Orthogonal basis for spherical monogenics by step two branching
The authors study higher dimensional function theory of monogenic functions; for references to the theory see [\textit{F. Brackx, R. Delanghe} and \textit{F. Sommen}, Clifford analysis. Research Notes in Mathematics, 76. Boston-London-Melbourne: Pitman Advanced Publishing Program (1982; Zbl 0529.30001); \textit{R. Delanghe, F. Sommen} and \textit{V. Souček}, Clifford algebra and spinor-valued functions. A function theory for the Dirac operator. Related REDUCE software by F. Brackx and D. Constales. Mathematics and its Applications (Dordrecht). 53. Dordrecht etc.: Kluwer Academic Publishers (1992; Zbl 0747.53001); \textit{J. Gilbert} and \textit{M. A. M. Murray}, Clifford algebras and Dirac operators in harmonic analysis. Cambridge (UK): Cambridge University Press (1991; Zbl 0733.43001)]. The aim of this publication is to study the decomposition of spherical harmonics and monogenics under the action of \(\text{Spin}(p)\times \text{Spin}(q)\). The authors also obtain an orthonormal basis for spherical harmonics and monogenics. The key tool is the Fischer decomposition. Another construction of \(\text{Spin}(p)\times \text{Spin}(q)\)-invariant spaces of monogenic functions is given in [\textit{A. K. Common} and \textit{F. Sommen}, J. Math. Anal. Appl. 185, No. 1, 189--206 (1994; Zbl 0822.30003); \textit{G. Jank} and \textit{F. Sommen}, Complex Variables, Theory Appl. 13, No. 3--4, 195--212 (1990; Zbl 0703.30044); \textit{F. Sommen}, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 21, 295--323 (1989; Zbl 0709.35077)].
scientific article