Approximation by Convolution Polyanalytic Operators in the Complex and Quaternionic Compact Unit Balls

In this paper, by using the convolution method, we obtain quantitative results in terms of various moduli of smoothness for approximation of polyanalytic functions by polyanalytic polynomials in the complex unit disc. Then, by introducing the polyanalytic Gauss-Weierstrass operators of a complex variable, we prove that they form a contraction semigroup on the space of polyanalytic functions defined on the compact unit disk. The quantitative approximation results in terms of moduli of smoothness are then extended to the case of slice p-polyanalytic functions on the quaternionic unit ball. Moreover, we show that also in the quaternionic case the Gauss-Weierstrass operators of a quaternionic variable form a contraction semigroup on the space of polyanalytic functions defined on the compact unit ball.