The book is an authoritative and up-to-date introduction to the field of Analysis and Potential Theory dealing with the distribution zeros of classical systems of polynomials such as orthogonal polynomials, Chebyshev, Fekete and Bieberbach polynomials, best or near-best approximating polynomials on compact sets and on the real line. The main feature of the book is the combination of potential theory with conformal invariants, such as module of a family of curves and harmonic measure, to derive discrepancy estimates for signed measures if bounds for their logarithmic potentials or energy integrals are known a priori. Classical results of Jentzsch and Szegรถ for the zero distribution of partial sums of power series can be recovered and sharpened by new discrepany estimates, as well as distribution results of Erdรถs and Turn for zeros of polynomials bounded on compact sets in the complex plane. Vladimir V. Andrievskii is Assistant Professor of Mathematics at Kent State University. Hans-Peter Blatt is Full Professor of Mathematics at Katholische Universitรคt Eichstรคtt
CONTENT
1 Auxiliary Facts -- 2 Zero Distribution of Polynomials -- 3 Discrepancy Theorems via Twoโ{128}{148}Sided Bounds for Potentials -- 4 Discrepancy Theorems via One-Sided Bounds for Potentials -- 5 Discrepancy Theorems via Energy Integrals -- 6 Applications of Jentzschโ{128}{148}Szegรถ and Erdรถsโ{128}{148}Turรกn Type Theorems -- 7 Applications of Discrepancy Theorems -- 8 Special Topics -- A Conformally Invariant Characteristics of Curve Families -- A.1 Module and Extremal Length of a Curve Family -- A.2 Reduced Module -- B Basics in the Theory of Quasiconformal Mappings -- B.1 Quasiconformal Mappings -- B.2 Quasiconformal Curves and Arcs -- C Constructive Theory of Functions of a Complex Variable -- C.1 Jackson Type Kernels -- C.2 Polynomial Kernels Approximating the Cauchy Kernel -- C.3 Inverse Theorems -- C.4 Polynomial Approximation in Domains with Smooth Boundary -- D Miscellaneous Topics -- D.1 The Regularized Distance -- D.2 Greenโ{128}{153}s Function for a System of Intervals -- Notation
