Based on variational properties, we generalize the approximation properties of the univariate natural cubic spline to splines in arbitrary dimensions. In one dimension, the solution of the variational problem { given a = x1 < · · · < xN = b and values f(x1), . . . , f(xN ) in R, find a function s with s(xi)=f(xi), i=1, . . . ,N , that minimizes ‖s‖L2([a,b]) is given by the natural univariate cu...

A numerical scheme, based on the cubic B-spline wavelets for solving fractional integro-differential equations is presented. The fractional derivative of these wavelets are utilized to reduce the fractional integro-differential equation to system of algebraic equations. Numerical examples are provided to demonstrate the accuracy and efficiency and simplicity of the method.

Of concern here is the numerical solution of singular integral equations of Cauchy type; i.e., equations involving principal value integrals. The unknown function is expressed as the product of an appropriate weight function and a cubic spline. The problem is reduced to a system of linear algebraic equations which is solved for the approximate values of the function at the knots. An estimate is...

In this paper, Quintic spline in o -step points is used for the solution of fourth-order boundary value problems. Spline relations and boundary formulas are developed and the convergence analysis of the given method is investigated. Numerical illustrations are given to show the applicability and e ciency of our method.

Chaiken's algorithm is a procedure for inserting new knots into uniform quadratic Bspline curves by doubling the control points and taking two successive averages. Lane and Riesenfeld showed that Chaiken's algorithm extends to uniform B-spline curves of arbitrary degree. By generalizing the notion of successive averaging, we further extend Chaiken's algorithm to B-spline curves of arbitrary deg...

Most shapes are simply too complicated to define using a single Bézier curve. A spline curve is a sequence of curve segments that are connected together to form a single continuous curve. For example, a piecewise collection of Bézier curves, connected end to end, can be called a spline curve. Overhauser curves are another example of splines. The word “spline” can also be used as a verb, as in “...

The problem of interpolation of discrete set of data on the interval [a, b] representing the function f is investigated. A family of C splines with one free generating function is introduced in order to solve this problem. Cubic C splines belong to this family. The required conditions which must satisfy the generating function in order to obtain explicit interpolants are presented and examples ...

The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory’s rational cubic C splines as well as related rational quintic C splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range rest...