Shape-preserving interpolation by cubic splines

We consider the problem of shape-preserving interpolation by cubic splines. We propose a unified approach to the derivation of sufficient conditions for the k-monotonicity of splines (the preservation of the sign of any derivative) in interpolation of k-monotone data for k = 0, …, 4.     

Birkhoff interpolation by Chebyshevian splines

This paper deals with the interpolation of the function and its derivative values at scatted points, so-called Birkhoff Interpolation, by piecewise Chebyshevian spline. Research supported in part by NSERC Canada under Grant ≠A7687. This research formed part of a Thesis written for the Degree of Master of Science at the University of Alberta undr the supervision of Professor S.D. Riemenschneider.     

Lagrange and hermite interpolation by bivariate splines

We develop methods for constructing sets of points which admit Lagrange and Hermite type interpolation by spaces of bivariate splines on rectangular and triangular partitions which are uniform, in general. These sets are generated by building up a net of lines and by placing points on these lines which satisfy interlacing properties for univariate spline spaces.     

EfficientL 2 approximation by splines

LetS _N ^k (t) be the linear space ofk-th order splines on [0, 1] having the simple knotst _i determined from a fixed functiont by the rulet _i=t(i/N). In this paper we introduce sequences of operators {Q _N } _N =1 fromC ^k [0, 1] toS _N ^k (t) which are computationally simple and which, asN, give essentially the best possible approximations tof and its firstk–1 derivatives, in the norm ofL ₂[0, 1]. Precisely, we show thatN ^k–1((f–Q _N f) ⁱ –dist₂(f ⁽¹⁾,S _N ^k–1 (t)))0 fori=0, 1, ...,k–1. Several numerical examples are given.The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816     

Multivariate approximation by PDE splines

This work deals with an approximation method for multivariate functions from data constituted by a given data point set and a partial differential equation (PDE). The solution of our problem is called a PDE spline. We establish a variational characterization of the PDE spline and a convergence result of it to the function which the data are obtained. We estimate the order of the approximation error and finally, we present an example to illustrate the fitting method.     

Approximation and interpolation by convexity-preserving rational splines

A discussion and algorithm for combined interpolation and approximation by convexity-preserving rational splines is given.     

Monotone and convex interpolation by weighted cubic splines

Algorithms for interpolating by weighted cubic splines are constructed with the aim of preserving the monotonicity and convexity of the original discrete data. The analysis performed in this paper makes it possible to develop two algorithms with the automatic choice of the shape-controlling parameters (weights). One of them preserves the monotonicity of the data, while the other preserves the convexity. Certain numerical results are presented.     

Interpolation and approximation by monotone cubic splines

We study the reconstruction of a function defined on the real line from given, possibly noisy, data values and given shape constraints. Based on two abstract minimization problems characterization results are given for interpolation and approximation (in the euclidean norm) under monotonicity constraints. We derive from these results Newton-type algorithms for the computation of the monotone spline approximant.     

Monotone and convex interpolation by weighted quadratic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted C ¹ quadratic splines in such a way that the monotonicity and convexity of the data are preserved. The analysis culminates in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Weighted C ¹ quadratic B-splines and control point approximation are also considered.     

Shape-preserving interpolation by splines using vector subdivision

We give a local convexity preserving interpolation scheme using parametricC ² cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives. This is also adapted to give a scheme which is both local convexity and local monotonicity preserving when the data values are strictly increasing in thex-direction.     

A remark on interpolation by bivariate splines

We study the determining set for bivariate spline spacesS _k ^o on type-1 triangulation of square using B-net techniques. We further construct the interpolation schemes for these spline spaces that are unisolvent for any function f of C^σ.     

Constructing smooth interpolation splines

The authors present formulas for constructing continuous splines and smooth splines which are exact on any power of a given function.     

Lacunary interpolation by quartic splines on uniform meshes

The interpolation of a discrete set of data on the interval [0, 1], representing the first and the second derivatives (except at 0) of a smooth function f is investigated via quartic C²-splines. Error bounds in the uniform norm for ∥s⁽ⁱ⁾ − f⁽ⁱ⁾∥, i=0(1)2, if f ∈ C^l[0, 1], l=3, 5 and ⁽³⁾ ∈ BV[0, 1], together with computational examples will also be presented.     

Constrained approximation by splines with free knots

In this paper, a method that combines shape preservation and least squares approximation by splines with free knots is developed. Besides the coefficients of the spline a subset of the knot sequence, the so-calledfree knots, is included in the optimization process resulting in a nonlinear least squares problem in both the coefficients and the knots. The original problem, a special case of aconstrained semi-linear least squares problem, is reduced to a problem that has only the knots of the spline as variables. The reduced problem is solved by a generalized Gauss-Newton method. Special emphasise is given to the efficient computation of the residual function and its Jacobian. Dedicated to our colleague and teacher Prof. Dr. J. W. Schmidt on the occasion of his 65th birthday Research of the first author was supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1,2-2.     

Bivariate segment approximation and splines

The problem to determine partitions of a given rectangle which are optimal for segment approximation (e.g., by bivariate piecewise polynomials) is investigated. We give criteria for optimal partitions and develop algorithms for computing optimal partitions of certain types. It is shown that there is a surprising relationship between various types of optimal partitions. In this way, we obtain good partitions for interpolation by tensor product spline spaces. Our numerical examples show that the methods work efficiently.     

Uniform approximation by some Hermite interpolating splines

In this paper some upper bound for the error ∥ s-f ∥_∞ is given, where f ε C¹[a,b], but s is a so-called Hermite spline interpolant (HSI) of degree 2q ?1 such that f(x_i) = s(x_i), f′(rmx_i) = s′(x_i), s^(j) (x_i) = 0 (i = 0, 1, …, n; j = 2, 3, …, q ?1; n > 0, q > 0) and the knots x_i are such that a = x₀ < x₁ < … < x_n = b. Necessary and sufficient conditions for the existence of convex HSI are given and upper error bound for approximation of the function fε C¹[a, b] by convex HSI is also given.