We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

A novel algorithm for ODF (orientation density function) estimation from diffraction pole figures is presented which is especially well suited for sharp textures and high-resolution pole figures measured with respect to arbitrarily scattered specimen directions, e.g. by area detectors. The estimated ODF is computed as the solution of a minimization problem which is based on a model of the diffraction counts as a Poisson process. The algorithm applies discretization by radially symmetric functions and fast Fourier techniques to guarantee smooth approximation and high performance. An implementation of the algorithm is freely available as part of the texture analysis software MTEX.

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A novel pole figure inversion method: specification of the MTEX algorithm

Algorithms for polynomials in Bernstein form

On the numerical condition of polynomials in Berstein form

The Lagrange representation of the interpolating polynomial can be rewritten in two more computationally attractive forms: a modified Lagrange form and a barycentric form. We give an error analysis of the evaluation of the interpolating polynomial using these two forms. The modified Lagrange formula is shown to be backward stable. The barycentric formula has a less favourable error analysis, but is forward stable for any set of interpolating points with a small Lebesgue constant. Therefore the barycentric formula can be significantly less accurate than the modified Lagrange formula only for a poor choice of interpolating points. This analysis provides further weight to the argument of Berrut and Trefethen that barycentric Lagrange interpolation should be the polynomial interpolation method of choice.

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The numerical stability of barycentric Lagrange interpolation

Introduction.- 1. Basic Principles and Facts.- 2. Gegenbauer Polynomials.- 3. Multivariate Polynomials.- 4. Polynomials on Sphere and Ball.- 5. Approximation Methods.- 6. Approximation on the Sphere.- 7. Approximation on the Ball.- 8. Tomography.- Appendices.- Bibliography.

Multivariate Polynomial Approximation

Hyperinterpolation on the Sphere at the Minimal Projection Order

On multivariate polynomials of least deviation from zero on the unit cube

We introduce a new family of multivariate polynomials by means of a very simple generating function which reduces to the generating function of the Chebyshev polynomials of the first kind in case of one single variable. The family solves several problems of best approximation on the unit ball. Let IP~, / ~ N o , denote the space of all real polynomials in r variables (x 1 . . . . , x r ) = x of degree not exceeding/~. For convenience, define IP" 1:= [01 and x m : ~ XTI mr : ...x~ , Ira1 = m l + ' " + m , . ,

Manfred Reimer

Papers

Multivariate Polynomial Approximation

Hyperinterpolation on the Sphere at the Minimal Projection Order

On multivariate polynomials of least deviation from zero on the unit cube

On Multivariate Polynomials of Least Deviation from Zero on the Unit Ball

Constructive theory of multivariate functions : with an application to tomography