Global Smoothness Preservation Property Research Articles

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.

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.

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.

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.

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.

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.

Fuzzy approximation by fuzzy convolution type operators

Here we introduce and study four sequences of naturally arising fuzzy integral operatorsof convolution type that are integral analogs of known fuzzy wavelet type operators, defined via a scaling function. Their fuzzy convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity. Also, their high-order fuzzy approximation is given similarly by involving the fuzzy modulus of continuity of the N th order ( N ≥ 1) H-fuzzy derivative of the engaged fuzzy number valued function. The fuzzy global smoothness preservation property of these operators is demonstrated also.