Constructing cubature formulae by the method of reproducing kernel

Summary. We examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently. Several new cubature formulae are derived. Received April 15, 1998 / Revised version received November 24, 1998 / Published online January 27, 2000     

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous function g ₁∈C ₀ [0,1]² with support in the rectangle [0,1] × [0,?] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1] × [?,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer. Received: 21 December 1995 / Revised version: 5 October 1996     

Transfinite interpolation on the medians of a triangle and best L 1-approximation

Let be a triangle in and let be the set of its three medians. We construct interpolants to smooth functions using transfinite (or blending) interpolation on The interpolants are of type f(₁)+g(₂)+h(₃), where (₁,₂,₃) are the barycentric coordinates with respect to the vertices of . Based on an error representation formula, we prove that the interpolant is the unique best L¹-approximant by functions of this type subject the function to be approximated is from a certain convexity cone in C³().Received: 17 December 2003     

Multiplicity of holomorphic functions

The main result of this paper concerns an analytic version of Birkhoff's ergodic theorem. It allows us to define asymptotic multiplicities given any algebraically stable rational map . This is the key for our study of the singularities of the Green current T associated to f. We characterize the points where the Lelong number of T is strictly positive. Received: 30 July 1999     

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     

High dimensional integration of smooth functions over cubes

Summary. We construct a new algorithm for the numerical integration of functions that are defined on a -dimensional cube. It is based on the Clenshaw-Curtis rule for and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function. We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands and . Received April 3, 1995 / Revised version received November 27, 1995     

On the Zero-Divergence of Equidistant Lagrange Interpolation

 In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0. Moreover, we show that the rate of divergence attains almost the maximal possible rate. (Received 2 February 2000)     

Sampling expansions associated with Kamke problems

The present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels. The situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form where N and P are two differential expressions of orders n and p respectively, and is the eigenvalue parameter. Both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular. Received February 16, 1998; in final form March 15, 1999     

Pointwise Remez- and Nikolskii-type inequalities for exponential sums

Let So is the collection of all n + 1 term exponential sums with constant first term. We prove the following two theorems. Theorem 1 (Remez-type inequality for $E_n$ at 0). Let $s \in \left( 0, \frac 12 \right]\,.$ There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that where the supremum is taken for all $f \in E_n$ satisfying Theorem 2 (Nikolskii-type inequality for $E_n$ ). There are absolute constants $c_1 > 0$ and $c_2 > 0$ such that for every $a < y < b$ and $q > 0\,.$ It is quite remarkable that, in the above Remez- and Nikolskii-type inequalities, behaves like , where denotes the collection of all algebraic polynomials of degree at most n with real coefficients. Received: 4 November 1998 / in final form: 2 March 1999     

Asymptotic error expansion of wavelet approximations of smooth functions II

Summary. We generalize earlier results concerning an asymptotic error expansion of wavelet approximations. The properties of the monowavelets, which are the building blocks for the error expansion, are studied in more detail, and connections between spline wavelets and Euler and Bernoulli polynomials are pointed out. The expansion is used to compare the error for different wavelet families. We prove that the leading terms of the expansion only depend on the multiresolution subspaces and not on how the complementary subspaces are chosen. Consequently, for a fixed set of subspaces , the leading terms do not depend on the fact whether the wavelets are orthogonal or not. We also show that Daubechies' orthogonal wavelets need, in general, one level more than spline wavelets to obtain an approximation with a prescribed accuracy. These results are illustrated with numerical examples. Received May 3, 1993 / Revised version received January 31, 1994     

A class of bivariate orthogonal polynomials and cubature formula

Summary. The existence of Gaussian cubature for a given measure depends on whether the corresponding multivariate orthogonal polynomials have enough common zeros. We examine a class of orthogonal polynomials of two variables generated from that of one variable. Received February 9, 1993 / Revised version received January 18, 1994     

Sur l'erreur d'approximation par éléments finis en utilisant la méthode des plaquettes splines

Résumé. On établit des majorations de l'erreur d'approximation par éléments finis à partir de données de Lagrange pour des fonctions appartenant à un espace de Sobolev d'ordre convenable, lorsque les degrés de liberté sont approchés à l'aide de la méthode des plaquettes splines introduite par A. Le Méhauté (cf. [13], [14], [15]). Les résultats obtenus s'appliquent notamment à la construction de surfaces de classe . Received May 29, 1995 / Revised version received August 20, 1995     

Best uniform approximation by harmonic functions on subsets of Riemannian manifolds

We investigate best uniform approximations to bounded, continuous functions by harmonic functions on precompact subsets of Riemannian manifolds. Applications to approximation on unbounded subsets ofR ² are given.Communicated by J. Milne Anderson.     

Exact order of Hoffman’s error bounds for elliptic quadratic inequalities derived from vector-valued Chebyshev approximation

In this paper, we introduce the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities. Elliptic quadratic inequalities are closely related to Chebyshev approximation of vector-valued functions (including complex-valued functions). The set of Chebyshev approximations of a vector-valued function defined on a finite set is shown to be Hausdorff strongly unique of order exactly 2 ^s for some nonnegative integer s. As a consequence, the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 ^-s for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated by the elliptic quadratic inequalities. Received: April 15, 1999 / Accepted: February 21, 2000?Published online July 20, 2000     

Quadrature formulae with free nodes for periodic functions

Summary. The problem of existence and uniqueness of a quadrature formula with maximal trignonometric degree of precision for 2-periodic functions with fixed number of free nodes of fixed different multiplicities at each node is considered. Our approach is based on some properties of the topological degree of a mapping with respect to an open bounded set and a given point. The explicit expression for the quadrature formulae with maximal trignometric degree of precision in the 2-periodic case of multiplicities is obtained. An error analysis for the quadrature with maximal trigonometric degree of precision is given. Received April 16, 1992/Revised version received June 21, 1993     

Extrapolation techniques for ill-conditioned linear systems

Summary. In this paper, the regularized solutions of an ill–conditioned system of linear equations are computed for several values of the regularization parameter . Then, these solutions are extrapolated at by various vector rational extrapolations techniques built for that purpose. These techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition. Numerical results illustrate the effectiveness of the procedures. Received June 23, 1997 / Revised version received October 24, 1997