Approximation in compact balls by convolution operators of quaternion and paravector variable (Q372710)

Firstly, a convolution operator of a quaternion variable and the classical Vallée-Poussin kernel are introduced. For \(W\)-analytic functions in the unit ball an approximation result of Voronovskaja-type is obtained. With the help of Vallee-Poisson polynomials \(W\)-analytic functions can be approximated in a special way. Such type of functions are a little bit more general than slice regular functions. An effective error estimation is given. Similar results can be also obtained for other kernels (Fejer, Riesz, Zygmund, Jackson etc.) Furthermore, the Gauss-Weierstrass kernel is used for approximations. It is proved that the right/left Gauss-Weierstrass convolution of a quaternion variable defines a contraction semigroup in a corresponding quaternionic right/left Banach space. The results can be extended to \(W\)-analytic functions of a paravector variable in the Clifford algebra framework. The restriction to paravectors is caused because of the non-multiplicativity of the norm in a Clifford algebra.