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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)

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)

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)

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)