## Journal of the Mathematical Society of Japan

### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 52, Number 3 (2000), 545-559.

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107286

Digital Object Identifier
doi:10.2969/jmsj/05230545

Mathematical Reviews number (MathSciNet)
MR1760604

Zentralblatt MATH identifier
0964.44002

Subjects
Primary: 40E05: Tauberian theorems, general
Secondary: 44A15: Special transforms (Legendre, Hilbert, etc.)

#### Citation

H. BINGHAM, Nicholas; INOUE, Akihiko. Extension of the Drasin-Shea-Jordan theorem. J. Math. Soc. Japan 52 (2000), no. 3, 545--559. doi:10.2969/jmsj/05230545. https://projecteuclid.org/euclid.jmsj/1213107286