Semilocal Smoothing Splines

The paper is aimed at periodic and nonperiodic semilocal smoothing splines, or S-splines of class C p, formed by polynomials of degree n. The first p?+?1 coefficients of each polynomial are determined by the values of the preceding polynomial and its first p derivatives at the glue-points, while the remaining n???p coefficients of the higher derivatives of the polynomial are found by the method of least squares. These conditions are supplemented with the initial conditions (nonperiodic case) or the periodicity condition on the spline-function on the segment where it is defined. A linear system of equations is obtained for the coefficients of the polynomials constituting the spline. Its matrix has a block structure. Existence and uniqueness theorems are proved and it is shown that that the convergence of the splines to the original function depends on the eigenvalues of the stability matrix. Examples of stable S-splines are given.     

Interpolation Scheme for Planar Cubic <Emphasis Type="Italic">G</Emphasis><Superscript>2</Superscript> Spline Curves

In this paper a method for interpolating planar data points by cubic G ² splines is presented. A spline is composed of polynomial segments that interpolate two data points, tangent directions and curvatures at these points. Necessary and sufficient, purely geometric conditions for the existence of such a polynomial interpolant are derived. The obtained results are extended to the case when the derivative directions and curvatures are not prescribed as data, but are obtained by some local approximation or implied by shape requirements. As a result, the G ² spline is constructed entirely locally.     

On spline Galerkin methods for singular integral equations with piecewise continuous coefficients

Summary This article analizes the convergence of the Galerkin method with polynomial splines on arbitrary meshes for systems of singular integral equations with piecewise continuous coefficients inL ² on closed or open Ljapunov curves. It is proved that this method converges if and, for scalar equations and equidistant partitions, only if the integral operator is strongly elliptic (in some generalized sense). Using the complete asymptotics of the solution, we provide error estimates for equidistant and for special nonuni-form partitions.     

A Family of Spline Quasi-Interpolants on the Sphere

In this paper, we construct a local quasi-interpolant Q for fitting a function f defined on the sphere S. We first map the surface S onto a rectangular domain and next, by using the tensor product of polynomial splines and 2-periodic trigonometric splines, we give the expression of Qf. The use of trigonometric splines is necessary to enforce some boundary conditions which are useful to ensure the C ² continuity of the associated surface. Finally, we prove that Q realizes an accuracy of optimal order.     

Some Properties of Minimal Splines

S. G. Mikhlin was the first to construct systematically coordinate functions on an equidistant grid solving a system of approximate equations (called “fundamental relations”, see [5]; Goel discussed some special cases earlier in 1969; see also [1, 4, 6]). Further, the idea was developed in the case of irregular grids (which may have finite accumulation points, see [1] ). This paper is devoted to the investigation of A-minimal splines, introduced by the author; they include polynomial minimal splines which have been discussed earlier. Using the idea mentioned above, we give necessary and sufficient conditions for existence, uniqueness and g-continuity of these splines. The application of these results to polynomial splines of m-th degree on an equidistant grid leads us, in particular, to necessary and sufficient conditions for the continuity of their i-th derivative (i = 1, ?, m). These conditions do not exclude discontinuities of other derivatives (e.g. of order less than i). This allows us to give a certain classification of minimal spline spaces. It turns out that the spline classes are in one-to-one-correspondence with certain planes contained in a hyperplane.     

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.     

Galerkin methods with splines for singular integral equations over (0, 1)

Summary In this paper a convergence analysis of Galerkin methods with splines for strongly elliptic singular integral equations over the interval (0, 1) is given. As trial functions we utilize smoothest polynomial splines on arbitrary meshes and continuous splines on special nonuniform partitions, multiplied by a weight function. Using inequalities of Gårding type for singular integral operators in weightedL ² spaces and the complete asymptotics of solutions at the endpoints, we provide error estimates in certain Sobolev norms.     

Higher integrality conditions, volumes and Ehrhart polynomials

A polytope is integral if all of its vertices are lattice points. The constant term of the Ehrhart polynomial of an integral polytope is known to be 1. In previous work, we showed that the coefficients of the Ehrhart polynomial of a lattice-face polytope are volumes of projections of the polytope. We generalize both results by introducing a notion of k-integral polytopes, where 0-integral is equivalent to integral. We show that the Ehrhart polynomial of a k-integral polytope P has the properties that the coefficients in degrees less than or equal to k are determined by a projection of P, and the coefficients in higher degrees are determined by slices of P. A key step of the proof is that under certain generality conditions, the volume of a polytope is equal to the sum of volumes of slices of the polytope.     

Lösung von Differentialgleichungen mit Splinefunktionen. Eine Störungstheorie

Summary In this paper non-linear splines (depending onn+1 parameters) are used to patch up the solution of an initial value problem in intervals of stepsizeh. The elements of the solution are fixed byq smoothness conditions andd conditions derived from the differential equation in an appropriate setup. The feasibility of the method can be connected to that of the polynomial spline method by a perturbation type argument. Thus the question of convergence forh0 is closely connected to the linear (polynomial) case.A new elementary prove is given for divergence of the polynomial splines ifq is larger thand+1, as was done by Mülthei [4] with other techniques.A byproduct is an extention of the famous result for polynomial interpolation by Runge on equidistant grids that interpolation of a given function by splines of too high smoothness can cause divergence forh0.

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereiches 72 entstanden     

Twice continuously differentiable S-splines

Twice continuously differentiable S-splines consisting of fifth degree polynomials are constructed, uniqueness and existence theorems are proved, stability conditions are established for such splines. The first three coefficients of each polynomial are determined by conditions of smooth gluing, the others are determined by the least squares method. This provides the ability to smooth initial data. The peculiarity of these splines is their semilocal property, i.e., each polynomial implicitly depends on function values determining previous polynomials and does not depend on values determining subsequent polynomials. It turns out that in this case the stability conditions are fulfilled under some very strong restrictions. Under there conditions and other ones ensuring sufficient closeness of the first polynomial and its derivatives to values of the function and its derivatives it is proved that this closeness is retained on the whole given interval.     

The Problem of Topologies of Grothendieck and the Class of Fréchet T-Spaces

We examine the interpolation with periodic polynomial splines of degree d and defect r (d ≦ r) on equidistant partitions of the real axis and generalize known results for r = 0. We prove necessary and sufficient conditions for the existence and a certain L²-stability of the interpolants as well as their approximation properties in the scale of the periodic SOBOLEV spaces.     

Relations between the support of a compactly supported function and the exponential-polynomials spanned by its integer translates

The interrelation between the shape of the support of a compactly supported function and the space of all exponential-polynomials spanned by its integer translates is examined. The results obtained are in terms of the behavior of these exponential-polynomials on certain finite subsets ofZ ^s , which are determined by the support of the given function. Several applications are discussed. Among these is the construction of quasi-interpolants of minimal support and the construction of a piecewise-polynomial whose integer translates span a polynomial space which is not scale-invariant. As to polynomial box splines, it is proved here that in many cases a polynomial box spline admits a certain optimality condition concerning the space of the total degree polynomials spanned by its integer translates: This space is maximal compared with the spaces corresponding to other functions with the same supportCommunicated by Klaus Höllig.     

Shape-Preserving C 3 Interpolation: The Curve Case

We present a new method for the construction of shape-preserving curves interpolating a given set of 3D data. The interpolating functions are obtained using quintic-like spaces of polynomial splines with variable degrees. These splines are of class C ³ and are therefore curvature and torsion continuous and possess a very simple geometric structure, which permits to easily handle the shape-constraints.     

Dimension of -splines on type-6 tetrahedral partitions

We consider a linear space of piecewise polynomials in three variables which are globally smooth, i.e. trivariate C¹-splines of arbitrary polynomial degree. The splines are defined on type-6 tetrahedral partitions, which are natural generalizations of the four-directional mesh. By using Bernstein–Bézier techniques, we analyze the structure of the spaces and establish formulae for the dimension of the smooth splines on such uniform type partitions.     

Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines

Summary A method is presented for fitting a function defined on a general smooth spherelike surfaceS, given measurements on the function at a set of scattered points lying onS. The approximating surface is constructed by mapping the surface onto a rectangle, and using a tensor-product of polynomial splines with periodic trigonometric splines. The use of trigonometric splines allows a convenient solution of the problem of assuring that the resulting surface is continuous and has continuous tangent planes at all points onS. Two alternative algorithms for computing the coefficients of the tensor fit are presented; one based on global least-squares, and the other on the use of local quasi-interpolators. The approximation order of the method is established, and the numerical performance of the two algorithms is compared.Supported in part by the National Science Foundation under Grant DMS-8902331 and by the Alexander von Humboldt Foundation     

Scattered Data Fitting by Direct Extension of Local Polynomials to Bivariate Splines

We present a new scattered data fitting method, where local approximating polynomials are directly extended to smooth (C ¹ or C ²) splines on a uniform triangulation Δ (the four-directional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of Δ. This construction allows to use discrete polynomial least squares approximations to the local portions of the data directly as parts of the approximating spline. The remaining Bernstein–Bézier coefficients are efficiently computed by extension, i.e., using the smoothness conditions. To obtain high quality local polynomial approximations even for difficult point constellations (e.g., with voids, clusters, tracks), we adaptively choose the polynomial degrees by controlling the smallest singular value of the local collocation matrices. The computational complexity of the method grows linearly with the number of data points, which facilitates its application to large data sets. Numerical tests involving standard benchmarks as well as real world scattered data sets illustrate the approximation power of the method, its efficiency and ability to produce surfaces of high visual quality, to deal with noisy data, and to be used for surface compression.     

Dimension elevation for Chebyshevian splines

Dimension elevation refers to the Chebyshevian version of the classical degree elevation process for polynomials or polynomial splines. In this paper, we consider the case of splines. The original spline space is based on a given Extended Chebsyhev space \mathbbE{\mathbb{E}} contained in another Extended Chebsyhev space \mathbbE^*{\mathbb{E}}^* of dimension increased by one. The original spline space, based on \mathbbE{\mathbb{E}}, is then embedded in a larger one, based on \mathbbE^*\mathbb{E}^*. Thanks to blossoms we show how to compute the new poles of any spline in the original spline space in terms of its initial poles.     

Shape-Preserving Approximation of Spatial Data

We present a new method for the construction of shape-preserving curves approximating a given set of 3D data, based on the space of quintic like polynomial splines with variable degrees recently introduced in [7]. These splines – which are C ³ and therefore curvature and torsion continuous – possess a very simple geometric structure, which permits to easily handle the shape-constraints.     

On the variational characterization and convergence of bivariate splines

It is shown that bivariate interpolatory splines defined on a rectangleR can be characterized as being unique solutions to certain variational problems. This variational property is used to prove the uniform convergence of bivariate polynomial splines interpolating moderately smooth functions at data which includes interpolation to values on a rectangular grid. These results are then extended to bivariate splines defined on anL-shaped region.This research was supported by a University of Kansas General Research Grant.     

Spline fitting discontinuous functions given just a few Fourier coefficients

This paper considers the use of polynomial splines to approximate periodic functions with jump discontinuities of themselves and their derivatives when the information consists only of the first few Fourier coefficients and the location of the discontinuities. Spaces of splines are introduced which include, members with discontinuities at those locations. The main results deal with the orthogonal projection of such a spline space on spaces of trigonometric polynomials corresponding to the known coefficients. An approximation is defined based on inverting this projection. It is shown that when discontinuities are sufficiently far apart, the projection is invertible, its inverse has norm close to 1, and the approximation is nearly as good as directL ₂ approximation by members of the spline space. An example is given which illustrates the results and which is extended to indicate how the approximation technique may be used to provide smoothing which which accurately represents discontinuities.