An improved order of approximation for thin-plate spline interpolation in the unit disc

Summary. We show that the -norm of the error in thin-plate spline interpolation in the unit disc decays like , where , under the assumptions that the function to be approximated is and that the interpolation points contain the finite grid . Received February 13, 1998 / Published online September 24, 1999     

Gregory type quadrature based on quadratic nodal spline interpolation

Summary. Using a method based on quadratic nodal spline interpolation, we define a quadrature rule with respect to arbitrary nodes, and which in the case of uniformly spaced nodes corresponds to the Gregory rule of order two, i.e. the Lacroix rule, which is an important example of a trapezoidal rule with endpoint corrections. The resulting weights are explicitly calculated, and Peano kernel techniques are then employed to establish error bounds in which the associated error constants are shown to grow at most linearly with respect to the mesh ratio parameter. Specializing these error estimates to the case of uniform nodes, we deduce non-optimal order error constants for the Lacroix rule, which are significantly smaller than those calculated by cruder methods in previous work, and which are shown here to compare favourably with the corresponding error constants for the Simpson rule. Received July 27, 1998/ Revised version received February 22, 1999 / Published online January 27, 2000     

Convex interval interpolation using a three-term staircase algorithm

Motivated by earlier considerations of interval interpolation problems as well as a particular application to the reconstruction of railway bridges, we deal with the problem of univariate convexity preserving interval interpolation. To allow convex interpolation, the given data intervals have to be in (strictly) convex position. This property is checked by applying an abstract three-term staircase algorithm, which is presented in this paper. Additionally, the algorithm provides strictly convex ordinates belonging to the data intervals. Therefore, the known methods in convex Lagrange interpolation can be used to obtain interval interpolants. In particular, we refer to methods based on polynomial splines defined on grids with additional knots. Received September 22, 1997 / Revised version received May 26, 1998     

An always successful method in univariate convex C^2 interpolation

Summary. We consider convex interpolation with cubic splines on grids built by adding two knots in each subinterval of neighbouring data sites. The additional knots have to be variable in order to get a chance to always retain convexity. By means of the staircase algorithm we provide computable intervals for the added knots such that all knots from these intervals allow convexity preserving spline interpolation of continuity. Received May 31, 1994 / Revised version received December 22, 1994     

The uniform convergence of thin plate spline interpolation in two dimensions

Summary. Let be a function from to that has square integrable second derivatives and let be the thin plate spline interpolant to at the points in . We seek bounds on the error when is in the convex hull of the interpolation points or when is close to at least one of the interpolation points but need not be in the convex hull. We find, for example, that, if is inside a triangle whose vertices are any three of the interpolation points, then is bounded above by a multiple of , where is the length of the longest side of the triangle and where the multiplier is independent of the interpolation points. Further, if is any bounded set in that is not a subset of a single straight line, then we prove that a sequence of thin plate spline interpolants converges to uniformly on . Specifically, we require , where is now the least upper bound on the numbers and where , , is the least Euclidean distance from to an interpolation point. Our method of analysis applies integration by parts and the Cauchy--Schwarz inequality to the scalar product between second derivatives that occurs in the variational calculation of thin plate spline interpolation. Received November 10, 1993 / Revised version received March 1994     

Exact bounds for the uniform approximation by spline interpolants

Summary. This paper completes a result of Reimer (1984) concerning -th-degree cardinal and -periodic interpolation. The method of proof is not restricted to the case of and being odd and seems to be more elementary. Received February 1, 1993 / Revised version received September 14, 1993     

Piecewise linear prewavelets on arbitrary triangulations

This paper studies locally supported piecewise linear prewavelets on bounded triangulations of arbitrary topology. It is shown that a concrete choice of prewavelets form a basis of the wavelet space when the degree of the vertices in the triangulation is not too high. Received December 29, 1997 / Revised version received April 14, 1998     

Composition based staircase algorithm and constrained interpolation with boundary conditions

Summary. Weakly coupled systems of inequalities arise frequently in the consideration of so-called direct methods for shape preserving interpolation. In this paper, a composition based staircase algorithm for bidiagonal systems subject to boundary conditions is developed. Using the compositions of the corresponding relations instead of their projections, we are able to derive a necessary and sufficient solvability criterion. Further, all solutions of the system can be constructed in a backward pass. To illustrate the general approach, we consider in detail the problem of convex interpolation by cubic splines. For this problem, an algorithm of the complexity O(n) in the number n of data points is obtained. Received August 4, 1998 / Revised version received February 5, 1999 / Published online January 27, 2000     

Polynomial interpolation of minimal degree

Summary. Minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which Lagrange interpolation with respect to the given points is uniquely solvable and degree reducing. This is a generalization of the concept of least interpolation introduced by de Boor and Ron. This paper investigates the behavior of Lagrange interpolation with respect to these spaces, giving a Newton interpolation method and a remainder formula for the error of interpolation. Moreover, a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view. Received June 9, 1995 / Revised version received June 26, 1996     

Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

Error bounds for minimal energy bivariate polynomial splines

Summary. We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms. Received March 28, 2000 / Revised version received June 23, 2000 / Published online March 8, 2002 RID="*" ID="*" Supported by the National Science Foundation under grant DMS-9870187 RID="**" ID="**" Supported by the National Science Foundation under grant DMS-9803340 and by the Army Research Office under grant DAAD-19-99-1-0160     

Asymptotic formula for the error in cardinal interpolation

Summary. We give the asymptotic formula for the error in cardinal interpolation. We generalize the Mazur Orlicz Theorem for periodic function. Received February 22, 1999 / Revised version received October 15, 1999 / Published online March 20, 2001     

Spaces of distributions, interpolation by translates of a basis function and error estimates

Summary. Interpolation with translates of a basis function is a common process in approximation theory. The most elementary form of the interpolant consists of a linear combination of all translates by interpolation points of a single basis function. Frequently, low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it is often the solution of a variational problem. In this paper we concentrate on developing a wide variety of spaces for which a variational theory is available. For each of these spaces, we show that there is a natural choice of basis function. We also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates. Received December 10, 1996 / Revised version received August 29, 1997     

On the optimal stability of bases of univariate functions

Summary. This paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation. The only bases considered are those whose elements have no sign changes. Among these, an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval. A uniqueness result and many examples of such optimally stable bases are also provided. Received May 26, 2000 / Published online August 17, 2001     

General interpolation on the lattice h{\Bbb Z}^s: Compactly supported fundamental solutions

Summary. In this paper we combine an earlier method developed with K. Jetter on general cardinal interpolation with constructions of compactly supported solutions for cardinal interpolation to gain compactly supported fundamental solutions for the general interpolation problem. The general interpolation problem admits the interpolation of the functional and derivative values under very weak restrictions on the derivatives to be interpolated. In the univariate case, some known general constructions of compactly supported fundamental solutions for cardinal interpolation are discussed together with algorithms for their construction that make use of MAPLE. Another construction based on finite decomposition and reconstruction for spline spaces is also provided. Ideas used in the latter construction are lifted to provide a general construction of compactly supported fundamental solutions for cardinal interpolation in the multivariate case. Examples are provided, several in the context of some general interpolation problem to illustrate how easy is the transition from cardinal interpolation to general interpolation. Received May 11, 1993 / Revised version received August 16, 1994     

Quadrilateral finite elements of FVS type and class C^\rho

Summary. Let be some partition of a planar polygonal domain into quadrilaterals. Given a smooth function , we construct piecewise polynomial functions of degree for odd, and for even on a subtriangulation of . The latter is obtained by drawing diagonals in each , and is a composite quadrilateral finite element generalizing the classical cubic Fraeijs de Veubeke and Sander (or FVS) quadrilateral. The function interpolates the derivatives of up to order at the vertices of . Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements. Received April 30, 1992 / Revised version received June 3, 1994     

Bivariate Hermite-Birkhoff interpolation and Vandermonde determinants

The concepts of Vandermonde determinant and confluent Vandermonde determinant are extended to the multidimensional setting by relating them to multivariate interpolation problems. With an approach different from that of other recent papers on this subject, the values of these determinants are computed, recovering and extending the results of those papers.Partially supported by Research Grant PS900121 DGICYT.     

Least supported bases and local linear independence

Summary. We introduce the concept of least supported basis, which is very useful for numerical purposes. We prove that this concept is equivalent to the local linear independence of the basis. For any given locally linearly independent basis we characterize all the bases of the space sharing the same property. Several examples for spline spaces are given. Received December 4, 1992 / Revised version received March 2, 1993     

Smooth macro-elements based on Clough-Tocher triangle splits

Summary. Macro-elements of smoothness on Clough-Tocher triangle splits are constructed for all . These new elements are improvements on elements constructed in [11] in that (disproving a conjecture made there) certain unneeded degrees of freedom have been removed. Numerical experiments on Hermite interpolation with the new elements are included. Received September 6, 2000 / Revised version received November 15, 2000 / Published online July 25, 2001     

Approximation of boundary element matrices

Summary. This article considers the problem of approximating a general asymptotically smooth function in two variables, typically arising in integral formulations of boundary value problems, by a sum of products of two functions in one variable. From these results an iterative algorithm for the low-rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed. This algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance. Received June 21, 1999 / Revised version received December 6, 1999 / Published online June 8, 2000