Discrepancy of Signed Measures and Polynomial Approximation

ISBN-13: 9780387986524 / Angielski / Twarda / 2001 / 438 str.

V. V. Andrievskii; Hans-Peter Blatt; H. P. Blatt

Opis

In many situations in approximation theory the distribution of points in a given set is of interest. For example, the suitable choiee of interpolation points is essential to obtain satisfactory estimates for the convergence of interpolating polynomials. Zeros of orthogonal polynomials are the nodes for Gauss quadrat ure formulas. Alternation points of the error curve char- acterize the best approximating polynomials. In classieal complex analysis an interesting feature is the location of zeros of approximants to an analytie function. In 1918 R. Jentzsch 91] showed that every point of the circle of convergence of apower series is a limit point of zeros of its partial sums. This theorem of Jentzsch was sharpened by Szego 170] in 1923. He proved that for apower series with finite radius of convergence there is an infinite sequence of partial sums, the zeros of whieh are "equidistributed" with respect to the angular measure. In 1929 Bernstein 27] stated the following theorem. Let f be a positive continuous function on -1, 1]; if almost all zeros of the polynomials of best 2 approximation to f (in a weighted L -norm) are outside of an open ellipse c with foci at -1 and 1, then f has a continuous extension that is analytic in c.

Recenzje

From the reviews of the first edition:

"Distributions of certain point sets are in many respects important in approximation theory. ... Many classical theorems ... deal with such problems. This book collects a number of generalizations of these theorems. ... The book is written with great care. ... This monograph is a valuable addition to the library of any researcher in approximation theory. The topic is rather specialized, but the style and the importance of potential theory in the discipline, makes it also suited for an advanced course in approximation theory." (Simon Stevin Bulletin, Vol. 11 (1), 2004)

"This book is devoted to discrepancy estimates for the zero of polynomials and for signed measures. ... A remarkable feature of the book is that most of the results in it are shown to be sharp. ... This work is a valuable monograph on a field that has attracted considerable interest in the recent past, and which has various applications in approximation theory and orthogonal polynomials." (Vilmos Totik, Bulletin of the London Mathematical Society, Vol. 35, 2003)

"This book discusses in detail the discrepancy of signed measures ... . The detailed proofs and the rich reference make this book eligible for a self-study textbook and a reference book, too." (Béla Nagy, Acta Scientiarum Mathematicarum, Vol. 68, 2002)

Spis treści

* Preface * 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-Szegö and Erdös-Turán 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 *

Blurb

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 Szegö 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 Erdös 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 Universität Eichstätt.