Approximation and shape preserving properties of the nonlinear Favardsza-Szász-Mirakjan operator of max-product kind

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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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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A Family of 6-Point n-Ary Interpolating Subdivision Schemes

Mehran University Research Journal of Engineering and Technology ◽

10.22581/muet1982.1804.03 ◽

2018 ◽

Vol 37 (4) ◽

pp. 481-490

Author(s):

Robina Bashir ◽

Ghulam Mustafa

Keyword(s):

Divided Difference ◽

Interpolation Property ◽

Approximation Order ◽

Visual Performance ◽

Shape Preserving ◽

Computer Aided ◽

Specific Position ◽

Step Algorithm ◽

Shape Preserving Properties ◽

Geometric Designs

We derive three-step algorithm based on divided difference to generate a class of 6-point n-ary interpolating sub-division schemes. In this technique second order divided differences have been calculated at specific position and used to insert new vertices. Interpolating sub-division schemes are more attractive than approximating schemes in computer aided geometric designs because of their interpolation property. Polynomial generation and polynomial reproduction are attractive properties of sub-division schemes. Shape preserving properties are also significant tool in sub-division schemes. Further, some significant properties of ternary and quaternary sub-division schemes have been elaborated such as continuity, degree of polynomial generation, polynomial reproduction and approximation order. Furthermore, shape preserving property that is monotonicity is also derived. Moreover, the visual performance of proposed schemes has also been demonstrated through several examples.

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Some geometric operations on binary pictures and their shape preserving properties

Pattern Recognition Letters ◽

10.1016/0167-8655(90)90045-4 ◽

1990 ◽

Vol 11 (5) ◽

pp. 355-361 ◽

Cited By ~ 1

Author(s):

Swapan K. Parui

Keyword(s):

Shape Preserving ◽

Shape Preserving Properties

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Lower Bounds on OBDD Proofs with Several Orders

ACM Transactions on Computational Logic ◽

10.1145/3468855 ◽

2021 ◽

Vol 22 (4) ◽

pp. 1-30

Author(s):

Sam Buss ◽

Dmitry Itsykson ◽

Alexander Knop ◽

Artur Riazanov ◽

Dmitry Sokolov

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Communication Complexity ◽

Multiparty Communication ◽

Constant Degree ◽

Polynomial Size ◽

Explicit Constant ◽

Open Question ◽

System 1 ◽

Exponential Size

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

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On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

Axioms ◽

10.3390/axioms9020061 ◽

2020 ◽

Vol 9 (2) ◽

pp. 61

Author(s):

Francesca Pitolli

Keyword(s):

Fractional Derivative ◽

Boundary Value Problems ◽

Numerical Experiments ◽

Boundary Value ◽

Bernstein Operator ◽

Explicit Formulas ◽

Shape Preserving ◽

Engineering Control ◽

Shape Preserving Properties ◽

Fractional Boundary Value Problems

Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate.

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

International Journal of Mathematical Analysis ◽

10.12988/ijma.2013.37188 ◽

2013 ◽

Vol 7 ◽

pp. 2491-2502

Author(s):

Sergei Sidorov

Keyword(s):

Positive Measure ◽

Order Of Approximation ◽

Shape Preserving ◽

Linear Shape

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Voronovskaja’s theorem, shape preserving properties and iterations for complex q-Bernstein polynomials

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2010.1156 ◽

2011 ◽

Vol 48 (1) ◽

pp. 23-43 ◽

Cited By ~ 3

Author(s):

Sorin Gal

Keyword(s):

Convergence Theorem ◽

Analytic Functions ◽

Quantitative Estimate ◽

Bernstein Polynomials ◽

Exact Order ◽

Approximation Properties ◽

Shape Preserving ◽

Voronovskaja’S Theorem ◽

Shape Preserving Properties ◽

Compact Disks

In this paper, first we prove Voronovskaja’s convergence theorem for complex q-Bernstein polynomials, 0 < q < 1, attached to analytic functions in compact disks in ℂ centered at origin, with quantitative estimate of this convergence. As an application, we obtain the exact order in approximation of analytic functions by the complex q-Bernstein polynomials on compact disks. Finally, we study the approximation properties of their iterates for any q > 0 and we prove that the complex qn-Bernstein polynomials with 0 < qn < 1 and qn → 1, preserve in the unit disk (beginning with an index) the starlikeness, convexity and spiral-likeness.

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