| Logarithmic Potentials with External Fields 1997 Edition Contributor(s): Saff, Edward B. (Author), Totik, Vilmos (Author) |
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ISBN: 3540570780 ISBN-13: 9783540570783 Publisher: Springer OUR PRICE: $52.24 Product Type: Hardcover - Other Formats Published: October 1997 Annotation: This treatment of potential theory emphasizes the effects of an external field (or weight) on the minimum energy problem. Several important aspects of the external field problem (and its extension to signed measures) justify its special attention. The most striking is that it provides a unified approach to seemingly different problems in constructive analysis. These include the asymptotic analysis of orthogonal polynomials, the limited behavior of weighted Fekete points; the existence and construction of fast decreasing polynomials; the numerical conformal mapping of simply and doubly connected domains; generalization of the Weierstrass approximation theorem to varying weights; and the determination of convergence rates for best approximating rational functions. |
| Additional Information |
| BISAC Categories: - Mathematics | Calculus - Mathematics | Mathematical Analysis - Mathematics | Functional Analysis |
| Dewey: 515.9 |
| LCCN: 97028048 |
| Series: Springer Series in Computational Physics |
| Physical Information: 1.13" H x 6.14" W x 9.21" (2.02 lbs) 505 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: In recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real line). Most of the applications are based on an exten- sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and includes: creation of the theory of non-classical orthogonal polynomials with re- spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully develop the classical theory of logarithmic potentials. |