This proceeding is intended to be a first introduction to spectral methods. It is written around some simple problems that are solved explicitly and in details and that aim at demonstrating the power of those methods. The mathematical foundation of the spectral approximation is first introduced, based on the Gauss quadratures. The two usual basis of Legendre and Chebyshev polynomials are then p...

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix r new rows and columns, so that the original Jacobi matrix is shifted downward. The r new...

This paper develops a family of preconditioners for pseudospectral approximations of pth-order linear differential operators subject to various types of boundary conditions. The approximations are based on ultraspherical polynomials with special attention being paid to Legendre and Chebyshev polynomial methods based on Gauss–Lobatto quadrature points. The eigenvalue spectrum of the precondition...

In the topological classification of phase portraits no distinctions are made between a focus and a node and neither are they made between a strong and a weak focus or between foci of different orders. These distinctions are however important in the production of limit cycles close to the foci in perturbations of the systems. The distinction between the one direction node and the two directions...

It is often the case that the exact moments of a statistic of the continuous type can be explicitly determined, while its density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. The density approximants discussed in this article are based on the first n exact moments of the corresponding distributions. A unified semiparametric approach to...

We give two recursive expressions for both MacWilliams and Chebyshev matrices. The expressions give rise to simple recursive algorithms for constructing the matrices. In order to derive the second recursion for the Chebyshev matrices we find out the Krawtchouk coefficients of the Discrete Chebyshev polynomials, a task interesting on its own.

We introduce the so-called Clifford–Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related ...

Some integral properties of zonal spherical functions, hypergeometric functions and invariant polynomials are studied for real normed division algebras.

We study interpolation polynomials based on the points in [−1, 1]× [−1, 1] that are common zeros of quasi-orthogonal Chebyshev polynomials and nodes of near minimal degree cubature formula. With the help of the cubature formula we establish the mean convergence of the interpolation polynomials. 1991 Mathematics Subject Classification: Primary 41A05, 33C50.