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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)

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 Min-Mod and UNO. Under certain conditions, convergence and error… (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)

Let ω k ϕ (f, δ) w,L q be the Ditzian–Totik modulus with weight w, M k be the cone of k-monotone functions on (−1, 1), i.e., those functions whose kth divided differences are nonnegative for all selections of k + 1 distinct points in is the set of algebraic polynomials of degree at most n. Additionally, let w α,β (x) := (1 + x) α (1 − x) β be the classical… (More)

For r ≥ 3, n ∈ N and each 3-monotone continuous function f on [a, b] (i.e., f is such that its third divided differences [x 0 , x 1 , x 2 , x 3 ] f are nonnegative for all choices of distinct points x 0 ,. .. , x 3 in [a, b]), we construct a spline s of degree r and of minimal defect (i.e., s ∈ C r −1 [a, b]) with n − 1 equidistant knots in (a, b), which is… (More)