An Additive Decomposition of Harmonic Functions in $\rm{I\!R}^{3}$

Additive decompositions of harmonic functions play an important role in function theory and for the solution of partial differential equations. One of the best known results is the decomposition of harmonic functions as a sum of a holomorphic and an anti-holomorphic function. This decomposition can be generalized also to the analysis of quaternion valued harmonic function, where the summands are then monogenic or anti-monogenic, repectively. For paravector-valued functions, sometimes called A-valued functions, this decomposition is not possible. Fortunately, it was shown in previous articles that harmonic functions can be presented by a linear combination of monogenic, antimonogenic and ψ-hyperholomorphic functions with ψ = {1, e2, −e1}, called a structural set. The question for more general set ψ will be studied in this paper. Mathematics Subject Classification (2010). 30G35, 42C05, 33E10.