On a three dimensional analogue to the holomorphic z-powers: power series and recurrence formulae

Abstract

The main objective of this contribution is a constructive generalization of the holomorphic power and Laurent series expansions in ℂ to dimension 3 using the framework of hypercomplex function theory. This first article on hand deals with generalized Fourier and Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in ℝ³ and their series coefficients can be explicitly (one-to-one) linked with each other. Finally, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented.

Keywords:

• complete orthonormal systems
• solid spherical monogenics
• Fourier series
• Taylor series
• recurrence formulae
• closed-form representations

AMS Subject Classifications::

• 30G35
• 32A05
• 42C05
• 65Q30

View correction statement:

On a three-dimensional analogue to the holomorphic z-powers: power series and recurrence formulae

Acknowledgement

The author expresses his gratitude to his mentor Prof K. Gürlebeck for the helpful discussions.