On the order of approximation by linear shape-preserving operators on subsets of [0,1] with positive measure

Starting from the study of the Shepard nonlinear operator of max-prod type in [6], [7], in the book [8], Open Problem 5.5.4, pp. 324-326, the Favard-Sz?sz-Mirakjan max-prod type operator is introduced and the question of the approximation order by this operator is raised. In the recent paper [1], by using a pretty complicated method to this open question an answer is given by obtaining an upper pointwise estimate of the approximation error of the form C?1(f;?x/?n) (with an unexplicit absolute constant C>0) and the question of improving the order of approximation ?1(f;?x/?n) is raised. The first aim of this note is to obtain the same order of approximation but by a simpler method, which in addition presents, at least, two advantages : it produces an explicit constant in front of ?1(f;?x/?n) and it can easily be extended to other max-prod operators of Bernstein type. Also, we prove by a counterexample that in some sense, in general this type of order of approximation with respect to ?1(f;?) cannot be improved. However, for some subclasses of functions, including for example the bounded, nondecreasing concave functions, the essentially better order ?1 (f;1/n) is obtained. Finally, some shape preserving properties are obtained.

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The Rate of Convergence for Linear Shape-Preserving Algorithms

Concrete Operators ◽

10.1515/conop-2015-0008 ◽

2015 ◽

Vol 2 (1) ◽

Author(s):

Dmitry Boytsov ◽

Sergei Sidorov

Keyword(s):

Rate Of Convergence ◽

Upper Bound ◽

Explicit Methods ◽

Shape Preserving ◽

Rate Of Approximation ◽

Quantitative Results ◽

Linear Shape

AbstractWe prove some results which give explicit methods for determining an upper bound for the rate of approximation by means of operators preserving a cone. Thenwe obtain some quantitative results on the rate of convergence for some sequences of linear shape-preserving operators.

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Non-Linear Shape Preserving Electron-Beams

CLEO: 2014 ◽

10.1364/cleo_qels.2014.fm3d.7 ◽

2014 ◽

Author(s):

Maor Mutzafi ◽

Ido Kaminer ◽

Gal Harari ◽

Mordechai Segev

Keyword(s):

Electron Beams ◽

Shape Preserving ◽

Non Linear ◽

Linear Shape

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C1Rational Quadratic Trigonometric Interpolation Spline for Data Visualization

Mathematical Problems in Engineering ◽

10.1155/2015/983120 ◽

2015 ◽

Vol 2015 ◽

pp. 1-20 ◽

Cited By ~ 1

Author(s):

Shengjun Liu ◽

Zhili Chen ◽

Yuanpeng Zhu

Keyword(s):

Trigonometric Interpolation ◽

Data Sets ◽

Order Of Approximation ◽

Shape Parameters ◽

Interpolation Spline ◽

Shape Preserving ◽

Shape Preserving Interpolation ◽

Trigonometric Spline ◽

The Given ◽

Data Constraints

A newC1piecewise rational quadratic trigonometric spline with four local positive shape parameters in each subinterval is constructed to visualize the given planar data. Constraints are derived on these free shape parameters to generate shape preserving interpolation curves for positive and/or monotonic data sets. Two of these shape parameters are constrained while the other two can be set free to interactively control the shape of the curves. Moreover, the order of approximation of developed interpolant is investigated asO(h3). Numeric experiments demonstrate that our method can construct nice shape preserving interpolation curves efficiently.

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Approximation and Shape Preserving Properties of the Bernstein Operator of Max-Product Kind

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2009/590589 ◽

2009 ◽

Vol 2009 ◽

pp. 1-26 ◽

Cited By ~ 16

Author(s):

Barnabás Bede ◽

Lucian Coroianu ◽

Sorin G. Gal

Keyword(s):

Nonlinear Operator ◽

Approximation Error ◽

Approximation Order ◽

Order Of Approximation ◽

Shape Preserving ◽

Explicit Constant ◽

Open Question ◽

Shape Preserving Properties ◽

Upper Estimate ◽

Better Than

Starting from the study of theShepard nonlinear operator of max-prod typeby Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324–326, theBernstein max-prod-type operatoris introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form (with an unexplicit absolute constant ) and the question of improving the order of approximation is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions including, for example, that of concave functions, we find the order of approximation , which for many functions is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.

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