The convolution of associated homogeneous distributions on R–Part II

Abstract

This is the second in a series of two papers in which we construct a convolution product for the set ℋ′ (R) of associated homogeneous distributions (AHDs) with support in R. In Part I we showed 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, if the resulting degree of homogeneity a + b−1 ∉ N. In this article, we develop a functional extension process, based on the Hahn–Banach theorem, to give a meaning to the convolution product of two AHDs of degrees a − 1 and b − 1, in the critical case that a + b − 1 ∈ N. With respect to this construction, the structure (ℋ′(R), *) is shown to be closed.

Keywords:

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

AMS Subject Classifications::

• 46F10
• 46F12
• 46F30