Global Smoothness Preservation Research Articles

The second order modulus revisited: remarks, applications, problems

Several questions concerning the second order modulus of smoothness are addressed in this note. The central part is a refined analysis of a construction of certain smooth functions by Zhuk and its application to several problems in approximation theory, such as degree of approximation and the preservation of global smoothness. Lower bounds for some optimal constants introduced by Sendov are given as well. We also investigate an alternative approach using quadratic splines studied by Sendov.

On a new approach in the space of measurable functions

In this paper, we present a new modulus of continuity for locally integrable function spaces which is effected by the natural structure of the L_{p} space. After basic properties of it are expressed, we provide a quantitative type theorem for the rate of convergence of convolution type integral operators and iterates of them. Moreover, we state their global smoothness preservation property including the new modulus of continuity. Finally, the obtained results are performed to the Gauss-Weierstrass operators.

Approximation by multivariate generalized Poisson–Cauchy type singular integral operators

This research and survey work deals exclusively with the study of the approximation of generalized multivariate Poisson–Cauchy type singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. These operators are not in general positive linear operators. In particular we study the rate of convergence of these integral operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these integral operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of approximation. The above properties are studied with respect to $$L_{p}$$ norm, $$1\le p\le \infty $$ .

Complete Approximations by Multivariate Generalized Gauss-Weierstrass Singular Integrals

Abstract This research and survey article deals exclusively with the study of the approximation of generalized multivariate Gauss-Weierstrass singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. The multivariate generalized Gauss-Weierstrass operators are not in general positive linear operators. In particular we study the rate of convergence of these operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of multivariate approximation. The above properties are studied with respect to Lp norm, 1 ≤ p ≤ ∞.

A note on the modified Picard integral operators

This study is a natural continuation of modified Picard operators, defined by Agratini et al, preserving an exponential function. Herein, we first show that these operators are approximation processes in the setting of large classes of weighted spaces. Then, we obtain weighted uniform convergence of the operators via exponential weighted modulus of smoothness. Finally, we give, by using the weighted modulus of continuity, the result regarding global smoothness preservation properties for the generalized Picard operators, which based in Agratini et al.

The Picard and Gauss–Weierstrass Singular Integrals in (p, q)-Calculus

The vast development of the techniques in both the quantum calculus and the post-quantum calculus leads to a significant increase in activities in approximation theory due to applications in computational science and engineering. Herein, we introduce (p, q)-Picard and (p, q)-Gauss–Weierstrass integral operators in terms of the (p, q)-Gamma integral. We give a general formula for the monomials under both (p, q)-Picard and (p, q)-Gauss–Weierstrass operators as well as some special cases. We discuss the uniform convergence properties of them. We show that both operators have optimal global smoothness preservation property via usual modulus of continuity. Finally, we establish the rate of approximation using the weighted modulus of smoothness. Depending on the choices of parameters p and q in the integrals, we are able to obtain better error estimation than classical ones.

Quantitative approximation by shift invariant multivariate sublinear-Choquet operators

Abstract A very general multivariate positive sublinear Choquet integral type operator is given through a convolution-like iteration of another multivariate general positive sublinear operator with a multivariate scaling type function. For it, sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Furthermore, two examples of very general multivariate specialized operators are presented fulfilling all the above properties; the higher order of multivariate approximation of these operators is also studied.

Approximation by Shift Invariant Univariate Sublinear-Shilkret Operators

A very general positive sublinear Shilkret integral type operator is given through a convolution-like iteration of another general positive sublinear operator with a scaling type function.For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates. Additionally, two examples of very general specialized operators are presented fulfilling all the above properties, the higher order of approximation of these operators is also considered.

Global smoothness and approximation by generalized discrete singular operators

In this article we continue with the study of generalized discrete singular operators over the real line regarding their simultaneous global smoothness preservation property with respect to \(L_{p}\) norm for \(
 1\leq p\leq \infty ,\) by involving higher order moduli of smoothness.
 Additionally we study their simultaneous approximation to the unit operator with rates involving the modulus of smoothness. The Jackson type inequalities produced in this article are almost sharp, containing neat 
 constants, and they reflect the high order of differentiability of involved
 function.

Global smoothness preservation with second order modulus of smoothness

Abstract We establish the global smoothness preservation of a function f by the sequence of linear positive operators. Our estimate is in terms of the second order Ditzian-Totik modulus of smoothness. Application is given to the Bernstein operator.

General theory of global smoothness and approximation by smooth singular operators

In this article we continue with the study of smooth general singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞ , by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of smoothness. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. We finish with applications to trigonometric singular integral operators.

Global smoothness preservation and simultaneous approximation for multivariate general singular integral operators

In this article we continue with the study of multivariate smooth general singular integral operators over R N , N ≥ 1 , regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞ , by involving multivariate higher order moduli of smoothness. Also we study their multivariate simultaneous approximation to the unit operator with rates. The multivariate Jackson type inequalities obtained are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. In the uniform case of global smoothness we prove optimality. At the end we list the multivariate Picard, Gauss–Weierstrass, Poisson–Cauchy and Trigonometric singular integral operators as applicators of our general theory.

Global smoothness and uniform convergence of smooth Poisson–Cauchy type singular operators

In this article we introduce the smooth Poisson–Cauchy type singular integral operators over the real line. Here we study their simultaneous global smoothness preservation property with respect to the L p norm, 1 ⩽ p ⩽ ∞ , by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.

General King-Type Operators

We introduce sequences of operators Vn of King’s type and study them in regard to uniform convergence, global smoothness preservation, the approximation of decreasing and convex functions, the validity of a Voronovskaja- type theorem and a recursion formula generalizing a corresponding result for the classical Bernstein operators.

Global smoothness and uniform convergence of smooth Gauss–Weierstrass singular operators

In this article we continue with the study of smooth Gauss–Weierstrass singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞ , by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.

$q$-GENERALIZATIONS OF THE PICARD AND GAUSS-WEIERSTRASS SINGULAR INTEGRALS

Introducing a higher order modulus of smoothness based on $q$-integers, in this paper first we obtain Jackson-type estimates in approximation by Jackson-type generalizations of the $q$-Picard and $q$-Gauss-Weierstrass singular integrals and give their global smoothness preservation property with respect to the uniform norm. Then, we study approximation and geometric properties of the complex variants for these $q$ -singular integrals attached to analytic functions in compact disks. Finally, we prove approximation properties of these $q$-singular integrals attached to vector-valued functions.

Generalized Picard singular integrals

In this article, we introduce and study a new type of Picard singular integral operators on R n constructed by means of the concept of the nonisotropic β -distance and the q -exponential functions. The central role here is played by the concept of nonisotropic β -distance, which allows one to improve and generalize the results given for classical Picard and q -Picard singular integral operators. In order to obtain the rate of convergence we introduce a new type of modulus of continuity depending on the nonisotropic β -distance with respect to the uniform norm. Then we give the definition of β -Lebesque points depending on nonisotropic β -distance and a pointwise approximation result shown at these points. Furthermore, we study the global smoothness preservation property of these new type Picard singular integral operators and prove a sharp inequality.

$q$-GENERALIZATIONS OF THE PICARD AND GAUSS-WEIERSTRASS SINGULAR INTEGRALS

Introducing a higher order modulus of smoothness based on $q$-integers, in this paper first we obtain Jackson-type estimates in approximation by Jackson-type generalizations of the $q$-Picard and $q$-Gauss-Weierstrass singular integrals and give their global smoothness preservation property with respect to the uniform norm. Then, we study approximation and geometric properties of the complex variants for these $q$ -singular integrals attached to analytic functions in compact disks. Finally, we prove approximation properties of these $q$-singular integrals attached to vector-valued functions.

Best constants in preservation of global smoothness for Szász–Mirakyan operators

We obtain the best possible constants in preservation inequalities concerning the usual first modulus of continuity for the classical Szász–Mirakyan operator. The probabilistic representation of this operator in terms of the standard Poisson process is used.

Global smoothness and uniform convergence of smooth picard singular operators

In this article, we continue with the study of smooth Picard singular integral operators over the real line regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞, by involving higher-order moduli of smoothness. Also, we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp, containing elegant constants, and they reflect the high order of differentiability of the engaged function.