scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

scholarly journals On the Convergence of a Family of Chlodowsky Type Bernstein-Stancu-Schurer Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Lian-Ta Shu ◽  
Guorong Zhou ◽  
Qing-Bo Cai
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Continuous Functions ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Approximation Theorems ◽  
New Family ◽  
Weighted Modulus

We construct a new family of univariate Chlodowsky type Bernstein-Stancu-Schurer operators and bivariate tensor product form. We obtain the estimates of moments and central moments of these operators, obtain weighted approximation theorem, establish local approximation theorems by the usual and the second order modulus of continuity, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. For the bivariate case, we give the rate of convergence by using the weighted modulus of continuity. We also give some graphs and numerical examples to illustrate the convergent properties of these operators to certain functions and show that the new ones have a better approximation to functions f for one dimension.

scholarly journals On Durrmeyer Type λ -Bernstein Operators via ( p , q )-Calculus

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Qing-Bo Cai ◽  
Guorong Zhou
Keyword(s):  
Convergence Theorem ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Numerical Examples ◽  
Korovkin Type Approximation Theorem ◽  
Lipschitz Continuous ◽  
Type Approximation ◽  
Second Moments

In the present paper, Durrmeyer type λ -Bernstein operators via ( p , q )-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in some λ cases than Durrmeyer type ( p , q )-Bernstein operators.

scholarly journals Shape-Preserving and Convergence Properties for theq-Szász-Mirakjan Operators for Fixedq∈(0,1)

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Heping Wang ◽  
Fagui Pu ◽  
Kai Wang
Keyword(s):  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operator ◽  
Convergence Properties ◽  
Shape Preserving ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Good Shape ◽  
Variation Diminishing ◽  
Shape Preserving Properties

We introduce aq-generalization of Szász-Mirakjan operatorsSn,qand discuss their properties for fixedq∈(0,1). We show that theq-Szász-Mirakjan operatorsSn,qhave good shape-preserving properties. For example,Sn,qare variation-diminishing, and preserve monotonicity, convexity, and concave modulus of continuity. For fixedq∈(0,1), we prove that the sequence{Sn,qf}converges toB∞,q(f)uniformly on[0,1]for eachf∈C[0, 1/(1-q)], whereB∞,qis the limitq-Bernstein operator. We obtain the estimates for the rate of convergence for{Sn,qf}by the modulus of continuity off, and the estimates are sharp in the sense of order for Lipschitz continuous functions.

scholarly journals A new metric between distributions of point processes

2008 ◽  
Vol 40 (03) ◽  
pp. 651-672 ◽  
Author(s):  
Dominic Schuhmacher ◽  
Aihua Xia
Keyword(s):  
Point Processes ◽  
Relative Difference ◽  
Continuous Functions ◽  
Upper And Lower Bounds ◽  
Point Pattern ◽  
Multiple Point ◽  
Finite Point ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Comprehensive Collection

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅ 1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅ 1 and its induced Wasserstein metric d̅ 2 for point process distributions are given, including examples of useful d̅ 1-Lipschitz continuous functions, d̅ 2 upper bounds for the Poisson process approximation, and d̅ 2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅ 1 in applications.

scholarly journals Imbeddings of Brezis-Wainger type. The case of missing derivatives

2001 ◽  
Vol 131 (3) ◽  
pp. 667-700
Author(s):  
M. Krbec ◽  
Hans-Jürgen Schmeisser
Keyword(s):  
Continuous Functions ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Mixed Derivatives

We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

scholarly journals Approximation properties of the new type generalized Bernstein-Kantorovich operators

2021 ◽  
Vol 7 (3) ◽  
pp. 3826-3844
Author(s):  
Mustafa Kara ◽  
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Lipschitz Function ◽  
Type Theorem ◽  
Numerical Results ◽  
Bernstein Operators ◽  
Approximation Properties ◽  
Voronovskaya Type Theorem ◽  
New Type ◽  
Kantorovich Operators

<abstract><p>In this paper, we introduce new type of generalized Kantorovich variant of $ \alpha $-Bernstein operators and study their approximation properties. We obtain estimates of rate of convergence involving first and second order modulus of continuity and Lipschitz function are studied for these operators. Furthermore, we establish Voronovskaya type theorem of these operators. The last section is devoted to bivariate new type $ \alpha $-Bernstein-Kantorovich operators and their approximation behaviors. Also, some graphical illustrations and numerical results are provided.</p></abstract>

scholarly journals Clarke critical values of subanalytic Lipschitz continuous functions

2005 ◽  
Vol 87 ◽  
pp. 13-25 ◽  
Author(s):  
Jérôme Bolte ◽  
Aris Daniilidis ◽  
Adrian Lewis ◽  
Masahiro Shiota
Keyword(s):  
Continuous Functions ◽  
Critical Values ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous
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