Learn More
To deene spline subdivision schemes for general compact sets, we use the representation of spline subdivision schemes in terms of repeated averages, and replace the usual average (convex combination) by a binary averaging operation between two compact sets, introduced in 1] and termed here the \metric average". These schemes are shown to converge in the(More)
A new class of Godunov-type numerical methods for solving nonlinear scalar conservation laws in one space dimension is introduced. This new class of methods , called weakly non-oscillatory (WNO), is a generalization of the classical non-oscillatory schemes. Under certain conditions, convergence and error estimates for the methods are proved. Examples of(More)
Let SN;r be the (nonlinear) space of free knot splines of degree r ? 1 with at most N pieces in a; b], and let M k be the class of all k-monotone functions on (a; b), i.e., those functions f for which the kth divided diierence x0; : : : ; x k ]f is nonnegative for all choices of (k +1) distinct points x0; : : : ; x k in (a; b). In this paper, we solve the(More)
Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some(More)
We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, " shape " refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function. It is rather well known that it is possible to approximate a function by algebraic(More)
Several results on equivalence of moduli of smoothness of univari-ate splines are obtained. For example, it is shown that, for any 1 ≤ k ≤ r + 1, 0 ≤ m ≤ r − 1, and 1 ≤ p ≤ ∞, the inequality n −ν ω k−ν (s (ν) , n −1) p ∼ ω k (s, n −1) p , 1 ≤ ν ≤ min{k, m + 1}, is satisfied, where s ∈ C m [−1, 1] is a piecewise polynomial of degree ≤ r on a quasi-uniform(More)