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

Abstract

Attaching to a compact ball $\overline{\mathbb{B}_{r}}$ in the quaternion field $\mathbb{H}$ and to analytic functions in Weierstrass sense (slice regular functions on $\overline{\mathbb{B}_{r}}$) some convolution operators, the exact orders of approximation in $\overline{\mathbb{B}_{r}}$ for these operators are obtained. The results in this paper extend to quaternionic variables those in the case of approximation of analytic functions of a complex variable in disks by convolution operators of a complex variable, extensively studied in the Chapter 3 of the book [5]. More in general, the results extend also to the setting of analytic functions of paravector variable with coefficients in a Clifford algebra.