## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Miniconference on Harmonic Analysis. M Cowling, C Meaney, and W Moran, eds. Proceedings of the Centre for Mathematical Analysis, v. 15. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1987), 21 - 33

### An introduction to the abel transform

#### Abstract

This paper is intended as an introduction to the Abel transform for the non·· specialist. Nevertheless it contains some of the essential ideas which enables us to present some recent results on this transform in the last section.

The Abel transform plays an important role in the theory of the spherical Fourier transform on synm1e'tric spaces of 'the noncompac't 'type. Its role is analogous t:o the role of 'the Radon 'transform in the theory of the ordinary Fourier transform on IRn. Therefore we first present the example of the ordin.ary Fourier 'transform in sec'tion l. Then we 'turn to the of SL(2,1R) (sections 2 and 3). Here we give an explicit expression for H'le Abel transform and review some of the results and applications. This will serve as motivation and as prototype for further research. In the last section we present some recent results.

#### Article information

**Dates**

First available in Project Euclid:
18
November
2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416336451

#### Citation

Beerends, R.J. An introduction to the abel transform. Miniconference on Harmonic Analysis, 21--33, Centre for Mathematical Analysis, The Australian National University, Canberra AUS, 1987.https://projecteuclid.org/euclid.pcma/1416336451