Manfred Reimer

Finite Difference Forms Containing Derivatives of Higher Order

Stability and error bounds in numerical integration of ordinary differential equations, determining highest possible degree of stable finite difference form

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

Abstract In the family of all r -variable real polynomials with total degree not exceeding μ and with maximum norm on the unit-cube not exceeding 1, any of the leading coefficients is maximum for a special product of one-variable Chebyshev polynomials of the first kind. This is a consequence of an even more general result on polynomials of least deviation from zero on the unit cube.

Extremal spline bases

Abstract In the space of all polynomial splines on an infinite equidistant grid with fixed odd degree and with fixed period, the Lagrangians belonging to the grid-points as nodes form an extremal basis with respect to the supremum-norm. This means that no element of the basis can be approximated within the span of the other elements better than by zero. The result is carried over to the nonperiodic cardinal-spline case. Moreover, an intrinsic insight into the behaviour of the Lagrangian splines is obtained.

A Remez-type algorithm for the calculation of extremal fundamental systems for polynomial spaces on the sphere

Extremal fundamental systems are nodal systems for which the Lagrangians have minimal norm which then is one. Hence they consist of points not distinguished one against another by definition, they are “equidistributed”, in some sense. The calculation of extremal fundamental systems can be performed by a Remez-type algorithm in combination with Newtons method, the method is applied to the most interesting polynomial spaces.ZusammenfassungExtremale Fundamentalsysteme sind Knotensysteme für welche die Lagrange-Elemente minimale Norm annehmen, die dann Eins ist. Der Definition nach sind ihre Punkte also nicht unterschieden, diese sind also in einem gewissen Sinn „gleichverteilt”. Der Berechnung extremaler Fundamentalsysteme dient ein Algorithmus vom Remez-Typ in Verbindung mit dem Newton-Verfahren, die Methode wird auf die wichtigsten Polynomräume angewendet.

Interpolation on the sphere and bounds for the Lagrangian square sums

SummaryFor spaces of polynomial functions on the sphere which are invariant against rotation, the square-sums of the Lagrangians can be estimated by means of the smallest eigenvalue of a positive definite system matrix defined by the reproducing kernel and the nodes used. As a consequence, bounds for the corresponding Lebesgueconstants are obtained. There are examples where the method leads to an estimation of the square-sum by one, which cannot be improved. In this case the Lagrangians perform an extremal basis.

The main roots of the Euler-Frobenius polynomials

Abstract We investigate the behavior of the largest root ⩽ −1 of an Euler-Frobenius polynomial. This root determines the convergence/divergence of a cardinal Lagrange spline series. Asymptotic representations are obtained in the most important cases.

Bounds for the Horner sums.

Estimation of absolute values of Horner sum, using Chebyshev polynomials as maximizing polynomials

The numerical stability of evaluation schemes for polynomials based on the lagrange interpolation form

For evaluation schemes based on the Lagrangian form of a polynomial with degreen, a rigorous error analysis is performed, taking into account that data, computation and even the nodes of interpolation might be perturbed by round-off. The error norm of the scheme is betweenn2 andn2+(3n+7)λn, where λn denotes the Lebesgue constant belonging to the nodes. Hence, the error norm is of least possible orderO(n2) if, for instance, the nodes are chosen to be the Chebyshev points or the Fekete points.

The radius of convergence of a cardinal Lagrange spline series of odd degree

Abstract We give a sharp criterion for the convergence of a Lagrangian cardinal spline series for the integer grid in terms of a “radius of convergence.”

An elementary algebraic representation of polynomial spline interpolants for equidistant lattices and its condition

SummaryA customary representation formula for periodic spline interpolants contains redundance which, however, can be eliminated by a transcendent method. We use an elementary indentity for the generalized Euler-Frobenius-polynomials, which seems to be unknown until now, in order to derive the theory by purely algebraic arguments. The general cardinal spline interpolation theory can be obtained from the periodic case by a simple approach to the limit. Our representation has minimum condition for odd/even degree if the interpolation points are the lattice (mid-)points. We evaluate the corresponding condition numbers and give an asymptotic representation for them.