Generalized q-Baskakov operators

2011 ◽  
Vol 61 (4) ◽  
Author(s):  
Ali Aral ◽  
Vijay Gupta
Keyword(s):  
Rate Of Convergence ◽  
Weighted Norm ◽  
Shape Preserving ◽  
Baskakov Operators ◽  
Shape Preserving Properties

AbstractIn the present paper we propose a generalization of the Baskakov operators, based on q integers. We also estimate the rate of convergence in the weighted norm. In the last section, we study some shape preserving properties and the property of monotonicity of q-Baskakov operators.

scholarly journals Baskakov-Kantorovich operators reproducing affine functions

2021 ◽  
Vol 66 (4) ◽  
pp. 739-756
Author(s):  
Jorge Bustamante ◽  
Keyword(s):  
Rate Of Convergence ◽  
Weighted Spaces ◽  
Weighted Norm ◽  
Baskakov Operators ◽  
Affine Functions ◽  
Upper Estimate ◽  
Kantorovich Operators

We present a new Kantorovich modi cation of Baskakov operators which reproduce a ne functions. We present an upper estimate for the rate of convergence of the new operators in polynomial weighted spaces and characterize all functions for which there is convergence in the weighted norm.

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

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

Axioms ◽  
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.

Rate of Convergence of Modified Baskakov Operators

1997 ◽  
Vol 30 (2) ◽  
pp. 339-346
Author(s):  
Vijay Gupta ◽  
D. Kumar
Keyword(s):  
Rate Of Convergence ◽  
Baskakov Operators

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

2011 ◽  
Vol 48 (1) ◽  
pp. 23-43 ◽  
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.

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

Filomat ◽  
2010 ◽  
Vol 24 (3) ◽  
pp. 55-72 ◽  
Author(s):  
Barnabás Bede ◽  
Lucian Coroianu ◽  
Sorin Gal
Keyword(s):  
Nonlinear Operator ◽  
Approximation Error ◽  
Pointwise Estimate ◽  
Approximation Order ◽  
Order Of Approximation ◽  
Bernstein Type ◽  
Shape Preserving ◽  
Explicit Constant ◽  
Open Question ◽  
Shape Preserving Properties

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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