Error estimates in Sobolev spaces for interpolating thin plate splines under tension

This paper discusses _Lp$L_p$-error estimates for interpolation by thin plate spline under tension of a function in the classical Sobolev space on an open bounded set with a Lipschitz-continuous boundary. A property of convergence is also given when the set of interpolating points becomes more and more dense.     

Thinning algorithms for scattered data interpolation

Multistep interpolation of scattered data by compactly supported radial basis functions requires hierarchical subsets of the data. This paper analyzes thinning algorithms for generating evenly distributed subsets of scattered data in a given domain in ℝ ^d .     

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.     

A nodal spline interpolant for the Gregory rule of even order

Summary The Gregory rule is a well-known example in numerical quadrature of a trapezoidal rule with endpoint corrections of a given order. In the literature, the methods of constructing the Gregory rule have, in contrast to Newton-Cotes quadrature,not been based on the integration of an interpolant. In this paper, after first characterizing an even-order Gregory interpolant by means of a generalized Lagrange interpolation operator, we proceed to explicitly construct such an interpolant by employing results from nodal spline interpolation, as established in recent work by the author and C.H. Rohwer. Nonoptimal order error estimates for the Gregory rule of even order are then easily obtained.     

Error estimates in W 2,∞-semi-norms for discrete interpolating D 2-splines

We derive error estimates in W^2,∞-semi-norms for multivariate discrete D²-splines that interpolate an unknown function at the vertices of given triangulations. These results are widely based on the construction of approximation operators and linear projectors onto piecewise polynomial spaces having weakly stable local bases.     

Computational aspects of multivariate polynomial interpolation: Indexing the coefficients

An algorithm is derived for generating the information needed to pass efficiently between multi-indices of neighboring degrees, of use in the construction and evaluation of interpolating polynomials and in the construction of good bases for polynomial ideals. This revised version was published online in June 2006 with corrections to the Cover Date.     

Identities for piecewise polynomial spaces determined by connection matrices

Summary TwoB-spline results — Marsden's identity and the de Boor-Fix dual functionals — are extended to geometrically continuous curves determined by connection matrices.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Radial basis functions under tension

We discuss multivariate interpolation with some radial basis function, called radial basis function under tension (RBFT). The RBFT depends on a positive parameter which provides a convenient way of controlling the behavior of the interpolating surface. We show that our RBFT is conditionally positive definite of order at least one and give a construction of the native space, namely a semi-Hilbert space with a semi-norm, minimized by such an interpolant. Error estimates are given in terms of this semi-norm and numerical examples illustrate the behavior of interpolating surfaces.     

Construction of spherical spline quasi-interpolants based on blossoming

A general theory of quasi-interpolants based on quadratic spherical Powell-Sabin splines on spherical triangulations of a sphere-like surface S is developed by using polar forms. As application, various families of discrete and differential quasi-interpolants reproducing quadratic spherical Bézier-Bernstein polynomials or the whole space of the spherical Powell-Sabin quadratic splines of class C¹ are presented.     

A parallel algorithm for discrete least squares rational approximation

Summary A new method for discrete least squares linearized rational approximation is presented. It generalizes the algorithm of Rutishauser-Gragg-Harrod-Reichel for discrete least squares polynomial approximation to the rational case. The algorithm is fast in the sense that it requires orderm computation time wherem is the number of data points and is the degree of the approximant. We describe how this algorithm can be implemented in parallel.     

Lagrange interpolation for continuous piecewise smooth functions

This note is devoted to Lagrange interpolation for continuous piecewise smooth functions. A new family of interpolatory functions with explicit approximation error bounds is obtained. We apply the theory to the classical Lagrange interpolation.     

Error bounds for a convexity-preserving interpolation and its limit function

Error bounds between a nonlinear interpolation and the limit function of its associated subdivision scheme are estimated. The bounds can be evaluated without recursive subdivision. We show that this interpolation is convexity preserving, as its associated subdivision scheme. Finally, some numerical experiments are presented.     

Frames for vector spaces and affine spaces

A finite frame for a finite dimensional Hilbert space is simply a spanning sequence. We show that the linear functionals given by the dual frame vectors do not depend on the inner product, and thus it is possible to extend the frame expansion (and other elements of frame theory) to any finite spanning sequence for a vector space. The corresponding coordinate functionals generalise the dual basis (the case when the vectors are linearly independent), and are characterised by the fact that the associated Gramian matrix is an orthogonal projection. Existing generalisations of the frame expansion to Banach spaces involve an analogue of the frame bounds and frame operator.The potential applications of our results are considerable. Whenever there is a natural spanning set for a vector space, computations can be done directly with it, in an efficient and stable way. We illustrate this with a diverse range of examples, including multivariate spline spaces, generalised barycentric coordinates, and vector spaces over the rationals, such as the cyclotomic fields.     

Two interpolation operators on irregularly distributed data in inner product spaces

Two interpolation operators in inner product spaces for irregularly distributed data are compared. The first is a well-known polynomial operator, which in a certain sense generalizes the classical Lagrange interpolation polynomial. The second can be obtained by modifying the first so as to get a partition-of-unity interpolant. Numerical tests and considerations on errors show that the two operators have very different approximation performances, and that by suitable modifications both can provide acceptable results, working in particular from R^m to Rⁿ and from C[−π,π] to R.     

A generalized Taylor factorization for Hermite subdivision schemes

In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so-called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For these schemes (and others) we provide the concept of a generalized Taylor factorization and show how it can be used to obtain convergence criteria for the Hermite scheme by means of factorization and contractivity.     

A new formal approach to the rational interpolation problem

Summary An elegant and fast recursive algorithm is developed to solve the rational interpolation problem in a complementary way compared to existing methods. We allow confluent interpolation points, poles, and infinity as one of the interpolation points. Not only one specific solution is given but a nice parametrization of all solutions. We also give a linear algebra interpretation of the problem showing that our algorithm can also be used to handle a specific class of structured matrices.     

Pseudo-polyharmonic vectorial approximation for div-curl and elastic semi-norms

Vector field reconstruction is a problem arising in many scientific applications. In this paper, we study a div-curl approximation of vector fields by pseudo-polyharmonic splines. This leads to the variational smoothing and interpolating spline problems with minimization of an energy involving the curl and the divergence of the vector field. The relationship between the div-curl energy and elastic energy is established. Some examples are given to illustrate the effectiveness of our approach for a vector field reconstruction.     

On orders of approximation of plane curves by parametric cubic splines

The aim of the present paper is to show that the convergence rate of the parametric cubic spline approximation of a plane curve is of order four instead of order three. For the first and second derivatives, the rates are of order three and two, respectively. Finally some numerical examples are given to illustrate the predicted error behaviour.