Kurt Jetter

Error estimates for scattered data interpolation on spheres

We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

Approximation with polynomial kernels and SVM classifiers

Abstract This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted L1 space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels.

Cardinal Interpolation by Multivariate Splines

The purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials. In particular, the notion of a commutator is introduced and its connection with the Marsden identity is observed. The order of a commutator is shown to be equivalent to the Strang and Fix conditions that arise in the study of the local approxima- tion orders using quasi-interpolants. We also prove that scaled cardinal interpolants give these local approximation orders. Introduction. Cardinal interpolation by bivariate box splines was first studied by de Boor, Hollig, and Riemenschneider (3). The purpose of our paper is to investigate the cardinal interpolation problem from a different point of view. In particular, the notion of a commutator is introduced. It will be shown that this notion generalizes the Marsden identity for univariate splines to the multivariate setting. The order of a commutator will be shown to be equivalent to the Strang and Fix conditions used in the study of the order of controlled approximation by Dahmen and Micchelli (8) or local approximation by de Boor and Jia (4). An application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper. 1. Preliminaries. This section consists of preliminary material for multivariate cardinal interpolation. Our approach is motivated by the work in (5) and (13) where cardinal interpolation in l2 was connected with certain convolution operators Lr Let Z denote the set of integers and Z+, the nonnegative ones. For any given complex sequence = ( by

Scalar multivariate subdivision schemes and box splines

We study scalar d-variate subdivision schemes, with dilation matrix 2I, satisfying the sum rules of order k. Using the results of Moller and Sauer, stated for general expanding dilation matrices, we characterize the structure of the mask symbols of such schemes by showing that they must be linear combinations of shifted box spline generators of some polynomial ideal. The directions of the corresponding box splines are columns of certain unimodular matrices. The ideal is determined by the given order of the sum rules or, equivalently, by the order of the zero conditions. The results presented in this paper open a way to a systematic study of subdivision schemes, since box spline subdivisions turn out to be the building blocks of any reasonable multivariate subdivision scheme. As in the univariate case, the characterization we give is the proper way of matching the smoothness of the box spline building blocks with the order of polynomial reproduction of the corresponding subdivision scheme. However, due to the interaction of the building blocks, convergence and smoothness properties may change, if several convergent schemes are combined. The results are illustrated with several examples.

An identity for multivariate Bernstein polynomials

We prove an identity for multivariate Bernstein polynomials on simplices, which may be considered a pointwise orthogonality relation. Its integrated version provides a new representation for the polynomial dual basis of Bernstein polynomials. An identity for the reproducing kernel is used to define quasi-interpolants of arbitrary order.

A new subdivision method for bivariate splines on the four-directional mesh

We present a new bivariate subdivision scheme based on two generators of a four-directional spline space. In particular, we are dealing with piecewise linear, continuous splines, and with piecewise cubic, continuously differentiable splines. The subdivision schemes rely on a matrix mask of small support. In addition, we present some results on related quasi-interpolating operators, and on approximation order of the underlying shift-invariant spaces.

Algorithms for cardinal interpolation using box splines and radial basis functions

SummaryWe describe an algorithm for (bivariate) cardinal interpolation which can be applied to translates of “basis functions” which include box splines or radial basis functions. The algorithm is based on a representation of the Fourier transform of the fundamental interpolant, hence Fast Fourier Transform methods are available. In numerical tests the 4-directional box spline (transformed to the characteristical submodule of ℤ2), the thin plate spline, and the multiquadric case give comparably equal and good results.

Multivariate Bernstein-Durrmeyer operators with arbitrary weight functions

In this paper we introduce a class of Bernstein-Durrmeyer operators with respect to an arbitrary measure @r on the d-dimensional simplex, and a class of more general polynomial integral operators with a kernel function involving the Bernstein basis polynomials. These operators generalize the well-known Bernstein-Durrmeyer operators with respect to Jacobi weights. We investigate properties of the new operators. In particular, we study the associated reproducing kernel Hilbert space and show that the Bernstein basis functions are orthogonal in the corresponding inner product. We discuss spectral properties of the operators. We make first steps in understanding convergence of the operators.

New polynomial preserving operators on simplices: direct results

A new class of differential operators on the simplex is introduced, which define weighted Sobolev norms and whose eigenfunctions are orthogonal polynomials with respect to Jacobi weights. These operators appear naturally in the study of quasi-interpolants which are intermediate between Bernstein-Durrmeyer operators and orthogonal projections on polynomial subspaces. The quasi-interpolants satisfy a Voronovskaja-type identity and a Jackson-Favard-type error estimate. These and further properties follow from a spectral analysis of the differential operators. The results are based on a pointwise orthogonality relation of Bernstein polynomials that was recently discovered by the authors.

Polynomial Reproduction in Subdivision

We study conditions on the matrix mask of a vector subdivision scheme ensuring that certain polynomial input vectors yield polynomial output again. The conditions are in terms of a recurrence formula for the vectors which determine the structure of polynomial input with this property. From this recurrence, we obtain an algorithm to determine polynomial input of maximal degree. The algorithm can be used in the design of masks to achieve a high order of polynomial reproduction.