Book Description: The topic of this book Orthogonal Polynomials and Special Functions (OPSF), has a very rich history, going back to 19th century when mathematicians and physicists tried to solve the most important di_erential equations of mathematical physics. Hermite-PadŽe approximation was also introduced in that time, to prove the transcendence of the remarkable constant e (the basis of the natural logarithm). Since then OPSF has developed to a standard subject within mathematics, which is driven by applications. The applications are numerous, both within mathematics (e.g. statistics, combinatorics, harmonic analysis, number theory) and other sciences, such as physics, biology, computer science, chemistry. The main reason for the fact that OPSF has been so successful over the centuries is its usefulness in other branches of mathematics and physics, as well as other sciences. There are many di_erent aspects of OPSF. Some of the most importantdevelopments for OPSF are related to the theory of rational approximation of analytic functions, in particular the extension to simultaneous rational approximation to a system of functions. Important tools for rational approximation are Riemann-Hilbert problems, the theory of orthogonal polynomials, logarithmic potential theory, and operator theory for di_erence operators. This new book presents the latest research in the field.
There were seven series of lectures within sixty hours. Christian Berg discusses matrix polynomials orthogonality: complex measures and matrix measures, compact sets of positive matrix measures on the real line, Kreins theorem characterizing matrix moment sequences, matrix inner products and orthonormal matrix polynomials, and some consequences of the three-term recurrence relation. Guillermo LŽopez discusses the classical constructive theory of approximation, which is basic for the understanding of several important and classical results from this theory of orthogonal polynomials and the spectral theory of infinite dimensional matrices. Francisco MarcellŽan presents a self-contained survey of recent results on Sobolev orthogonal polynomials. The topics covered are standard orthogonal polynomials, Sobolev inner products: the multiplication operator, coherent pairs of measures, analytic properties of Sobolev orthogonal polynomials. Franz Peherstorfer discusses inverse images of polynomial mappings are the basis for investigations in polynomial iteration, which leads often directly to an understanding of the behaviour of the iterates and the extremal properties of the iterates. The purpose of these notes is to give basic material on the structure and the geometry of inverse polynomial images and on the behaviour of extremal polynomials on such sets, including several intervals or arcs, lemniscates, Julia sets of Cantor types, dendrites,. . . . In particular, many examples are given. Walter van Assche discusses some aspects of analytic number theory, in particular rational approximation of irrational numbers, irrationality proofs and transcendence proofs. Quite often the construction of rational approximants to real numbers is by means of continued fractions, PadŽe approximation or Hermite-PadŽe approximation, showing that rational approximation and special functions have interesting applications in analytic number theory. Semyon Yakubovich give a short introduction to the theory of integral transforms in Lebesgue spaces, which are associated with hypergeometric functions as their kernels. We deal with a class of the so-called Kontorovich-Lebedev type integral transforms, which includes, in particular, the familiar Kontorovich-Lebedev, Mehler-Fock, Olevskii and Lebedev transforms. Joaquin Bustoz discusses Hausdorff summability, q-theory of Hausdorff summability, and potential applications to unsolved problems. The lectures notes are aimed at graduate students and post-docs, or anyone who wants to have an introduction to (and learn about) the subjects mentioned. Each of the contributions is self-contained, and contains up to date references to the literature so that anyone who wants to apply the results to his own advantage has a good starting point. The knowledge required for the lectures is real and complex analysis, some basic notions of algebra and discrete mathematics, and some elementary facts of orthogonal polynomials. A computer equipped with Maple software is useful to work on the exercises. So having mastered the lectures notes gives a good level to read research papers in this field, and to start doing research as well. |