Interactive shape preserving interpolation by curvature continuous rational cubic splines

A scheme is described for interactively modifying the shape of convexity preserving planar interpolating curves. An initial curve is obtained by patching together rational cubic and straight line segments. This scheme has, in general, geometric continuity of order 2(G² continuity) and preserves the local convexity of the data. A method for interactively modifying such curves, while maintaining their desirable properties, is discussed in detail. In particular, attention is focused upon local changes to the curve, while retaining G² continuity and shape preserving properties. This is achieved by interactive adjustment of the Bézier control points, followed by automatic adjustment of the values of weights and curvatures in a prescribed manner. A number of examples are presented.     

A shape preserving area true approximation of histogram by rational splines

For a given histogram, we consider an application of a simple rational spline to a shape preserving area true approximation of the histogram. An algorithm for determination of the spline is as easy as one with a quadratic polynomial spline, while the latter does not always preserve the shape of the histogram. Some numerical examples are given at the end of the paper.     

Non-uniform exponential tension splines

We describe explicitly each stage of a numerically stable algorithm for calculating with exponential tension B-splines with non-uniform choice of tension parameters. These splines are piecewisely in the kernel of D ²(D ²–p ²), where D stands for ordinary derivative, defined on arbitrary meshes, with a different choice of the tension parameter p on each interval. The algorithm provides values of the associated B-splines and their generalized and ordinary derivatives by performing positive linear combinations of positive quantities, described as lower-order exponential tension splines. We show that nothing else but the knot insertion algorithm and good approximation of a few elementary functions is needed to achieve machine accuracy. The underlying theory is that of splines based on Chebyshev canonical systems which are not smooth enough to be ECC-systems. First, by de Boor algorithm we construct exponential tension spline of class C ¹, and then we use quasi-Oslo type algorithms to evaluate classical non-uniform C ² tension exponential splines.      

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.     

Convexity preserving C 0 splines

Linear conditions for a C ⁰ spline to be convex are developed and used to create some convexity preserving interpolation and approximation methods.     

Construction of hyperbolic interpolation splines

The problem of constructing a hyperbolic interpolation spline can be formulated as a differential multipoint boundary value problem. Its discretization yields a linear system with a five-diagonal matrix, which may be ill-conditioned for unequally spaced data. It is shown that this system can be split into diagonally dominant tridiagonal systems, which are solved without computing hyperbolic functions and admit effective parallelization.     

Positive interpolation with rational splines

A method is presented for the construction of positive rational splines of continuity classC ².     

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.     

Chebyshev splines and shape parameters

A parametric spline curve is defined whose restriction to each sub-interval belongs to a 4-dimensional piecewise Chebyshev subspace depending on coefficients which play the role of shape parameters. This revised version was published online in August 2006 with corrections to the Cover Date.     

Extremal functional interpolation and approximation by splines

This article is the author's abstract of his dissertation for the degree Doctor of Physico-mathematical Science. The dissertation was defended on April 22, 1974 at the meeting of the Academic Council for the award of higher degrees in mathematics and mechanics at the Novosibirsk State University. Official opponents were: Academician N. N. Yanenko, Corresponding Member of the Academy of Sciences of the Ukrainian SSR, Doctor of Physicomathematical Sciences; Professor N. P. Korenchuk, Doctor of Physicomathematical Sciences; Professor G. Sh. Rubinshtein.Translated from Matematicheskie Zametki, Vol. 16, No. 5, pp. 843–854, November, 1974.     

Approximation and interpolation by convexity-preserving rational splines

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

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.     

Local exponential splines with arbitrary knots

We construct local L-splines that have an arbitrary arrangement of knots and preserve the kernel of a linear differential operator L of order r with constant coefficients and real pairwise distinct roots of the characteristic polynomial.     

Convex interval interpolation with cubic splines

A necessary and sufficient criterion is presented under which the problem of the convex interval interpolation with cubicC ¹-splines has at least one solution. The criterion is given as an algorithm which turns out to be effective.Dedicated to Professor Julius Albrecht on the occasion of his 60th birthday.     

Cardinal hermite interpolation with box splines

The study of cardinal interpolation (CIP) by the span of the lattice translates of a box spline has met with limited success. Only the case of interpolation with the box spline determined by the three directionsd ₁=(1, 0),d ₂=(0, 1), andd ₃=(1, 1) inR ² has been treated in full generality [2]. In the case ofR ^d,d ≥ 3, the directions that define the box spline must satisfy a certain determinant condition [6], [9]. If the directions occur with even multiplicities, then this condition is also sufficient. For Hermite interpolation (CHIP) both even multiplicities and the determinant condition for the directions does not prevent the linear dependence of the basis functions. This leads to singularities in the characteristic multiplier when using the standard Fourier transform method. In the case of derivatives in one direction, these singularities can be removed and a set of fundamental splines can be given. This gives the existence of a solution to CHIP inL ^p (R ^d) for data inl ^p (Z ^d), 1≤p≤2.