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...

Let φ be an orthonormal scaling function with approximation degree p−1, and let Bn be the B-spline of order n. Then, spline type scaling functions defined by f̄n = f ∗Bn (n = 1, 2, . . . ) possess higher approximation order, p+n−1, and compact support. The corresponding biorthogonal wavelet functions are also constructed. This technique is extended to the case of biorthogonal scaling function sy...

In this paper, G continuous cubic spline interpolation of data points in R, based on a discrete approximation of the strain energy, is studied. Simple geometric conditions on data are presented that guarantee the existence of the interpolant. The interpolating spline is regular, loop-, cuspand fold-free.

Abstract: This study proposes new C rational cubic spline interpolant of the form cubic/quadratic with three shape parameters to preserves the geometric properties of the given data sets. Sufficient conditions for the positivity and data constrained modeling of the rational interpolant are derived on one parameter while the remaining two parameters can further be utilized to change and modify t...

Based on analysis of basic cubic spline interpolation, the clamped cubic spline interpolation is generalized in this paper. The methods are presented on the condition that the first derivative and second derivative of arbitrary node are given. The Clamped spline and Curvature-adjusted cubic spline are also generalized. The methods are presented on the condition that the first derivatives of arb...

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 spli...

In this paper, a classical problem of the construction of a cubic Hermite G1 continuous spline curve is considered. The only data given are interpolation points, while tangent directions are unknown. They are constructed in such a way that a particular minimization of the strain energy of the spline curve is applied. The resulting spline curve is regular, cusp-, loopand fold-free. Even more, it...

In this article, a fourth order quintic spline method has been developed to obtain numerical solutions for second order boundary value problems with Dirichlet boundary conditions. The developments of the quintic spline method and convergence analysis were presented. Three test problems have been considered for comparison purposes. The numerical results showed that the quintic spline method is m...

Based on analysis of cubic spline interpolation, the differentiation formulas of the cubic spline interpolation on the three boundary conditions are put up forward in this paper. At last, this calculation method is illustrated through an example. The numerical results show that the spline numerical differentiations are quite effective for estimating first and higher derivatives of equally and u...

In this paper a pair of wavelets are constructed on the basis of Hermite cubic splines. These wavelets are in C1 and supported on [−1, 1]. Moreover, one wavelet is symmetric, and the other is antisymmetric. These spline wavelets are then adapted to the interval [0, 1]. The construction of boundary wavelets is remarkably simple. Furthermore, global stability of the wavelet basis is established. ...