Applications and Computation of Orthogonal Polynomials

The workshop on Applications and Computation of Orthogonal Polynomials took place March 22-28, 1998 at the Oberwolfach Mathematical Research Institute. It was the first workshop on this topic ever held at Oberwolfach. There were 46 participants from 13 countries, more than half coming from Germany and the United States, and a substantial number from Italy. A total of 23 plenary lectures were presented and 4 short informal talks. Open problems were discussed during an evening session. This volume contains refereed versions of 18 papers presented at, or submitted to, the conference. The theory of orthogonal polynomials, as a branch of classical analysis, is well established. But orthogonal polynomials play also an important role in many areas of scientific computing, such as least squares fitting, numerical integration, and solving linear algebraic systems. Though the basic tenets have their roots in 19th century mathematics, the use of modern computers has required the development and study of new algorithms that are accurate and robust. The computational methods and applications represented in this volume, of necessity, are incomplete, yet sufficiently varied to convey an impression of current activities in this area.

The sensitivity of least squares polynomial approximation. - Transpose-free look-ahead algorithms for Lanczos method. - Applications of anti-Gauss quadrature rules in linear algebra. - Stieltjes polynomials and the error of Gauss-Kronrod quadrature formulas. - Fast solution of confluent Vandermonde-like linear systems using polynomial arithmetic. - On discrete polynomial least-squares approximation in moving time windows. - Quadrature rules based on s-orthogonal polynomials for evaluating integrals with strong singularities. - Gegenbauer weight functions admitting L2 Duffin and Schaeffer type inequalities. - Questions related to Gaussian quadrature formulas and two-term recursions. - Construction and computation of a new set of orthogonal polynomials. - Fourier transforms of orthogonal polynomials of singular continuous spectral measures. - On a sequence of fast decreasing polynomial operators. - Müntz orthogonal polynomials and their numerical evaluation. - Positivity of Gauss-Kronrod formulae for a certain ultraspherical weight function. - A Christoffel-Darboux-type formula for Szegö polynomials and polynomial evaluation. - Applications of tensor-valued tri-variate Hermite polynomials and spherical harmonics in the kinetic theory of gases. - Indeterminate moment problems and a conjecture on the growth of the entire functions in the Nevanlinna parametrization. - Spectral methods based on nonclassical orthogonal polynomials. - Author index.