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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)

Let f 2 C ?1; 1] change its convexity nitely many times in the interval, say s times, at Y s : ?1 < y 1 < < y s < 1. We estimate the degree of approximation of f by polynomials of degree n, which change convexity exactly at the points Y s. We show that provided n is suuciently large, depending on the location of the points Y s , the rate of approximation is… (More)

- Kirill Kopotun, Marian Neamtu, Bojan Popov
- Math. Comput.
- 2003

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)

- Kirill Kopotun, Alexei Shadrin
- SIAM J. Math. Analysis
- 2003

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)

- K Kopotun, D Leviatan
- 1997

Let a function f 2 L p ?1; 1], 0 < p 1 have 1 r < 1 changes of monotonicity. For all suuciently large n, we construct algebraic polynomials p n of degree n which are comonotone with f, and such that kf ? p n k Lp?1; 1] C(r)! ' 2 (f; n ?1) p , where ! ' 2 (f; n ?1) p denotes the Ditzian-Totik second modulus of smoothness in L p metric.

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)

- Kirill Kopotun
- Math. Comput.
- 2007

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)

The paper deals with approximation of a continuous function, on a finite interval, which changes convexity finitely many times, by algebraic polynomials which are coconvex with it. We give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian– Totik moduli of smoothness.

- Kirill Kopotun
- Journal of Approximation Theory
- 2006