Nodal splines on compact intervals

In this paper, we demonstrate that the nodal spline interpolation operator on compact intervals can be regarded as a discretized version of the Martensen operator. This approach gives an interesting alternative perspective on the nodal splines on compact intervals. The origin of the Martensen operator can be found in a two-point Hermite spline interpolation problem.     

Martensen Splines

This paper is concerned with the construction of the fundamental functions associated with a two-point Hermite spline interpolation scheme used by Martensen in the context of the remainder of the Gregory quadrature rule. We derive both a recursive construction and an explicit representation in terms of the underlying B-Splines which can easily be deduced using Marsden’s identity. We can make use of these functions in order to introduce a local interpolation scheme which reproduces all splines. Finally, we examine the error of this interpolant to a sufficiently smooth function and realize that it behaves like in the case of splines of degree n. AMS subject classification (2000) 65D05, 65D07, 41A15     

An efficient algorithm for periodic Hermite spline interpolation with shifted nodes

Generalized Hermite spline interpolation with periodic splines of defect 2 on an equidistant lattice is considered. Then the classic periodic Hermite spline interpolation with shifted interpolation nodes is obtained as a special case.By means of a new generalization of Euler-Frobenius polynomials the symbol of the considered interpolation problem is defined. Using this symbol, a simple representation of the fundamental splines can be given. Furthermore, an efficient algorithm for the computation of the Hermite spline interpolant is obtained, which is mainly based on the fast Fourier transform.     

一类带参数的有理三次三角Hermite插值样条

给出一种带有参数的有理三次三角Hermite插值样条,具有标准三次Hermite插值样条相似的性质.利用参数的不同取值不但可以调控插值曲线的形状,而且比标准三次Hermite插值样条更好地逼近被插曲线.此外,选择合适的控制点,该种插值样条可以精确表示星形线和四叶玫瑰线等超越曲线.     

Riesz Bases of Splines and Regularized Splines with Multiple Knots

This paper deals withL₂(R)-norm and Sobolev-norm stability of polynomial splines with multiple knots, and with regularized versions thereof. An essential ingredient is a result on Hölder continuity of the shift operator operating on a B-spline series. The stability estimates can be reformulated in terms of a Riesz basis property for the underlying spline spaces. These can also be employed to derive a result on stable Hermite interpolation on the real line. We point to the connection with the problem of symmetric preconditioning of bi-infinite interpolation matrices.     

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves, the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed. The quartic Hermite spline curves not only have the same interpolation and conti-nuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C2 continuity by the shape parameters when the interpolation conditions are fixed.     

On multivariate Hermite interpolation

We study the problem of Hermite interpolation by polynomials in several variables. A very general definition of Hermite interpolation is adopted which consists of interpolation of consecutive chains of directional derivatives. We discuss the structure and some aspects of poisedness of the Hermite interpolation problem; using the notion of blockwise structure which we introduced in [10], we establish an interpolation formula analogous to that of Newton in one variable and use it to derive an integral remainder formula for a regular Hermite interpolation problem. For Hermite interpolation of degreen of a functionf, the remainder formula is a sum of integrals of certain (n + 1)st directional derivatives off multiplied by simplex spline functions.     

An energy-minimization framework for monotonic cubic spline interpolation

This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C² continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. It is possible for a set of monotonically increasing (or decreasing) data points to yield a curve that is not monotonic, i.e., the spline may oscillate. In such cases, it is necessary to sacrifice some smoothness in order to preserve monotonicity.The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. The conditions are simplified and consolidated to yield a fast method for determining monotonicity. This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic C² cubic spline interpolation results are presented. Extensions to shape preserving splines and data smoothing are described.     

Error estimates for Hermite interpolation on spheres

In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere S^d−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.     

Interpolation by splines satisfying mixed boundary conditions

We consider interpolation of Hermite data by splines of degreen withk given knots, satisfying boundary conditions which may involve derivatives at both end points (e.g., a periodicity condition). It is shown that, for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen?k. The method of proof is to show that any spline interpolating zero data vanishes identically, rather than the usual determinantal approach.     

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.     

Uniform bounds for cardinal Hermite spline operators with double knots

A method is given for computing the uniform norm of the cardinal Hermite spline operator. This is the operator that takes two bounded biinfinite sequences of numbers into the unique bounded spline of degree 2k − 1(k 2) with knots of multiplicity two at the integers and that interpolates the two given sequences for both functional and first derivative values at the integers. The computational schema relies on knowledge of the Bernoulli splines, while the theoretical aspects make use of some properties of zeros of periodic splines.     

Multivariate Bernoulli splines and the periodic interpolation problem

Periodic spline interpolation in Euclidian spaceR d is studied using translates of multivariate Bernoulli splines introduced in [25]. The interpolating polynomial spline functions are characterized by a minimal norm property among all interpolants in a Hilbert space of Sobolev type. The results follow from a relation between multivariate Bernoulli splines and the reproducing kernel of this Hilbert space. They apply to scattered data interpolation as well as to interpolation on a uniform grid. For bivariate three-directional Bernoulli splines the approximation order of the interpolants on a refined uniform mesh is computed.     

Construction of interpolation splines minimizing semi-norm in W_{2}^{(m,m-1)}(0,1) space

In the present paper using S.L. Sobolev’s method interpolation splines minimizing the semi-norm in a Hilbert space are constructed. Explicit formulas for coefficients of interpolation splines are obtained. The obtained interpolation spline is exact for polynomials of degree m?2 and e ^?x . Also some numerical results are presented.     

Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

Cubic Splines with Minimal Norm

Natural cubic interpolatory splines are known to have a minimal L ₂-norm of its second derivative on the C ² (or W ₂ ² ) class of interpolants. We consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic splines only. The cases of classical cubic splines with defect one (interpolation of function values) and of Hermite C ¹ splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.     

C 1 Quadratic and C 2 Quartic Macro-Elements on a Modified Morgan-Scott Triangulation

In this paper, we construct a quadratic composite finite element of class C ¹ and quartic composite finite element of class C ² on a new triangulation τ ₁₀ which is obtained by splitting each triangle of a given triangulation τ into ten smaller subtriangles. These new elements can be used for constructing spline spaces with local basis that can be applied for solving some Hermite interpolation problems with optimal approximation order.     

About measures and nodal systems for which the Hermite interpolants uniformly converge to continuous functions on the circle and interval

We obtain the Laurent polynomial of Hermite interpolation on the unit circle for nodal systems more general than those formed by the n-roots of complex numbers with modulus one. Under suitable assumptions for the nodal system, that is, when it is constituted by the zeros of para-orthogonal polynomials with respect to appropriate measures or when it satisfies certain properties, we prove the convergence of the polynomial of Hermite-Fejér interpolation for continuous functions. Moreover, we also study the general Hermite interpolation problem on the unit circle and we obtain a sufficient condition on the interpolation conditions for the derivatives, in order to have uniform convergence for continuous functions.Finally, we obtain some improvements on the Hermite interpolation problems on the interval and for the Hermite trigonometric interpolation.