Vladimir V. Andrievskii

Discrepancy of signed measures and polynomial approximation

Auxiliary Facts.- Zero Distribution of Polynomials.- Discrepancy Theorems via Two-Sided Bounds for Potentials.- Discrepancy Thoerems via One-Sided Bounds for Potentials.- Discrepancy Theorems via Energy Integrals.- Applications of Jentzsch-Szego and Erdos-Turan Type Theorems.- Applications of Discrepancy Theorems.- Special Topics.- Appendix A: Conformally Invariant Characteristics of Curve Families.- Appendix B: Basics in the Theory of Quasiconformal Mappings.- Appendix C: Constructive Theory of Functions of a Complex Variable.- Appendix D: Miscellaneous Topics.- Bibliography.- Glossary of Notation.- Index.

Polynomial approximation of analytic functions on a finite number of continua in the complex plane

The Dzjadyk-type theorem concerning the polynomial approximation of functions on a continuum in the complex plane C is generalized to the case of polynomial approximation of functions on a compact set in C which consists of a finite number of continua.

The highest smoothness of the Green function implies the highest density of a set

We investigate local properties of the Green function of the complement of a compact setEυ[0,1] with respect to the extended complex plane. We demonstrate, that if the Green function satisfies the 1/2-Hölder condition locally at the origin, then the density ofE at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]..

Maximal polynomial subordination to univalent functions in the unit disk

AbstractLet Ω⊂C be a simply connected domain, 0∈Ω, and let ℘n,n∈N, be the set of all polynomials of degree at mostn. By ℘n(Ω) we denote the subset of polynomials p ∈℘n withp(0)=0 andp(D)⊂Ω, whereD stands for the unit disk {z: |z|<1}, and by we denote the “maximal range” of these polynomials. Letf be a conformal mapping fromD onto Ω,f(0)=0. The main theme of this note is to relate Ωn (or some important aspects of it) to the imagesfs(D), wherefs(z):=f[(1−s)z], 0<s<1. For instance we prove the existence of a universal constantc0 such that, forn≥2c0,

On approximation of continuous functions by trigonometric polynomials

We generalize the classical Jackson-Bernstein constructive description of Holder classes of periodic functions on the interval [-@p,@p]. We approximate by trigonometric polynomials continuous functions defined on a compact set E@?[-@p,@p]. This set may consist of an infinite number of intervals.

A Note on a Remez-Type Inequality for Trigonometric Polynomials

We obtain sharp bounds, in the uniform norm along the unit circle T, of exponentials of logarithmic potentials, if the logarithmic capacity of the subset of T, where they are at most 1, is known.

Simultaneous approximation and interpolation of functions on continua in the complex plane

Abstract We construct polynomial approximations of Dzjadyk type (in terms of the k -th modulus of continuity, k ⩾1) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E . Furthermore, the error in this approximation is decaying as e − cn α strictly inside E , where c and α are positive constants independent of the degree n of the approximating polynomial.

A remez-type inequality in terms of capacity

We obtain sharp uniform bounds for exponentials of logarithmic potentials if the logarithmic capacity of the subset, where they are at most 1, is known.

Remez-Type Inequalities in Terms of Linear Measure

We obtain sharp uniform bounds for an exponential Q of a logarithmic potential on a quasi-smooth curve (in the sense of Lavrentiev) in terms of the linear measure of the subset of that curve on which Q is bounded by 1.

Angular Distribution of Zeros of the Partial Sums of ez via the Solution of Inverse Logarithmic Potential Problem

We continue the work of Szegő [18] on describing the angular distribution of the zeros of the normalized partial sum sn(nz) of ez, where