The convolution of associated homogeneous distributions on R–Part I

Abstract

This is the first in a series of two papers in which we construct a convolution product for associated homogeneous distributions (AHDs) with support in R. In this article, we show that if f ^a and g ^b are AHDs with degrees of homogeneity a − 1 and b − 1, the convolution f ^a * g ^b exists as an AHD, provided the resulting degree of homogeneity a + b − 1 is not a natural number. Under this restriction, it is found that the convolution product of AHDs is bilinear, bicontinuous and associative. New convolution products are derived for several basis AHDs such as half-line distributions, associated Riesz distributions and associated generalizations of Heisenberg distributions.

Keywords:

• generalized function
• tempered distribution
• associated homogeneous distribution
• convolution
• real line

AMS subject classifications::

• 46F10
• 46F12
• 46F30