Control curves and knot insertion for trigonometric splines

We introduce control curves for trigonometric splines and show that they have properties similar to those for classical polynomial splines. In particular, we discuss knot insertion algorithms, and show that as more and more knots are inserted into a trigonometric spline, the associated control curves converge to the spline. In addition, we establish a convex-hull property and a variation-diminishing result.     

Cones and recurrence relations for simplex splines

We prove some new relations between functions defined as shadows of cones (cone splines) and simplices (simplex splines). We use them to show how ans-variate simplex spline of some orderk can be written as a sum ofk+1 (s-l)-variate simplex splines of orderk-1. A recurrence relation on the spatial dimension of the simplex spline,s, is proposed as an interesting alternative to the recurrence relation in [17], where one uses the orderk for recursion, but not the spatial dimensions.     

A recurrence relation for chebyshevianB-splines

The fundamental recurrence relation for polynomialB-splines is generalized to ChebyshevianB-splines.     

Quasi-interpolation projectors for box splines

We consider box spline quasi-interpolants based on local linear functionals of point evaluator and integral type. The approximations are easy to compute, and reproduce the whole spline space in question.     

On the L 1-condition number of the univariate Bernstein basis

We show that the size of the 1-norm condition number of the univariate Bernstein basis for polynomials of degree n is O (2ⁿ / √n). This is consistent with known estimates [3], [5] for p = 2 and p = ∞ and leads to asymptotically correct results for the p-norm condition number of the Bernstein basis for any p with 1 ≤ p ≤ ∞.     

Discrete cubic spline interpolation

Defining equations, a best approximation property, and error bounds are given for a discrete cubic spline interpolant. Furthermore the distance between two cubic spline interpolants is estimated, and numerical examples are provided.