A Characterization of Analytic Functionals on the Sphere II

Abstract

Matsuzawa [4] started to characterize generalized functions (for example, distributions or hyperfunctions) on ℝⁿ as initial values of heat functions. His idea is valid even for quasi-analytic ultradistri- butions in $\varepsilon {{'}_{{\left\{ s \right\}}}}\left( {{{\mathbb{R}}^{n}}} \right)$ with s > 1/2. Note that functions in ε_{s}(ℝⁿ)are non-quasi-analytic if s > 1 but quasi-analytic if s ≤ 1. We propose a method to extend Matuzawa’s idea for wider range of s.

In the first part [7] we studied generalized functions on the one- dimensional sphere (that is, the circle). We study here generalized functions on the n-dimensional sphere ${\mathbb{S}^n}$ with n > 1, expanding them into the spherical harmonic functions.