The paper is focused on the understanding of the spline function as a modern tool in data analysis. We consider three numerical methods that offer framework for the spline function’s use as the interpolation, fitting and smoothing of the data. For each of these three methods, we present the corresponding spline and the conditions required by data for using the appropriate type of spline.

We present a method for hierarchical data approximation using curved quadratic simplicial elements for domain decomposition. Scientific data defined over twoor three-dimensional domains typically contain boundaries and discontinuities that are to be preserved and approximated well for data analysis and visualization. Curved simplicial elements make possible a better representation of curved geo...

We present a method for hierarchical data approximation using quadratic simplicial elements for domain decomposition and field approximation. Higher-order simplicial elements can approximate data better than linear elements. Thus, fewer quadratic elements are required to achieve similar approximation quality. We use quadratic basis functions and compute best quadratic simplicial spline approxim...

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...

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...

This paper proposes a method for approximating unorganized points in lR 3 with smooth B-spline surfaces. The method involves: meshless parameteri-zation; triangulation; shape-preserving reparameterization; and least squares spline approximation.

Splines are minimum-norm approximations of functions that interpolate the given data, (xl, f(xl)), l = 0, . . . , N − 1. In this paper, we consider the problem of approximating functions in certain reproducing kernel Hilbert spaces, with special choices of the points xl, l = 0, . . . , N−1. The designs considered here are node sets of integration lattices and digital nets for functions f define...