On derivatives of spherical monogenics

Abstract

The main objective of this article is to consider complete orthonormal systems of monogenic polynomials together with their hypercomplex derivatives. Desired is that the derivatives of the basis polynomials are again basis functions from the original system. Based on this result, we prove an orthogonal decomposition of the space of square integrable monogenic functions with respect to the derivatives of arbitrary order.

An orthogonal system of monogenic polynomials over prolate spheroids in R3

“…In our point of view the system can be seen as a refinement of the three-dimensional homogeneous monogenic polynomials recently exploited by several authors. For a general orientation the reader is suggested to check some of the existing pioneering works [10][11][12][13][14][15] and [11,16,14,[17][18][19], of which the first six deal entirely with the R 3 −→ R 4 case and the last six the R 3 −→ R 3 case. It should be noticed that the application of homotopy methods in such polynomials will not preserve their monogenicity, and therefore it will not allow the generation of the polynomials presented here.…”

Hypercomplex derivative bases of polynomials in Clifford analysis

“…In the present work we effect by the operator 1 2 D on any bases of special monogenic polynomials contrary to [25] where the authors consider uniquely a specified base of homogeneous monogenic polynomials.…”

“…We quote essentially the recent paper of Cacão et al [25] where they obtained that the hypercomplex derivative of the base homogeneous monogenic polynomials from a simple and explicitly orthonormal system of monogenic functions (studied in [26]) are again base functions from the original system. It should be observed that our approach to construct the hypercomplex derivative base is indeed different.…”

Bloch’s Theorem in the Context of Quaternion Analysis

“…, n + 1, are the associated Legendre functions. In [4] and [6], a special R-linear complete orthonormal system of A-valued homogeneous monogenic polynomials in the unit ball of R 3 is explicitly constructed by applying the operator 1 2 D to the system (3). Restricting the resulting solid spherical monogenics to the surface of the unit ball we obtain a system of spherical monogenics, denoted by…”