Rotation invariant ultradistributions

TL;DR

This paper characterizes rotation invariant ultradistributions as those equal to their spherical mean, providing a comprehensive study of their spherical representations applicable to both quasianalytic and non-quasianalytic cases.

Contribution

It establishes a necessary and sufficient condition for rotation invariance of ultradistributions via spherical means, extending the theory to all cases.

Findings

Rotation invariant ultradistributions equal their spherical mean

Spherical representations of ultradistributions are characterized

Results apply to both quasianalytic and non-quasianalytic cases

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 $\mathbb{R}^{n}$. Our results apply to both the quasianalytic and the non-quasianalytic case.