We apply a straightforward method to produce a distribution [([([D.sup.m.sub,z] [[absolute value of x].sup.z]).sub.e]).sub.z=-n-2p] on D ([R.sub.n]) that is a SAHD and coincides with the partial distribution [([D.sup.m.sub,z] [[absolute value of x].sup.z]).sub.z=-n-2p] on [D.sub.r] ([R.sup.n]).
Proof, (i) SAHD on [R.sup.n] [right arrow] ([T.sup.[lambda]]) * AHD on R.
(ii) [([T.sup.[lambda]]).sup.*] AHD on R [right arrow] SAHD on [R.sup.n].
We present a construction of spherical (i.e., O (n)-invariant) associated homogeneous distributions (SAHDs) based on [R.sup.n], as pullbacks of associated homogeneous distributions (AHDs) based on R.
An important subset of H' ([R.sup.n]) are the O (n)-invariant AHDs on [R.sup.n], called SAHDs and of which [r.sup.z], z [member of] C, is a well-known example, having degree of homogeneity z and order of association 0, see e.g., [11, p.
We show that the pullback [T.sup.*], along the particular scalar function T [??] [T.sup.1], of any AHD on R generates a distribution on [R.sup.n] that is a linear combination of distributions of the form [D.sup.m.sub.z] [[absolute value of x].sup.z], called basis SAHDs. We properly define the distributions [D.sup.m.sub,z] [[absolute value of x].sup.z], which are only briefly considered in [11, p.
This work extends and generalizes the treatment of SAHDs on [R.sup.n] in [11].
In section 5, the results from sections 3 and 4 are combined to generate SAHDs on [R.sup.n].