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- K Kopotun, D Leviatan, I A Shevchuk
- 1998

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)

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)

- 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.

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)

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)

- K Kopotun, D Leviatan, I A Shevchuk
- 2003

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)

- K Kopotun, D Leviatan, I A Shevchuk
- 2004

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.