp e 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 proo f is simpler than the counter-parts of the previous results and does not ev en use the Polya Peak Theorem which has been so essential before. The usefu lness of the extension is highlighted by an application to Hankel transform s.