Identities for trigonometric B-splines with an application to curve design

In this paper we investigate some properties of trigonometric B-splines. We establish a complex integral representation for these functions, which is in certain analogy to the polynomial case, but the proof of which has to be done in a different and more complicated way. Using this integral representation, we can prove some identities concerning the evaluation of a trigonometric B-spline, its derivative and its partial derivative w.r.t. the knots. Finally we show that—in the case of equidistant knots—the trigonometric B-splines of odd order form a partition of a constant, and therefore the corresponding B-spline curve possesses the convex-hull property. This is illustrated by a numerical example.     

B-splines with geometric knot spacings

In this paper we study B-splines when the intervals between consecutive knots are in geometric progression and obtain generalizations of the particularly simple properties of the uniform B-splines, where the knots are equally spaced.     

Generalized cardinal B-splines: Stability, linear independence, and appropriate scaling matrices

Generalized cardinal B-splines are defined as convolution products of characteristic functions of self-affine lattice tiles with respect to a given integer scaling matrix. By construction, these generalized splines are refinable functions with respect to the scaling matrix and therefore they can be used to define a multiresolution analysis and to construct a wavelet basis. In this paper, we study the stability and linear independence properties of the integer translates of these generalized spline functions. Moreover, we give a characterization of the scaling matrices to which the construction of the generalized spline functions can be applied.     

Riesz bounds of Wilson bases generated byB-splines

In this paper, we are concerned with biorthogonal Wilson bases having B-splines as well as powers of sinc functions as window functions. We prove properties of B-splines and exponential Euler splines and use these properties to estimate the Riesz bounds of the Wilson bases.     

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on Ω-splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.     

On a class of weak Tchebycheff systems

In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.     

On the structure of a table of multivariate rational interpolants

In the table of multivariate rational interpolants the entries are arranged such that the row index indicates the number of numerator coefficients and the column index the number of denominator coefficients. If the homogeneous system of linear equations defining the denominator coefficients has maximal rank, then the rational interpolant can be represented as a quotient of determinants. If this system has a rank deficiency, then we identify the rational interpolant with another element from the table using less interpolation conditions for its computation and we describe the effect this dependence of interpolation conditions has on the structure of the table of multivariate rational interpolants. In the univariate case the table of solutions to the rational interpolation problem is composed of triangles of so-called minimal solutions, having minimal degree in numerator and denominator and using a minimal number of interpolation conditions to determine the solution.Communicated by Dietrich Braess.     

On computational aspects of simplicial splines

Some new results on multivariate simplex B-splines and their practical application are presented. New recurrence relations are derived based on [2] and [15]. Remarks on boundary conditions are given and an example of an application of bivariate quadratic simplex splines is presented. The application concerns the approximation of a surface which is constrained by a differential equation.Communicated by Charles Micchelli.     

Using the refinement equation for the construction of pre-wavelets III: Elliptic splines

The purpose of this paper is to provide multiresolution analysis, stationary subdivision and pre-wavelet decomposition onL ²(R ^d ) based on a general class of functions which includes polyharmonic B-splines.The work of this author has been partially supported by a DARPA grant.The work of this author has been partially supported by Fondo Nacional de Ciencia y Technologia under Grant 880/89.     

Closed surfaces defined from biquadratic splines

In order to construct closed surfaces with continuous unit normal, this paper studies certain spaces of spline functions on meshes of four-sided faces. The functions restricted to the faces are biquadratic polynomials or, in certain special cases, bicubic polynomials. A basis is constructed of positive functions with small support which sum to 1 and reduce to tensor-product biquadratic B-splines away from certain singular vertices. It is also shown that the space is suitable for interpolating data at the midpoints of the faces.Communicated by Wolfgang Dahmen.     

Almost strictly totally positive matrices

A determinantal identity, frequently used in the study of totally positive matrices, is extended, and then used to re-prove the well-known univariate knot insertion formula for B-splines. Also we introduce a class of matrices, intermediate between totally positive and strictly totally positive matrices. The determinantal identity is used to show any minor of such matrices is positive if and only if its diagonal entries are positive. Among others, this class of matrices includes B-splines collocation matrices and Hurwitz matrices.This author acknowledges a sabbatical stay at IBM T.J. Watson Research Center in 1990, which was supported by a DGICYT grant from Spain.     

Optimality and uniqueness conditions in complex rational Chebyshev approximation with examples

It is well known that best complex rational Chebyshev approximants are not always unique and that, in general, they cannot be characterized by the necessary local Kolmogorov condition or by the sufficient global Kolmogorov condition. Recently, Ruttan (1985) proposed an interesting sufficient optimality criterion in terms of positive semidefiniteness of some Hermitian matrix. Moreover, he asserted that this condition is also necessary, and thus provides a characterization of best approximants, in a fundamental case.In this paper we complement Ruttan's sufficient optimality criterion by a uniqueness condition and we present a simple procedure for computing the set of best approximants in case of nonuniqueness. Then, by exhibiting an approximation problem on the unit disk, we point out that Ruttan's characterization in the fundamental case is not generally true. Finally, we produce several examples of best approximants on a real interval and on the unit circle which, among other things, give some answers to open questions raised in the literature.     

Basis of splines associated with some singular differential operators

Functions being piecewise in Ker (D ^k DpD) are a special case of Chebyshev splines having one nontrivial weight and also a special case of singular splines. An algorithm is designed which enables calculating with related B-splines and their derivatives. Ifp(t) is approximated by a piecewise constant, an interesting recurrence for calculating with polynomial B-splines is obtained.     

Orthonormality of Cardinal Chebyshev B-Spline Bases in Weighted Sobolev Spaces

Extending a recent result of Ulrich Reif on cardinal polynomial B-splines, we show that the cardinal Chebyshev B-spline basis associated with a linear differential operator with constant real coefficients is orthonormal with respect to a unique weighted Sobolev-type inner product.     

Cost-Reduction in Hyperinterpolation

Discretized Newman–Shapiro-operators furnish a generalized hyperinterpolation method on the sphere with valuable mathematical properties. Unfortunately the price is high numerical evaluation cost, which, however, can be reduced significantly, in a first step, by a truncation method. The remaining, relevant terms, now small in number, are values of a (zonal) kernel function with arguments near the pole. Here, and with respect to the degree, the kernel function satisfies an asymptotic formula. It is based on a generalized Mehler–Heine-type formula which concerns certain ‘divided’ Gegenbauer-polynomials and Bessel-functions. This formula is proved and used in order to reduce, in a second step, the evaluation cost once more, such that the discretized Newman–Shapiro-operators become a competitive direct numerical polynomial approximation method on the sphere. For example, the graph of a degree 160 approximation to a rather complicated spherical function has been calculated with a time (cost) reduction, in total, by a factor about 10⁻⁴.     

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.     

Regularity of multiwavelets

The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets. The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary L _p spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.     

Shape-preserving approximation inL p

We prove a direct theorem for shape preservingL _p -approximation, 0p, in terms of the classical modulus of smoothnessw ₂(f, t _p ¹ ). This theorem may be regarded as an extension toL _p of the well-known pointwise estimates of the Timan type and their shape-preserving variants of R. DeVore, D. Leviatan, and X. M. Yu. It leads to a characterization of monotone and convex functions in Lipschitz classes (and more general Besov spaces) in terms of their approximation by algebraic polynomials.Communicated by Ron DeVore.     

On the convergence properties of basic series representing special monogenic functions

In this paper, it is shown that certain classes of special monogenic functions cannot be represented by the basic series in the whole space. New definitions for the order of basis of special monogenic polynomials are given together with theorems on representation of classes of special monogenic functions in certain balls and at a point. Received: 8 January 2002     

Behavior of alternation points in best rational approximation

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when n (0 < 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659.