International Journal of Mathematics and Mathematical Sciences
Volume 21 (1998), Issue 4, Pages 695-700

The neutrix convolution product in Z(m) and the exchange formula

C. K. Li1 and E. L. Koh2

1Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta, Canada
2Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada

Received 18 November 1996; Revised 28 February 1997

Copyright © 1998 C. K. Li and E. L. Koh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


One of the problems in distribution theory is the lack of definition for convolutions and products of distribution in general. In quantum theory and physics (see e.g. [1] and [2]), one finds that some convolutions and products such as 1xδ are in use. In [3], a definition for product of distributions and some results of products are given using a specific delta sequence δn(x)=Cmnmρ(n2r2) in an m-dimensional space. In this paper, we use the Fourier transform on D(m) and the exchange formula to define convolutions of ultradistributions in Z(m) in terms of products of distributions in D(m). We prove a theorem which states that for arbitrary elements f˜ and g˜ in Z(m), the neutrix convolution f˜g˜ exists in Z(m) if and only if the product fg exists in D(m). Some results of convolutions are obtained by employing the neutrix calculus given by van der Corput [4].