Convex interpolating splines of arbitrary degree. III

For given data (x _i, f_i) _i=0 ⁿ (x ₀<x ₁<...<x _n) we consider the possibility of finding a spline functions of arbitrary degreek (k3) with preassigned smoothnessl, where 1l[(k-1)/2]. The splines should be such thats(x _i)=f _i (i=0, 1,...,n) ands is convex or nondecreasing and convex on [x ₀,x _n]. An explicit formula for this function as well as the conditions that guarantee the required properties are established. An algorithm for the determination of the splines and the error bounds is also included.     

An algorithm for computing shape-preserving interpolating splines of arbitrary degree

The aim of this paper is to describe an algorithm for computing co-monotone and/or co-convex splines of degree m and deficiency m − k at the knots (0 < k < m − k), which are interpolant or osculatory to a given set of data. The method is based upon some existence properties recently developed. Graphical examples and a listing of the FORTRAN code SPISP1 are given.     

Approximation by interpolating variational splines

In this paper we present an approximation problem of parametric curves and surfaces from a Lagrange or Hermite data set. In particular, we study an interpolation problem by minimizing some functional on a Sobolev space that produces the new notion of interpolating variational spline. We carefully establish a convergence result. Some specific cases illustrate the generality of this work.     

Convex interval interpolation with cubic splines,II

The problem of convex interval interpolation with cubicC ¹-splines has an infinite number of solutions, if it is solvable at all. For selecting one of the solutions a regularized mean curvature is minimized. The arising finite dimensional constrained program is solved numerically by means of a dualization approach.Dedicated to Professor Julius Albrecht on the occasion of his 65th birthday.     

Smooth,easy to compute interpolating splines

We present a system of interpolating splines with first-order and approximate second-order geometric continuity. The curves are easily computed in linear time by solving a diagonally dominant, tridiagonal system of linear equations. Emphasis is placed on the need to find aesthetically pleasing curves in a wide range of circumstances; favorable results are obtained even when the knots are very unequally spaced or widely separated. The curves are invariant under translation, rotation, and scaling, and the effects of a local change fall off exponentially as one moves away from the disturbed knot.Approximate second-order continuity is achieved by using a linear mock curvature function in place of the actual endpoint curvature for each spline segment and choosing tangent directions at knots so as to equalize these. This avoids extraneous solutions and other forms of undesirable behavior without seriously compromising the quality of the results.The actual spline segments can come from any family of curves whose endpoint curvatures can be suitably approximated, but we propose a specific family of parametric cubics. There is freedom to allow tangent directions and tension parameters to be specified at knots, and special curl parameters may be given for additional control near the endpoints of open curves.This research was supported in part by the National Science Foundation under grants IST-820-1926 and MCS-83-00984 and by the Systems Development Foundation.     

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.     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.     

Convergence of interpolating cardinal splines: Power growth

Letf(x) be the restriction to the real axis of an entire function of exponential typeτ<π and of power growth on the axis. Then thenth order cardinal spline,ℒ _nf(x), interpolatingf(x) at the integers converges uniformly on compacta tof(x). This is also true of the respective derivatives. An example shows that exponential typeπ is not necessarily permitted. The proof utilizes distribution theory and estimates on the derivatives of the Fourier transform of the fundamental splineL _n(x). This research is partially supported by Canadian National Research Council Grant A-7687.     

Bivariate interpolating polynomials and splines (I)

The multivariate splines which were first presented by de Boor as a complete theoretical system have intrigued many mathematicians who have devoted many works in this field which is still in the process of development. The author of this paper is interested in the area of interpolation with special emphasis on the interpolation methods and their approximation orders. But such B-splines (both univariate and multivariate) do not interpolated directly, so I approached this problem in another way which is to extend my interpolating spline of degree 2n-1 in univariate case (See[7]) to multivariate case. I selected triangulated region which is inspired by other mathematician’s works (e.g. [2] and [3]) and extend the interpolating polynomials from univariate to m-variate case (See [10])In this paper some results in the case m=2 are discussed and proved in more concrete details. Based on these polynomials, the interpolating splines (it is defined by me as piecewise polynomials in which the unknown partial derivatives are determined under certain continuous conditions) are also discussed. The approximation orders of interpolating polynomials and of cubic interpolating splines are inverstigated. We limited our discussion on the rectangular domain which is partitioned into equal right triangles. As to the case in which the rectangular domain is partitioned into unequal right triangles as well as the case of more complicated domains, we will discuss in the next paper.     

Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of order $k\geqslant 2$ spanned by $\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}$ on each subinterval $[x_i,x_{i+1}\rangle\subset [0,1]$ , i?=?0,1, ...l. Most of the paper deals with non-polynomial case m _i ,n _i ?∈?[4,?∞?), and polynomial splines known as VDP–splines are the special case when m _i , n _i are integers. We describe VDP–splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functions p(x), q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator $\frac{d}{dx} p \frac{d}{dx} q \frac{d^{k-2}} {dx^{k-2}}$ . Although the space of splines is not based on an Extended Chebyshev system, we argue that total positivity and variation diminishing still holds. Unlike the abstract results, constructive properties, like Marsden identity, recurrences for quasi-Bernstein polynomials and knot insertion algorithms may be more involved and we prove them only for VDP splines of orders 4 and 5.     

Explicit calculation of interpolating cubic splines on equi-distant knots

The coefficients of the cubic splines(x) interpolating to the functionf(x) on the equi-distant knots,x _i =ih(i=0(1)n andh=1/n) in the interval [0, 1], are determined explicitly in the cases whenf(x) is either periodic or has linear combinations of the first and second derivatives specified as boundary conditions.The effects of perturbations in the boundary conditions are analysed in closed form and exact results given for the ensuing changes in the spline fit. As illustration of the techniques a numerical example is given.     

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.     

Fitting data using optimal Hermite type cubic interpolating splines

In this work we obtain a new optimal property for cubic interpolating splines of Hermite type applied to data-fitting problems. The existence and uniqueness of the Hermite type cubic spline with minimal quadratic oscillation in average are proved.     

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.     

Notes on spline functions III: On the convergence of the interpolating cardinal splines as their degree tends to infinity

It is shown that for entire functionsf(x) defined by a Fourier-Stieltjes integral (9) the cardinal splineS _m (x) of the odd degree 2m−1, which interpolatesf(x) at all integers, converges tof(x) asm tends to infinity. Properties of the exponential Euler spline are used in the proof. Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

Construction ofC 2 Pythagorean-hodograph interpolating splines by the homotopy method

The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing aC ² PH quintic spline that interpolates a given sequence of pointsp ₀,p ₁,...,p _N and end-derivativesd ₀ andd _N to be reduced to solving a tridiagonal system ofN quadratic equations inN complex unknowns. The system can also be easily modified to incorporate PH-splineend conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2 ^N+1 distinct interpolants for each of several different data sets. We observe empirically that all but one of these interpolants exhibits undesirable looping behavior (which may be quantified in terms of theelastic bending energy, i.e., the integral of the square of the curvature with respect to arc length). The remaining good interpolant, however, is invariably afairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding ordinaryC ₂ cubic spline. Moreover, the PH spline has the advantage that its offsets arerational curves and its arc length is apolynomial function of the curve parameter.     

A necessary condition for convergence of interpolating parabolic and cubic splines

Let the sequence of nets _n be such that , where h_i ⁽ⁿ⁾ are the lengths of the segments of a net. The bound is necessary in order that interpolating parabolic and cubic splines converge for any function in C ( = 0) or C(0 < < 1), where C is the class of functions satisfying a Lipschitz condition of order. It is also shown that this bound cannot essentially be weakened.Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 165–178, February, 1976.The author thanks Yu. N. Subbotin for a useful discussion of the results obtained.     

Multidimensional natural splines of odd degree

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 65–68, February, 1990.