Walsh equiconvergence theorems in the quaternionic setting

Abstract

In this paper, we prove the quaternionic version of the result of Walsh stating that the difference between the partial sums of the Taylor expansion of an analytic function and its interpolation polynomial at the roots of unity converges in a larger disc than the disc of analyticity of the function. Our result holds for functions of a quaternionic variable which are slice regular in a ball and thus they admit a converging power series expansion. We also prove a generalization of this theorem as well as its converse. Because of the noncommutative setting, the results are nontrivial and require a notion of multiplication of functions (and of polynomials) which does not commute with the evaluation.