Approximation on compact sets of functions and all derivatives

TL;DR

This paper extends Mergelyan type approximation to include uniform approximation of all derivatives on compact sets, broadening the scope of polynomial and rational function approximation in complex analysis.

Contribution

It introduces a new approximation framework that ensures uniform convergence of all derivatives on compact sets, generalizing classical results.

Findings

Established conditions for uniform approximation of all derivatives on compact sets.

Extended classical approximation theorems to include derivative approximation.

Provided new methods for approximating functions with all derivatives uniformly.

Abstract

In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.