Quadrature and Orthogonal Polynomials, Volume Volume 5 (Numerical analysis 2000)

/homepage/sac/cam/na2000/index.html 7-Volume Set now available at special set price ! Orthogonal polynomials play a prominent role in pure, applied, and computational mathematics, as well as in the applied sciences. It is the aim of the present volume in the series "Numerical Analysis in the 20th Century" to review, and sometimes extend, some of the many known results and properties of orthogonal polynomials and related quadrature rules. In addition, this volume discusses techniques available for the analysis of orthogonal polynomials and associated quadrature rules. Indeed, the design and computation of numerical integration methods is an important area in numerical analysis, and orthogonal polynomials play a fundamental role in the analysis of many integration methods. The 20th century has witnessed a rapid development of orthogonal polynomials and related quadrature rules, and we therefore cannot even attempt to review all significant developments within this volume. We primarily have sought to emphasize results and techniques that have been of significance in computational or applied mathematics, or which we believe may lead to significant progress in these areas in the near future. Unfortunately, we cannot claim completeness even within this limited scope. nevertheless, we hope that the readers of the volume will find the papers of interest and many references to related work of help. We outline the contributions in the present volume. Properties of orthogonal polynomials are the focus of the papers by MarcellÁn and Álvarez-Nodarse and by Freund. The former contribution discusses "Favard's theorem", i.e., the question under which conditions the recurrence coefficients of a family of polynomials determine a measure with respect to which the polynomials in this family are orthogonal. Polynomials that satisfy a three-term recurrence relation as well as SzegÕ polynomials are considered. The measure is allowed to be signed, i.e., the moment matrix is allowed to be indefinite. Freund discusses matrix-valued polynomials that are orthogonal with respect to a measure that defines a bilinear form. This contribution focuses on breakdowns of the recurrence relations and discusses techniques for overcoming this difficulty. Matrix-valued orthogonal polynomials form the basis for algorithms for reduced-order modeling. Freund's contribution to this volume provides references to such algorithms and their application to circuit simulation. The contribution by Peherstorfer and Steinbauer analyzes inverse images of polynomial mappings in the complex plane and their relevance to extremal properties of polynomials orthogonal with respect to measures supported on a variety of sets, such as several intervals, lemniscates, or equipotential lines. Applications include fractal theory and Julia etc. Orthogonality with respect to Sobolev inner products has attracted the interest of many researchers during the last decade. The paper by Martinez discusses some of the recent developments in this area. The contribution by LÓpez Lagomasino, Pijeira, and Perez Izquierdo deals with orthogonal polynomials associated with measures supported on compact subsets of the complex plane. The location and asymptotic distribution of the zeros of the orthogonal polynomials, as well as the n th-root asymptotic behavior of these polynomials is analyzed, using methods of potential theory. Investigations based on spectral theory for symmetric operators can provide insight into the analytic properties of both orthogonal polynomials and the associated PadÉ approximants. The contribution by Beckermann surveys these results. Van Assche and Coussement study multiple orthogonal polynomials. These polynomials arise in simultaneous rational approximation; in particular, they form the foundation for simultaneous Hermite-Pad&eacuL. Reichel is the author of 'Quadrature and Orthogonal Polynomials, Volume Volume 5 (Numerical analysis 2000)', published 2001 under ISBN 9780444506153 and ISBN 0444506152.