Three classes of functions are introduced, corresponding to known results in the theory of uniform polynomial approximation of functions on arcs in the complex plane. The structural properties of functions in these classes are described from a unified point of view, which allows one to understand more fully their nature and the differences between them.

http://iopscience.iop.org/1468-4810/44/1/A10/pdf/1468-4810_44_1_A10.pdf

Constructive description of certain classes of functions on quasismooth arcs

We consider best polynomial approximation to functions analytic in a Jordan domain D and continuous on/5. We relate theorems of Jackson and Bernstein type to the HSlder continuity of the exterior conformal mapping functions for D.

/pdf/on-theorems-of-jackson-and-bernstein-type-in-the-complex-pzng9gfjda.pdf

On Theorems of Jackson and Bernstein Type in the Complex Plane

J-property of Jordan arcs

For a compact set K which is the closure of a Jordan domain, the Faber operator provides a well-known tool for deriving results on the error of uniform polynomial approximation on K We show that the corresponding methods also work for compact sets which are not the closure of a Jordan domain In particular, the cases of so-called touching domains and of Jordan arcs are considered

Polynomial Approximation on Compact Sets Bounded by Dini-Smooth Arcs

Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) e C(Γ) with the same order of approximation as that in Jackson Theorem 1 on real interval [−1, 1]. The accuracy of the order of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [−1, 1] are obtained.

Jackson type theorems on complex curves

Best Approximation on Smooth Arcs

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Best Approximation on Smooth Arcs