Minimal-norm-derivative spline function in interpolation and approximation

The paper is concerned with applications of quadratic splines with minimal derivative to approximation of functions in approximation and interpolation problems. A smooth spline is constructed on a uniform mesh so as the norm of the spline derivative is minimal; the nodes of the spline and the nodes of interpolations coincide. This approach allows construction of a spline from given values of the function on the mesh without additional assignment of the value of the function derivative at the initial point, because the derivative can be determined from the minimality condition for the norm of the spline derivative in L ₂.     

Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

On quadratic spline interpolation

Using the quadratic spline interpolates(x) fitting the data (x _i,y _i), 0in and satisfying the end conditions_o=y_o, we give formulae approximatingy andy at selected knots by orders up toO(h ⁴).     

On integro quartic spline interpolation

In this paper, we use quartic B-spline to construct an approximating function to agree with the given integral values of a univariate real-valued function over the same intervals. It is called integro quartic spline interpolation. Our interpolation method is new and easy to implement. Moreover, it can work successfully even without any boundary conditions. The interpolation errors are studied. The super convergence (sixth order and fourth order, respectively) in approximating function values and second-order derivative values at the knots is proved. Numerical examples illustrate that our method is very effective and our integro-interpolating quartic spline has higher approximation ability than others.     

On minimax cardinal spline interpolation

Statistical Inference for Stochastic Processes - It is shown that in the problem of cardinal interpolation, spline interpolants of various degrees are R-minimax, with respect to corresponding...     

Quadratic spline interpolation with coinciding interpolation and spline grids

In this paper the quadratic spline interpolation with coinciding interpolation and spline grids for continuous functions is considered. The theorems mainly concern error estimations which allow to formulate a convergence statement. To get such results it is assumed that the function to be interpolated is suitably smooth or possesses a special behavior. A best approximation property and a statement about the solution of boundary value problems using quadratic spline functions are added.     

On best cubature formulas and spline interpolation

Summary A general cubature formula with an arbitrary preassigned weight function is derived using monosplines and integration by parts. The problem of determining the best cubature is formulated in terms of monosplines of least deviation and a solution to the problem is given by Theorem 3 below. This theorem may also be viewed as an optimal property of a new kind of two-dimensional spline interpolation.This work was done while the author was working at CERN, Geneva, Switzerland     

C^n-CONTINUOUS B-TYPE SPLINE CURVES WITH ITS LOCALIZATION INTERPOLATION AND APPROXIMATION

This paper presents a class of C ⁿ -continuous B-type spline curves with some parametric factors. The length of their local support is equal to 4. Taking the different values of the parametric factors, the curves can become free-type curves or interpolate a set of given points even mix the both cases. When the parametric factors satisfy the certain conditions, the degrees of the curves can be decreased as low as possible. Besides, when all the parametric factors tend to zero, the curves globally approximate to the control polygon.     

On saturation for a spline approximation

We obtain a saturation theorem for a wavelet operator which is constructed by B-spline function.     

On spline approximation of sliced inverse regression

The dimension reduction is helpful and often necessary in exploring the nonparametric regression structure.In this area,Sliced inverse regression (SIR) is a promising tool to estimate the central dimension reduction (CDR) space.To estimate the kernel matrix of the SIR,we herein suggest the spline approximation using the least squares regression.The heteroscedasticity can be incorporated well by introducing an appropriate weight function.The root-n asymptotic normality can be achieved for a wide range choice of knots.This is essentially analogous to the kernel estimation.Moreover, we also propose a modified Bayes information criterion (BIC) based on the eigenvalues of the SIR matrix.This modified BIC can be applied to any form of the SIR and other related methods.The methodology and some of the practical issues are illustrated through the horse mussel data.Empirical studies evidence the performance of our proposed spline approximation by comparison of the existing estimators.     

An improved order of approximation for thin-plate spline interpolation in the unit disc

Summary. We show that the -norm of the error in thin-plate spline interpolation in the unit disc decays like , where , under the assumptions that the function to be approximated is and that the interpolation points contain the finite grid . Received February 13, 1998 / Published online September 24, 1999     

Discrete cubic spline interpolation

Defining equations, a best approximation property, and error bounds are given for a discrete cubic spline interpolant. Furthermore the distance between two cubic spline interpolants is estimated, and numerical examples are provided.     

Hyperbolic spline interpolation algorithms

Isogeometric interpolation by hyperbolic splines is formulated as a differential multipoint boundary value problem. A discretization of this problem results in the necessity of solving a linear system with a five-diagonal matrix. This system can be ill-conditioned if the data are nonuniformly distributed. It is shown that this system can be split into tridiagonal systems with the property of diagonal dominance. The latter do not require that hyperbolic functions be evaluated. Their solution is numerically stable and can be efficiently parallelized on the basis of the superposition principle. For quasiuniform grids, these systems have positive definite matrices. Algorithms for parallelizing calculations in the case of tri- and five-diagonal systems are given.     

Tensor spline approximation

The cyclic-shift tensor-factorization interpolation method recently described by de Boor can be used in particular for least-squares fitting of multivariate data on a rectangular grid and for evaluation of the resulting tensor-product splines, taking advantage of existing linear algebra and univariate spline software. We discuss the computational details of this method, pointing out variants and suggesting techniques for dealing with ill-conditioned least-squares problems.     

Discrete cubic spline interpolation

Summary In the present paper we study the existence, uniqueness and convergence of discrete cubic spline which interpolate to a given function at one interior point of each mesh interval. Our result in particular, includes the interpolation problems concerning continuous periodic cubic splines and discrete cubic splines with boundary conditions considered respectively in Meir and Sharma (1968) and Lyche (1976) for the case of equidistant knots.