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The paper features some observations about the behaviour of special monogenic polynomials in the unit ball of ℝn which are relevant to the development of general approximation methods in the framework of Clifford Analysis. It focuses on an estimate for the best approximation of monogenic functions by a set of polynomials in Sobolev spaces. We illustrate some rather surprising orthogonality properties between symmetric homogeneous polynomials of the same degree and also an elementary method of obtaining a variety of linearly independent homogeneous polynomials of such type for the case of functions defined in ℝ3 and with values in the algebra of quaternions. The proof of their generalizations to higher dimensions in a suitable way is possible but needs more complicated combinatorical calculations which exceed the scope of this communication.

Fifty years ago, on the 6th of August of 1950, Rudolf Fueter died in Brunnen, Switzerland, at the age of seventy. He wrote a remarkable chapter in the development of Swiss mathematics in the first half of the century and was an example of civil courage during the dark times of the Second World War. This paper intends to describe some less well known facts in Fueter's life, particularly his motivations for creating a new branch of mathematics which he used to call hypercomplex function theory.

In [2], [6], [7], methods are discussed for solving initial value problems $$\left\{ \begin{gathered} \frac{{\partial u}}{{\partial t}}\left( {t,x_0 ,x_1 } \right) = f\left( {t,x_0 ,x_1 ,u\left( {t,x_0 ,x_1 } \right),\frac{{\partial u}}{{\partial x_0 }},\frac{{\partial u}}{{\partial x_1 }}} \right) \hfill \\ u\left( {0,x_0 ,x_1 } \right) = u^{\left( 0 \right)} \left( {x_0 ,x_1 } \right) \hfill \\ \end{gathered} \right.$$ in certain scales of Banach spaces. The crucial point is to use suitable interior estimates for the complex valued functionu=u0+iu1. For holomorphic functionsu these estimates follow from Cauchy’s integral formula or from equivalent estimates for the harmonic partsu0 andu1.