Rotation Invariant Ultradistributions

Abstract

We prove that an ultradistribution is rotation invariant if and only if it coincides with its spherical mean. For it, we study the problem of spherical representations of ultradistributions on R n . Our results apply to both the quasianalytic and the non-quasianalytic case.

Distributions: a spherical co-ordinates approach

“…the angular part ∂ ang of the Dirac operator. The outcome should match result (10), which indeed it does seen the following commutative scheme…”

“…Apparently there seems to be no possibility to uniquely define the actions of the ∂ rad and ∂ ang operators on a standard distribution by singling out specific distributions in the equivalent classes ( 9) and (10), except for the following two special cases. This first special case is illustrated by the delta-distribution (see also [3]):…”

“…By Hahn-Banach's theorem the bounded linear functional T (r, ω) ∈ V ′ may be extended to the distribution T(r, ω) ∈ D ′ (R × S m−1 ); such an extension is called a spherical representation of the distribution T (see e.g. [10]). However as the subspace V is not dense in D(R × S m−1 ), the spherical representation of a distribution is not unique, but if T 1 and T 2 are two different spherical representations of the same distribution T , their restrictions to V coincide:…”

On the Radial Derivative of the Delta Distribution

“…By Hahn-Banach's theorem the bounded linear functional T (r, ω) ∈ V ′ may be extended to the distribution T(r, ω) ∈ D ′ (R × S m−1 ); such an extension is called a spherical representation of the distribution T (see e.g. [7]). As the subspace V is not dense in D(R× S m−1 ), the spherical representation of a distribution is not unique, but if T 1 and T 2 are two different spherical representations of the same distribution T , their restrictions to V coincide: for all test functions Ξ(r, ω) ∈ D(R×S m−1 ), and similar expressions for ∂ ω T, r T and ω T. However if T 1 and T 2 are two different spherical representations of the same distribution T ∈ D ′ (R m ), then, upon restricting to test functions ϕ(r, ω) ∈ V, we are stuck with − T 1 (r, ω), ∂ r ϕ(r, ω) = − T 2 (r, ω), ∂ r ϕ(r, ω) since ∂ r ϕ(r, ω) is an odd function in the variables (r, ω) and does no longer belong to V (and neither do ∂ ω ϕ(r, ω), r ϕ(r, ω) and ω ϕ(r, ω)).…”

“…By Hahn-Banach's theorem the bounded linear functional T (r, ω) ∈ V ′ may be extended to the distribution T(r, ω) ∈ D ′ (R × S m−1 ); such an extension is called a spherical representation of the distribution T (see e.g. [7]). As the subspace V is not dense in D(R× S m−1 ), the spherical representation of a distribution is not unique, but if T 1 and T 2 are two different spherical representations of the same distribution T , their restrictions to V coincide:…”