Voronovskaja-Type Results in Compact Disks for Quaternion q-Bernstein Operators, q ≥ 1

Abstract

Attaching to a compact disk \({\overline{\mathbb{D}_{r}}}\) in the quaternion field \({\mathbb{H}}\) and to some analytic function in Weierstrass sense on \({\overline{\mathbb{D}_{r}}}\) the so-called q-Bernstein operators with q ≥ 1, Voronovskaja-type results with quantitative upper estimates are proved. As applications, the exact orders of approximation in \({\overline{\mathbb{D}_{r}}}\) for these operators, namely \({\frac{1}{n}}\) if q = 1 and \({\frac{1}{q^{n}}}\) if q > 1, are obtained. The results extend those in the case of approximation of analytic functions of a complex variable in disks by q-Bernstein operators of complex variable in Gal (Mediterr J Math 5(3):253–272, 2008) and complete the upper estimates obtained for q-Bernstein operators of quaternionic variable in Gal (Approximation by Complex Bernstein and Convolution-Type Operators, 2009; Adv Appl Clifford Alg, doi:10.1007/s00006-011-0310-8, 2011).