Let K ⊂ R be a regular compact set and let g ( z ) = g C ¯ ∖ K ( z , ∞ ) be the Green function for C ¯ ∖ K with pole at infinity. For δ > 0, define G ( δ ) ≔ max { g ( z ) : z ∈ C , dist ( z , K ) ≤ 2 δ } . Let { x n } n = 0 ∞ be a Leja sequence of points of K. Then the uniform norm ‖ T n ‖ = Λ n , n = 1 , 2 , … of the associated interpolation operator T n, i.e., the nth Lebesgue constant, is bounded from above by min δ > 0 2 n diam ( K ) δ e n G ( δ ) 9 / 8 . In particular, when K is a uniformly perfect subset of R, the Lebesgue constants grow at most polynomially in n. To the best of our knowledge, the result is new even when K is a finite union of intervals.

This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through the application of the theory of Dzyadyk’s kernels combined with the methods and results from modern geometric function theory and the theory of quasiconformal mappings.

Let K be a quasidisk on the complex plane. We construct a sequence of monic polynomials \(p_n=p_n(\cdot ,K)\) with all their zeros on K such that \(\Vert p_n\Vert _K\le O(1)\text{ cap }(K)^n\) as \(n\rightarrow \infty \).

We prove an analogue of the classical Bernstein polynomial inequality on a compact subset E of the real line. The Lipschitz continuity of the Green function for the complement of E with respect to the extended complex plane and the differentiability at a point of E of a special, associated with E, conformal mapping of the upper half-plane onto the comb domain play crucial role in our investigation.

Let K be a compact set in the complex plane consisting of a finite number of quasidisks. We study the rate of approximation of K from the outside by lemniscates in terms of level lines of the Green function for the complement of K.

We prove an analogue of the classical Bernstein theorem concerning the rate of polynomial approximation of piecewise analytic functions on a compact subset of the real line.

We establish the exact (up to the constants) double inequality for the Christoffel function for a measure supported on a Jordan domain bounded by a quasiconformal curve. We show that this quasiconformality of the boundary cannot be omitted.

We establish sharp $$L_p$$ , $$1\le p<\infty $$ , weighted Remez- and Nikolskii-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) curve in the complex plane.

Using the theory of quasiconformal mappings, we simplify the proof of the recent result by Taylor and Totik (2010) on the behavior of the Lebesgue constants for interpolation points on a compact set in the complex plane.

The estimates of the uniform norm of the Chebyshev polynomials associated with a compact set K in the complex plane are established. These estimates are exact (up to a constant factor) in the case where K consists of a finite number of quasiconformal curves or arcs. The case where K is a uniformly perfect subset of the real line is also studied.

The estimates of the uniform norm of the Chebyshev polynomial associated with a compact set K consisting of a finite number of continua in the complex plane are established. These estimates are exact (up to a constant factor) in the case where the components of K are either quasismooth (in the sense of Lavrentiev) arcs or closed Jordan domains bounded by a quasismooth curve.

We discuss results concerning a conjecture on the rate of polynomial approximation on the compact set of the plane to a complex extension of the absolute value function. The conjecture was stated by Grothmann and Saff in 1988. In particular, we prove another conjecture, Gaier conjecture, on the polynomial approximation of piecewise analytic functions on a compact set consisting of two touching discs.

We prove a Jackson–Mergelyan type theorem on the uniform polynomial approximation of continuous polyharmonic functions on a set "without cusps on the boundary that point inside of the set". We apply this theorem to derive a harmonic counterpart of a result by Mezhevich and Shirokov on the analytic polynomial approximation of continuous functions on a set consisting of two parallel segments.

We prove a Dzjadyk type theorem for the approximation of a real-valued function f that changes its sign finitely many times on a Dini-smooth Jordan arc L in the complex plane. The approximation is by harmonic polynomials which are copositive with f on L.

We establish L p , 1≦p<∞, Bernstein-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) curve in the complex plane.

This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory.

We establish L p ,1≤p<∞ Markov–Bernstein-type inequalities for algebraic polynomials considered on a quasismooth (in the sense of Lavrentiev) arc in the complex plane.

The distribution of zeros and poles of best rational approximants is well understood for the functions f ( x ) = | x | α , α > 0. If f ∈ C [−1, 1] is not holomorphic on [−1, 1], the distribution of the zeros of best rational approximants is governed by the equilibrium measure of [−1, 1] under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, a ‐values, and poles of best real rational approximants of degree at most n to a function f ∈ C [−1, 1] that is real‐valued, but not holomorphic on [−1, 1]. Generalizations to the lower half of the Walsh table are indicated.

We prove a Jackson-Mergelyan-type theorem on approximation of a function by reciprocals of complex polynomials. The function is continuous on a quasi-smooth (in the sense of Lavrentiev) arc in the complex plane.