Edward B. Saff, Vilmos Totik

Logarithmic Potentials with External Fields (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Springer Berlin Heidelberg
• Seitenzahl: 505
• Erscheinungstermin: 11. November 2013
• Englisch
• ISBN-13: 9783662033296
• Artikelnr.: 53152269

Autorenporträt

¿Edward B. Saff received his B.S. in mathematics from the Georgia Institute of Technology and his Ph.D. from the University of Maryland, where he was a student of the renowned analyst Joseph L. Walsh. Saff's research areas include approximation theory, numerical analysis, and potential theory. He has published more than 290 mathematical research articles, co-authored 9 books, and co-edited 11 volumes.  Recognitions of his research include his election as a SIAM Fellow (Society for Industrial and Applied Mathematics) in 2023, as a Foreign Member of the Bulgarian Academy of Sciences in 2013, as a Fellow of the American Mathematical Society in 2013, as well as a Guggenheim Fellowship in 1978. Saff is co-Editor-in-Chief and Managing Editor of the research journal Constructive Approximation and serves on the editorial boards of Computational Methods and Function Theory and the Journal of Approximation Theory. He has mentored 18 Ph.D.'s as well as 13 post-docs. Saff is currently Distinguished Professor of Mathematics at Vanderbilt University. Vilmos Totik was educated in Hungary and was a professor of mathematics at the University of Szeged and the University of South Florida until his retirement. His main research interest is classical mathematical analysis, approximation theory, orthogonal polynomials and potential theory. He has published (partially with co-authors) 5 monographs, one problem book in set theory and about 220 research papers in various disciplines.

Inhaltsangabe

Part 1 Fundamentals. I Weighted Potentials.- II Recovery of Measures, Green Functions and Balayage.- III Weighted Polynomials.- IV Determination of the Extremal Measure.- Part 2 Applications and Generalizations.- V Extremal Point Methods.- VI Weights on the Real Line.- VII Applications Concerning Orthogonal Polynomials.- VIII Signed Measures.- IX Some Problems from Physics.- X Generalizations.- Part 3 Appendices.- A.I Basic Tools.- A.II The Dirichlet Problem and Harmonic Measures.- A.III Weighted approximation in .- A.IV Classical Logarithmic Potential Theory.