Kirill Kopotun

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To de ne 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 and termed here the metric average These schemes are shown to converge in the Hausdor metric(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)
A new class of Godunov-type numerical methods (called here weakly nonoscillatory or WNO) for solving nonlinear scalar conservation laws in one space dimension is introduced. This new class generalizes the classical nonoscillatory schemes. In particular, it contains modified versions of MinMod and UNO. Under certain conditions, convergence and error(More)
Several results on equivalence of moduli of smoothness of univariate 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−νωk−ν(s(ν), n−1)p ∼ ωk(s, n −1)p, 1 ≤ ν ≤ min{k,m + 1}, is satisfied, where s ∈ Cm[−1, 1] is a piecewise polynomial of degree ≤ r on a quasi-uniform (i.e., the ratio of(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)
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 be the class of all k-monotone functions on (a, b), i.e., those functions f for which the kth divided difference [x0, . . . , xk]f is nonnegative for all choices of (k+1) distinct points x0, . . . , xk in (a, b). In this paper, we solve the(More)
Our main interest in this paper is approximation of a continuous function, on a finite interval, which changes convexity finitely many times by algebraic polynomials which are coconvex with it. This topic has received much attention in recent years, and the purpose of this paper is to give final answers to open questions concerning the validity of Jackson(More)