Minimal-norm-derivative spline function in interpolation and approximation

The paper is concerned with applications of quadratic splines with minimal derivative to approximation of functions in approximation and interpolation problems. A smooth spline is constructed on a uniform mesh so as the norm of the spline derivative is minimal; the nodes of the spline and the nodes of interpolations coincide. This approach allows construction of a spline from given values of the function on the mesh without additional assignment of the value of the function derivative at the initial point, because the derivative can be determined from the minimality condition for the norm of the spline derivative in L ₂.     

C1 rational quadratic spline interpolation to convex data

For a set of convex data we find conditions for the convexity of the rational quadratic spline interpolation. As an application, we prove for several functions that if the data derives from one of these functions, then the above convexity conditions are verified.     

紧支撑样条小波插值及其应用

基于紧支撑样条小波函数插值与定积分的思想,给出了由紧支撑样条小波插值函数构造数值积分公式的方法．并将该方法应用于二次、三次、四次和五次紧支撑样条小波函数,得到了相应的数值积分公式．最后,通过数值例子验证,发现该方法得到的数值积分公式是准确的,且具有较高精度．     

On the choice of the initial value of a quadratic spline in positive interpolation

Any quadratic spline with coinciding interpolating points and breakpoints can be uniquely determined by one initial condition. We examine how to choose the initial value in positive interpolation by using a norm of the spline. Monotone interpolation is also considered. The results are illuminated by numerical examples and by an application.     

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.     

A two-dimensional minimum-derivative spline

We consider the construction of a C ^(1,1) interpolation parabolic spline function of two variables on a uniform rectangular grid, i.e., a function continuous in a given region together with its first partial derivatives which on every partial grid rectangle is a polynomial of second degree in x and second degree in y. The spline function is constructed as a minimum-derivative one-dimensional quadratic spline in one of the variables, and the spline coefficients themselves are minimum-derivative quadratic spline functions of the other variable.     

Spherical spline interpolation—basic theory and computational aspects

The purpose of the paper is to adapt to the spherical case the basic theory and the computational method known from surface spline interpolation in Euclidean spaces. Spline functions are defined on the sphere. The solution process is made simple and efficient for numerical computation. In addition, the convergence of the solution obtained by spherical spline interpolation is developed using estimates for Legendre polynomials.     

OnG 2 continuous cubic spline interpolation

In this paper the interpolation byG ² continuous planar cubic Bézier spline curves is studied. The interpolation is based upon the underlying curve points and the end tangent directions only, and could be viewed as an extension of the cubic spline interpolation to the curve case. Two boundary, and two interior points are interpolated per each spline section. It is shown that under certain conditions the interpolation problem is asymptotically solvable, and for a smooth curvef the optimal approximation order is achieved. The practical experiments demonstrate the interpolation to be very satisfactory. Supported in prat by the Ministry of Science and Technology of Slovenjia, and in part by the NSF and SF of National Educational Committee of China.     

Comparison of spline and Lagrangian interpolation

Errors in spline and Lagrangian methods of interpolation are compared for a range of functions, tabular intervals, and levels of data errors. For large step-sizes (an important case in practise) and fairly accurate data Lagrangian methods are considerably more accurate than splines.     

On integro quartic spline interpolation

In this paper, we use quartic B-spline to construct an approximating function to agree with the given integral values of a univariate real-valued function over the same intervals. It is called integro quartic spline interpolation. Our interpolation method is new and easy to implement. Moreover, it can work successfully even without any boundary conditions. The interpolation errors are studied. The super convergence (sixth order and fourth order, respectively) in approximating function values and second-order derivative values at the knots is proved. Numerical examples illustrate that our method is very effective and our integro-interpolating quartic spline has higher approximation ability than others.     

Cubic spline interpolation of functions with high gradients in boundary layers

The cubic spline interpolation of grid functions with high-gradient regions is considered. Uniform meshes are proved to be inefficient for this purpose. In the case of widely applied piecewise uniform Shishkin meshes, asymptotically sharp two-sided error estimates are obtained in the class of functions with an exponential boundary layer. It is proved that the error estimates of traditional spline interpolation are not uniform with respect to a small parameter, and the error can increase indefinitely as the small parameter tends to zero, while the number of nodes N is fixed. A modified cubic interpolation spline is proposed, for which O((ln N/N)⁴) error estimates that are uniform with respect to the small parameter are obtained.     

Optimal error bounds for the cubic spline interpolation of lower smooth functions (1)

In this paper, the kernel of the cubic spline interpolation is given. An optimal error bound for the cubic spline interpolation of lower smooth functions is obtained.     

Parabolic spline interpolation for functions with large gradient in the boundary layer

We consider the problem of Subbotin’s parabolic spline interpolation for functions with large gradient domains. In the case of the common piecewise uniform Shishkin’s mesh we obtain two-sided accuracy estimates for the class of functions with exponential boundary layer. The spline interpolation accuracy estimates are not uniform in a small parameter, while the error itself can grow unboundedly as the small parameter vanishes and the number N of nodes remains fixed. We include the results of some simulations.     

Chordal cubic spline interpolation is fourth-order accurate

^** Email: michaelf{at}ifi.uio.no It is well known that complete cubic spline interpolation offunctions with four continuous derivatives is fourth-order accurate.In this paper we show that this kind of interpolation, whenused to construct parametric spline curves through sequencesof points in any space dimension, is again fourth-order accurateif the parameter intervals are chosen by chord length. We alsoshow how such chordal spline interpolants can be used to approximatethe arc-length derivatives of a curve and its length.     

Overcoming the boundary effects in surface spline interpolation

Surface spline interpolation when the domain is all of R^d isknown to converge much faster to the data function f than inthe case when the domain is the unit ball. This difference isunderstood to be due to boundary effects which, as will be shown,also affect the size of the surface spline's coefficients. Wepropose a modified form of surface spline interpolation which,to a great extent, overcomes these boundary effects. This modifiedsurface spline interpolant uses only the values of f at thegiven interpolation points.     

On minimax cardinal spline interpolation

Statistical Inference for Stochastic Processes - It is shown that in the problem of cardinal interpolation, spline interpolants of various degrees are R-minimax, with respect to corresponding...     

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.     

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.     

Refined-mesh interpolation method for functions with a boundary-layer component

Linear and quadratic spline interpolation methods for a one-variable function with a boundary-layer component are examined. It is shown that the interpolation method for such a function leads to considerable errors when applied on a uniform mesh. The error of linear and quadratic spline interpolations on meshes that are refined in the boundary layer is estimated. Numerical results are presented.     

Cubic spline interpolation with quasiminimal B-spline coefficients

Summary The end conditions for cubic spline interpolation with equidistant knots will be defined so as to make the (slightly modified) B-spline coefficients minimal. This produces good approximation results as compared e.g. with the not-a-knot spline.