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A counterexample in comonotone approximation in L p space

Xiang Wu, Song Zhou (1993)

Colloquium Mathematicae

Refining the idea used in [24] and employing very careful computation, the present paper shows that for 0 < p ≤ ∞ and k ≥ 1, there exists a function f ∈ C [ - 1 , 1 ] k , with f ( k ) ( x ) ≥ 0 for x ∈ [0,1] and f ( k ) ( x ) ≤ 0 for x ∈ [-1,0], such that lim supn→∞ (en(k)(f)p) / (ωk+2+[1/p](f,n-1)p) = + ∞ where e n ( k ) ( f ) p is the best approximation of degree n to f in L p by polynomials which are comonotone with f, that is, polynomials P so that P ( k ) ( x ) f ( k ) ( x ) ≥ 0 for all x ∈ [-1,1]. This theorem, which is a particular case of a more general one, gives a complete solution...

Approximation and shape preserving properties of the nonlinear Bleimann-Butzer-Hahn operators of max-product kind

Barnabás Bede, Lucian Coroianu, Sorin G. Gal (2010)

Commentationes Mathematicae Universitatis Carolinae

Starting from the study of the Shepard nonlinear operator of max-prod type in (Bede, Nobuhara et al., 2006, 2008), in the book (Gal, 2008), Open Problem 5.5.4, pp. 324–326, the Bleimann-Butzer-Hahn max-prod type operator is introduced and the question of the approximation order by this operator is raised. In this paper firstly we obtain an upper estimate of the approximation error of the form ω 1 ( f ; ( 1 + x ) 3 2 x / n ) . A consequence of this result is that for each compact subinterval [ 0 , a ] , with arbitrary a > 0 , the order of uniform...

Left general fractional monotone approximation theory

George A. Anastassiou (2016)

Applicationes Mathematicae

We introduce left general fractional Caputo style derivatives with respect to an absolutely continuous strictly increasing function g. We give various examples of such fractional derivatives for different g. Let f be a p-times continuously differentiable function on [a,b], and let L be a linear left general fractional differential operator such that L(f) is non-negative over a closed subinterval I of [a,b]. We find a sequence of polynomials Qₙ of degree ≤n such that L(Qₙ) is non-negative over I,...

Meshless Polyharmonic Div-Curl Reconstruction

M. N. Benbourhim, A. Bouhamidi (2010)

Mathematical Modelling of Natural Phenomena

In this paper, we will discuss the meshless polyharmonic reconstruction of vector fields from scattered data, possibly, contaminated by noise. We give an explicit solution of the problem. After some theoretical framework, we discuss some numerical aspect arising in the problems related to the reconstruction of vector fields

Nearly Coconvex Approximation

Leviatan, D., Shevchuk, I. (2002)

Serdica Mathematical Journal

* Part of this work was done while the second author was on a visit at Tel Aviv University in March 2001Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are nearly coconvex with it, namely, polynomials and piecewise polynomials that preserve the convexity of f except perhaps in some small neighborhoods of the points where f changes its convexity. We obtain...

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