Approximation theorems for spherical monogenics of complex degree
Let \(\Gamma\) be the spherical Dirac operator on \(S^{m-1}\) in \(\mathbb{R}^{m}\), induced by the Dirac operator in the embedding space \(\mathbb{R}^{m}\). The eigenfunctions of \(\Gamma + \alpha\) for complex \(\alpha\) correspond to eigenfunctions of the Dirac operator on the sphere itself. The \(C^{1}\)-functions satisfying \((\Gamma + \alpha)f = 0\) in \(\Omega \subset S^{m-1}\) are called spherical monogenics of order \(\alpha\) in \(\Omega\), the right module of these functions is denoted by \(M^{\alpha}(\Omega)\) . Two Runge approximation theorems of the following type are proved: Let \(K\) be compact, \(\Omega\) open, \(K \subset \Omega \subset S^{m-1}\). Then the following conditions are equivalent: (i) \(M^{\alpha}(\Omega)\) is dense in \(M^{\alpha}(K) = \lim \text{ind}_{K \subset \Omega '}M^{\alpha}(\Omega ')\) (ii) \(\Omega \setminus K\) has no components of which the closure in \(S^{m-1}\) is contained in \(\Omega\). Also an interesting theorem of Mittag-Leffler type is proved.
scientific article; zbMATH DE number 1334503