Günther Nürnberger

Developments in bivariate spline interpolation

The aim of this survey is to describe developments in the field of interpolation by bivariate splines. We summarize results on the dimension and the approximation order of bivariate spline spaces, and describe interpolation methods for these spaces. Moreover, numerical examples are given.

Multivariate approximation and splines

This volume presents refereed papers covering a variety of topics in the growing field of multivariate approximation and slines.

Unicity and strong unicity in approximation theory

It is the object of this paper to discuss the following question: Is it possible to characterize unicity and strong unicity of elements of best approximation by modified Kolmogorov-criteria? Furthermore, we examine the relationship between these two properties. Let G be a nonempty set in a normed linear space E, and letfbe an element of E. Consider P(f) := P,(f) := (go E G: IIfg, I/ < IIf-g II, g E G}, i.e., the set of elements of best approximation off in G. The set-valued map P: E -+ 2G defined in this way is called the metric projection. The set G is called proximinal (respectively, semi-Chebyshev) if P(f) is nonempty (respectively, contains at most one element) for each fg E. If G is both proximinal and semi-Chebyshev, then it is called Chebyshev. For each f c E let S, : = {L E E’: I/ L 11 = 1, L(f) = Ilfil> and let Ef be the set of extreme points of S, in the a(E’, E)-topology. We say the pair (g,, , f), with g, E G and f E E\G, satisfies the Kolmogorov-criterion if, for each g E G,

Quasi-interpolation by quadratic piecewise polynomials in three variables

A quasi-interpolation method for quadratic piecewise polynomials in three variables is described which can be used for the efficient reconstruction and visualization of gridded volume data. The Bernstein-Bezier coefficients of the splines are immediately available from the given data values by applying a local averaging, where no prescribed derivatives are required. Since the approach does not make use of a particular basis or a subset spanning the spline spaces, we analyze the smoothness properties of the trivariate splines. We prove that the splines yield nearly optimal approximation order while simultaneously its piecewise derivatives provide optimal approximation of the derivatives for smooth functions. The constants of the corresponding error bounds are given explicitly. Numerical tests confirm the results and the efficiency of the method.

Lagrange Interpolation by C 1 Cubic Splines on Triangulated Quadrangulations.

We describe local Lagrange interpolation methods based on C1 cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.

Lagrange Interpolation by Bivariate C1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C1-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.

Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nurnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines Sqr(Δ1) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q ⩾ 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.

A Remez type algorithm for spline functions

SummaryA Remez type algorithm for computing best spline approximations of degreen withk fixed knots is developed. It is shown that the sequence constructed in the algorithm converges to a best approximation, ifk≦n+1, and at least to a nearly best approximation, ifk>n+1.In contrast to the standard interpolation methods no restriction to the position of the knots is required. This fact may be important to treat approximation problems for splines with free knots.

A local version of Haar's theorem in approximation theory

We give a complete characterization of those functions in Co(T), where T is a locally compact subset of the real line, which have a strongly unique best approximation from an n-dimensional weak Chebyshev subspace of Co(T) and apply this result to spline functions of degree n with k fixed knots. Furthermore we prove similar results for unique best approximations.

Uniqueness and strong uniqueness of best approximations by spline subspaces and other subspaces

Abstract A complete characterization is given of those functions in C¦a, b¦ which have a unique best approximation from the subspace of spline functions of degree n with k fixed knots. Also, the relationship between unique and strongly unique best approximations from arbitrary finitedimensional subspaces of C0(T) is investigated.