On convergence properties of 3D spheroidal monogenics (Q2846502)

The authors consider constructive questions in quaternion analysis. It is a very important question to investigate approximation properties for monogenic functions by suitable polynomial systems. Meanwhile there are many well-known results over the ball. It is a very sophistical problem getting results over domains different from the ball in higher dimensions.NEWLINENEWLINE The authors chose 3D-prolate spheroids. Relations of spheroidal functions as natural bridge to Legendre and Chebychev polynomials are studied. Furthermore, monogenic functions are used as refinement of harmonic functions. Some products of Ferrer's associated Legendre functions and Chebychev polynomials are important for the construction. A recurrence formula is obtained which seems to be an essential step towards an efficient numerical method.NEWLINENEWLINE The results are demonstrated in the case of the Riesz system and the Moisil-Toedorescu system. In Theorem 3.1 a complete orthonormal system over the described domains is deduced. The main results are formulated in Theorem 3.3. Only with an explicit value for the norm of such a spheroidal monogenic, a Fourier expansion makes sense.