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

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.

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

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.

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

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.

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

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.

A new metric between distributions of point processes

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.

A new metric between distributions of point processes

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 d̅1 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 d̅1 and its induced Wasserstein metric d̅2 for point process distributions are given, including examples of useful d̅1-Lipschitz continuous functions, d̅2 upper bounds for the Poisson process approximation, and d̅2 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 d̅1 in applications.

Quasimonotone embedding of one-sided Lipschitz continuous functions

We prove that one-sided Lipschitz continuous functions can be embedded into quasimonotone increasing functions, in a quite natural way. As a consequence of this result, quasimontonicity methods can be used to study initial and boundary value problems in Banach spaces if one-sided Lipschitz continuous functions are involved.