Natural bicubic spline fractal interpolation

Fractal Interpolation functions provide natural deterministic approximation of complex phenomena. Cardinal cubic splines are developed through moments (i.e. second derivative of the original function at mesh points). Using tensor product, bicubic spline fractal interpolants are constructed that successfully generalize classical natural bicubic splines. An upper bound of the difference between the natural cubic spline blended fractal interpolant and the original function is deduced. In addition, the convergence of natural bicubic fractal interpolation functions towards the original function providing the data is studied.     

Curvature variation minimizing cubic Hermite interpolants

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain G¹ cubic interpolatory spline.     

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.     

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.     

A class of piecewise interpolating functions based on barycentric coordinates

Piecewise interpolation methods, as spline or Hermite cubic interpolation methods, define the interpolant function by means of polynomial pieces and ensure that some regularity conditions are guaranteed at the break-points. In this work, we propose a novel class of piecewise interpolating functions whose expression depends on the barycentric coordinates and a suitable weight function. The underlying idea is to specialize to the 1D settings some aspects of techniques widely used in multi-dimensional interpolation, namely Shepard’s, barycentric and triangle-based blending methods. We show the properties of convergence for the interpolating functions and discuss how, in some cases, the properties of regularity that characterize the weight function are reflected on the interpolant function. Numerical experiments, applied to some case studies and real scenarios, show the benefit of our method compared to other interpolant models.     

Multivariate refinable Hermite interpolant

We introduce a general definition of refinable Hermite interpolants and investigate their general properties. We also study a notion of symmetry of these refinable interpolants. Results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable Hermite interpolants. The theory developed here is constructive and yields an easy-to-use construction method for multivariate refinable Hermite interpolants. Using this method, several new refinable Hermite interpolants with respect to different dilation matrices and symmetry groups are constructed and analyzed.

Some of the Hermite interpolants constructed here are related to well-known spline interpolation schemes developed in the computer-aided geometric design community (e.g., the Powell-Sabin scheme). We make some of these connections precise. A spline connection allows us to determine critical Hölder regularity in a trivial way (as opposed to the case of general refinable functions, whose critical Hölder regularity exponents are often difficult to compute).

While it is often mentioned in published articles that ``refinable functions are important for subdivision surfaces in CAGD applications", it is rather unclear whether an arbitrary refinable function vector can be meaningfully applied to build free-form subdivision surfaces. The bivariate symmetric refinable Hermite interpolants constructed in this article, along with algorithmic developments elsewhere, give an application of vector refinability to subdivision surfaces. We briefly discuss several potential advantages offered by such Hermite subdivision surfaces.

     

Comonotone and coconvex rational interpolation and approximation

Comonotonicity and coconvexity are well-understood in uniform polynomial approximation and in piecewise interpolation. The covariance of a global (Hermite) rational interpolant under certain transformations, such as taking the reciprocal, is well-known, but its comonotonicity and its coconvexity are much less studied. In this paper we show how the barycentric weights in global rational (interval) interpolation can be chosen so as to guarantee the absence of unwanted poles and at the same time deliver comonotone and/or coconvex interpolants. In addition the rational (interval) interpolant is well-suited to reflect asymptotic behaviour or the like.     

On the construction of interpolation mesh surfaces

A method for constructing two-dimensional interpolation mesh functions is proposed that is more flexible than the classical cubic spline method because it makes it possible to construct interpolation surfaces that fit the given function at specified points by varying certain parameters. The method is relatively simple and is well suited for practical implementation.     

Shape-Preserving Local Interpolation for Plotting Solutions of ODEs

We investigate the use of piecewise rational interpolants ofDelbourgo and Gregory in an important and widely occurring application.We propose the following algorithm for visually pleasing plotsof the solution of an ordinary differential equation (ODE):use piecewise cubic Hermite interpolation where it can be shownto preserve shape (monotonicity and/or convexity) and also wherethere is no shape to preserve, otherwise use the appropriateconvex or monotone piecewise rational interpolant. Bounds arederived which enable efficient plotting of the rational interpolants.This scheme should be useful in any context where both solutionand derivative of a function are available as data.     

Fundamental Sets of Fractal Functions

Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes of real mappings like, for instance, maps defined on sets that are not intervals, maps integrable but not continuous and may be defined on unbounded domains. In particular, based on fractal interpolation functions, we construct fractal Müntz polynomials that successfully generalize classical Müntz polynomials. The parameters of the fractal Müntz system enable the control and modification of the properties of original functions. Furthermore, we deduce fractal versions of classical Müntz theorems. In this way, the fractal methodology generalizes the fundamental sets of the classical approximation theory and we construct complete systems of fractal functions in spaces of continuous and p-integrable mappings on bounded domains. This work is supported by the project No: SB 2005-0199, Spain.     

Stability of affine coalescence hidden variable fractal interpolation functions

The stability of an affine coalescence hidden variable fractal interpolation function is proved in a general set up in the present work, by establishing that any small perturbation in the generalized interpolation data leads to a small perturbation in the corresponding affine coalescence hidden variable fractal interpolation function.     

On the Lebesgue constant of barycentric rational interpolation at equidistant nodes

Recent results reveal that the family of barycentric rational interpolants introduced by Floater and Hormann is very well-suited for the approximation of functions as well as their derivatives, integrals and primitives. Especially in the case of equidistant interpolation nodes, these infinitely smooth interpolants offer a much better choice than their polynomial analogue. A natural and important question concerns the condition of this rational approximation method. In this paper we extend a recent study of the Lebesgue function and constant associated with Berrut’s rational interpolant at equidistant nodes to the family of Floater–Hormann interpolants, which includes the former as a special case.     

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.     

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.     

Constrained interpolation using rational Cubic Spline with linear denominators

In this paper, a rational cubic interpolant spline with linear denominator has been constructed, and it is used to constrain interpolation curves to be bounded in the given region. Necessary and sufficient conditions for the interpolant to satisfy the constraint have been developed. The existence conditions are computationally efficient and easy to apply. Finally, the approximation properties have been studied.     

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.     

A new non‐polynomial univariate interpolation formula of Hermite type

A new C ^∞ interpolant is presented for the univariate Hermite interpolation problem. It differs from the classical solution in that the interpolant is of non‐polynomial nature. Its basis functions are a set of simple, compact support, transcendental functions. The interpolant can be regarded as a truncated Multipoint Taylor series. It has essential singularities at the sample points, but is well behaved over the real axis and satisfies the given functional data. The interpolant converges to the underlying real‐analytic function when (i) the number of derivatives at each point tends to infinity and the number of sample points remains finite, and when (ii) the spacing between sample points tends to zero and the number of specified derivatives at each sample point remains finite. A comparison is made between the numerical results achieved with the new method and those obtained with polynomial Hermite interpolation. In contrast with the classical polynomial solution, the new interpolant does not suffer from any ill conditioning, so it is always numerically stable. In addition, it is a much more computationally efficient method than the polynomial approach. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multivariate interpolation with increasingly flat radial basis functions of finite smoothness

In this paper, we consider multivariate interpolation with radial basis functions of finite smoothness. In particular, we show that interpolants by radial basis functions in ℝ ^d with finite smoothness of even order converge to a polyharmonic spline interpolant as the scale parameter of the radial basis functions goes to zero, i.e., the radial basis functions become increasingly flat.     

Simultaneous Approximation of Sobolev Classes by Piecewise Cubic Hermite Interpolation

For the approximation in $L_p$-norm, we determine the weakly asymptotic orders for the simultaneous approximation errors of Sobolev classes by piecewise cubic Hermite interpolation with equidistant knots. For $p = 1$, $∞$, we obtain its values. By these results we know that for the Sobolev classes, the approximation errors by piecewise cubic Hermite interpolation are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths. At the same time, the approximation errors of derivatives are weakly equivalent to the corresponding infinite-dimensional Kolmogorov widths.     

The Neville-like form of the Fitzpatrick algorithm for rational interpolation

The Fitzpatrick algorithm, which seeks a Gr?bner basis for the solution of a system of polynomial congruences, can be applied to compute a rational interpolant. Based on the Fitzpatrick algorithm and the properties of an Hermite interpolation basis, we present a Neville-like algorithm for multivariate osculatory rational interpolation. It may be used to compute the values of osculatory rational interpolants at some points directly without computing the rational interpolation function explicitly.