Global Smoothness Preservation Research Articles

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.

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.

Global smoothness preservation by bivariate interpolation operators

Extending the results of [4] in the univariate case, in this paper we prove that the bivariate interpolation polynomials of Hermite-Fejer based on the Chebyshev nodes of the first kind, those of Lagrange based on the Chebyshev nodes of second kind and ±1, and those of bivariate Shepard operators, have the property of partial preservation of global smoothness, with respect to various bivariate moduli of continuity.

On global smoothness preservation in complex approximation

On global smoothness preservation in complex approximation

Convolution-Type Integral Operators in Complex Approximation

In this paper we study classes of convolution-type complex operators for analytic functions on the open unit disk D which are continuous on \(\bar D\). Their simple forms present three advantages: firstly, estimates with rates in terms of higher moduli of smoothness (or of best approximation) and explicit constants can easily be obtained, secondly we can prove global smoothness preservation properties with (best) constants for them, and thirdly, some shape preserving properties hold.

On Global Smoothness Preservation by Bernstein Operators

We investigate the global smoothness preservation by Bernstein operators measured by second-order modulus of smoothness and give a partial answer for a conjecture raised by H. H. Gonska in 1998.

A Generalization of Bernstein–Kantorovič Operators

In this paper, we use a probabilistic setting to introduce a double sequence (L〈k〉n) of linear polynomial operators which includes, as particular cases, the classical Bernstein operators, the Kantorovič operators, and the operators recently introduced by Cao. For these operators, we discuss several approximation properties. In particular, we deal with the convergence properties according to the way in which the different parameters vary, and the preservation of global smoothness and classes of functions determined by concave moduli of continuity. A remarkable feature of our approach is that if f is differentiable, the approximation properties of both L〈k〉nf and its derivatives can be discussed simultaneously. Throughout the paper, probabilistic methods play an important role.

Global smoothness preservation by multivariate singular integrals

By using various kinds of moduli of smoothness, it is established that the multivariate variants of the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson-type generalisations satisfy the “global smoothness preservation” property. The results are extensions of those proved by the authors for the univariate case.

Best Constants in Global Smoothness Preservation Inequalities for Some Multivariate Operators

The Bernstein operator on the standardk-simplex and other analogousk-variate operators allow for a probabilistic representation in terms of the successive increments of a real valued superstationary stochastic process (a notion introduced in the paper) starting at the origin and having nondecreasing paths. For this class of operators, we obtain estimates of the best constants in preservation of the first modulus of continuity corresponding to thel1-norm, and in preservation of classes of functions defined by concave moduli of continuity. We also show that, in some special cases, such best constants do not depend upon the dimensionk. To show our results, we use probabilistic tools such as couplings and Wasserstein distances for multivariate probability distributions. The general results are applied to the computation of the aforementioned constants for several classical multivariate operators.

Global smoothness preservation and the variation-diminishing property

Global smoothness preservation and the variation-diminishing property

General Theory of Global Smoothness Preservation by Singular Integrals, Univariate Case

In this article it is established that the well-known singular integrals of Picard, Poisson–Cauchy, Gauss–Weierstrass, and their Jackson-type generalizations fulfill the “global smoothness preservation” property, i.e., they “ripple” less than the function they are applied on, that is, producing a nice and fit approximation to the unit. The related results are established over various spaces of functions and the associated inequalities involve different types of corresponding moduli of smoothness. Several times these inequalities are proved to be sharp, namely, they are attained.

Some Shift-Invariant Integral Operators, Univariate Case, Revisited

In recent articles the first author and H. Gonska [e.g., see G. Anastassiou, C. Cottin, and H. Gonska, Global smoothness of approximating functions, Analysis, 11, 43–57 (1991); G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] studied global smoothness preservation by some univariate and multivariate linear operators over compact domains and ℝ n , n ≥ 1. In particular, they studied a very general positive linear integral type operator [e.g., see G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)] over ℝ n that was introduced through a convolution-like integration of another general positive linear operator with a scaling-type function. In this article the authors, among others, extend and generalize [G. Anastassiou and H. Gonska, On some shift-invariant integral operators, univariate case, Ann. Pol. Math. LXI.3, 225–243 (1995)]. Also certain new similar but more general integral operators are introduced and studied. These operators arise in a natural way, and for all these sufficient conditions are given for shift invariance, preservation of higher-order global smoothness and sharpness of the related inequalities, convergence to the unit using the first modulus of continuity, shape preservation, and preservation of continuous probabilistic distribution functions. Several examples of very general specialized operators, old and new, are given that satisfy all the above properties.

Differentiated shift-invariant integral operators, univariate case

In a recent paper [1] the author, along with H. Gonska, introduced some wavelet type integral operators over the whole real line and studied their properties such as shift-invariance, global smoothness preservation, convergence to the unit, and preservation of probability distribution functions. These operators are very general and they are introduced through a convolution-like iteration of another general operator with a scaling type function. In this paper the author provides sufficient conditions, so that the derivatives of the above operators enjoy the same nice properties as their originals. A sufficient condition is also given so that the “global smoothness preservation” related inequality becomes sharp. At the end several applications are given, where the derivatives of the very general specialized operators are shown to fulfill all the above properties. In particular it is shown that they preserve continuous probability density functions.

Global smoothness preservation by singular integrals

Here is established that the well-known singular integrals of Picard, Poisson-Cauchy and Gauss- Weierstrass fulfill the "global smoothness preservation" property. J. e., they "ripple" less than the function they are applied on, that is producing a nice approximation to the unit. The associated inequalities are sharp.

Bernstein-Durrmeyer operators

In this paper, we consider the Durrmeyer-type modifications of the classical Bernstein, Szász and Baskakov operators, and deal with two different kind of problems. Firstly, we obtain new results concerning preservation of shape properties, Lipschitz constants and global smoothness, as well as monotonic convergence under convexity. To do this, we use a probabilistic approach based on representations of these operators in terms of stochastic processes having a.s. nondecreasing paths and satisfying a suitable martingale-type condition. Secondly, we show that the Szász-Durrmeyer operator is the limit, in an appropriate sense, of both the Bernstein-Durrmeyer and the Baskakov-Durrmeyer operators. We provide rates of convergence which are derived from the bounds for the total variation distance between the probability measures involved.

On a Problem of Gonska

This paper investigates global smoothness preservation by the Bernstein operators. When the smoothness is measured by the modulus of continuity, this problem is well understood. When it is measured by the second order modulus of smoothness, H. Gonska conjectured that the Lipschitz classes of second order keep invariate under the Bernstein operators. Here we present a counterexample to this conjecture. Then we introduce a new modulus of smoothness and show that the Lip-α(0 < α ≤ 1) classes measured by this modulus are invariate under the Bernstein operators. By means of this modulus we also improve some previous results concerning global smoothness preservation.

On some shift invariant integral operators, univariate case

In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convoluti