Hermite subdivision with shape constraints on a rectangular mesh

In 1999, Dubuc and Merrien introduced a Hermite subdivision scheme which gives C ¹-interpolants on a rectangular mesh. In this paper a two parameter version of this scheme is analyzed, and C ¹-convergence is proved for a range of the two parameters. By introducing a control grid the parameters in the scheme can be chosen so that the interpolant inherits positivity and/or directional monotonicity from the initial data. Several examples are given showing that a desired shape can be achieved even if only very crude estimates for the initial slopes are used. AMS subject classification (2000) 65D05, 65D17     

The Sensitivity of a Spline Function to Perturbations of the Knots

In this paper we study the sensitivity of a spline function, represented in terms of B-splines, to perturbations of the knots. We do this by bounding the difference between a primary spline and a secondary spline with the same B-spline coefficients, but different knots. We give a number of bounds for this difference, both local bounds and global bounds in general L ^p-spaces. All the bounds are based on a simple identity for divided differences.     

Preconditioned Iterative Methods for Scattered Data Interpolation

The scattered data interpolation problem in two space dimensions is formulated as a partial differential equation with interpolating side conditions. The system is discretized by the Morley finite element space. The focus of this paper is to study preconditioned iterative methods for the corresponding discrete systems. We introduce block diagonal preconditioners, where a multigrid operator is used for the differential equation part of the system, while we propose an operator constructed from thin plate radial basis functions for the equations corresponding to the interpolation conditions. The effect of the preconditioners are documented by numerical experiments.     

Chebyshevian multistep methods for ordinary differential equations

Summary In this paper some theory of linear multistep methods fory ^(r) (x)=f(x,y) is extended to include smooth, stepsize-dependent coefficients. Treated in particular is the case where exact integration of a given set of functions is desired.Work on this paper was supported in part by U.S. Army Research Office (Durham) Grant DA-ARO(D)-31-124-G1050 and National Science Foundation Grant GP-23655 with The University of Texas at Austin.     

Total positivity and the existence of piecewise exponential B-splines

We give necessary and sufficient conditions for the total positivity of certain connection matrices arising in piecewise exponential spline spaces. These total positivity conditions are sufficient for existence of B-splines in such spaces, but they are far from being necessary. We give a necessary and sufficient condition for existence of B-splines in the case of piecewise exponential spline spaces with only two differential operators, which eventually leads to a necessary condition for any piecewise exponential spline spaces. Dedicated to Professor Charles A. Micchelli for his 60th birthday Mathematics subject classifications (2000) 65D17, 65D07, 41A15, 41A50.     

A newton form for trigonometric Hermite interpolation

Trigonometric divided differences are used to derive a trigonometric analog of the Newton form of the Hermite interpolation polynomial.