One - Dimensional Associated Homogeneous Distributions

Let H′ (R) denote the set of Associated Homogeneous Distributions (AHDs) with support in R. The set H′ (R) consists of the distributional analogues of one-dimensional power-log functions. H′ (R) is an important subset of the tempered distributions S′ (R), because (i) it contains the majority of the (one-dimensional) distributions typically encountered in physics applications and (ii) recent work done by the author shows that H′ (R), as a linear space, can be extended to a convolution algebra and an isomorphic multiplication algebra. This paper (i) reviews the general properties enjoyed by AHDs, (ii) completes the list of properties of the various important basis AHDs by deriving many new and general expressions for their derivatives, Fourier transforms, Taylor and Laurent series with respect to the degree of homogeneity, etc., and (iii) introduces some useful distributional concepts, such as extensions of partial distributions, that play a natural role in the construction of AHD algebras.