Tina Bosner

Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of order

Non-uniform Exponential Tension Splines

k\geqslant 2

Knot Insertion Algorithms for Weighted Splines

spanned by

Quadratic convergence of approximations by CCC-Schoenberg operators

\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}

Original article: Singularly perturbed advection-diffusion-reaction problems: Comparison of operator-fitted methods

on each subinterval

On Calculating with Lower Order Chebyshev Splines

[x_i,x_{i+1}\rangle\subset [0,1]

Numerically stable algorithm for cycloidal splines

, i = 0,1, ...l. Most of the paper deals with non-polynomial case mi,ni ∈ [4, ∞ ), and polynomial splines known as VDP–splines are the special case when mi, ni are integers. We describe VDP–splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functions p(x), q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator

Basis of splines associated with singularly perturbed advection -diffusion problems

\frac{d}{dx} p \frac{d}{dx} q \frac{d^{k-2}} {dx^{k-2}}

Tension spline with application on image resampling

. Although the space of splines is not based on an Extended Chebyshev system, we argue that total positivity and variation diminishing still holds. Unlike the abstract results, constructive properties, like Marsden identity, recurrences for quasi-Bernstein polynomials and knot insertion algorithms may be more involved and we prove them only for VDP splines of orders 4 and 5.

Collocation by singular splines

We describe explicitly each stage of a numerically stable algorithm for calculating with exponential tension B-splines with non-uniform choice of tension parameters. These splines are piecewisely in the kernel of D2(D2–p2), where D stands for ordinary derivative, defined on arbitrary meshes, with a different choice of the tension parameter p on each interval. The algorithm provides values of the associated B-splines and their generalized and ordinary derivatives by performing positive linear combinations of positive quantities, described as lower-order exponential tension splines. We show that nothing else but the knot insertion algorithm and good approximation of a few elementary functions is needed to achieve machine accuracy. The underlying theory is that of splines based on Chebyshev canonical systems which are not smooth enough to be ECC-systems. First, by de Boor algorithm we construct exponential tension spline of class C1, and then we use quasi-Oslo type algorithms to evaluate classical non-uniform C2 tension exponential splines.