Localization results for generalized Baskakov/Mastroianni and composite operators

In this paper we study localization results for classical sequences of linear positive operators that are particular cases of the generalized Baskakov/Mastroianni operators and also for certain class of composite operators that can be derived from them by means of a suitable transformation. Amongst these composite operators we can find classical sequences like the Meyer-König and Zeller operators and the Bleimann, Butzer and Hahn ones. We extend in different senses the traditional form of the localization results that we find in the classical literature and we show several examples of sequences with different behavior to this respect.     

Pointwise approximation by Bézier variant of integrated MKZ operators

In this paper the pointwise approximation of Bézier variant of integrated MKZ operators for general bounded functions is studied. Two estimate formulas of this type approximation are obtained. The approximation of functions of bounded variation becomes a special case of the main result of this paper. In the case of functions of bounded variation, Theorem B of the paper corrects the mistake of Theorem 1 of the article [V. Gupta, Degree of approximation to functions of bounded variation by Bézier variant of MKZ operators, J. Math. Anal. Appl. 289 (2004) 292-300].     

On the iterates of positive linear operators preserving the affine functions

In this note we study the limit behavior of the iterates of a large class of positive linear operators preserving the affine functions and, as a byproduct of our result, we obtain the limit of the iterates of Meyer-König and Zeller operators.     

Strong converse inequalities for Meyer-König and Zeller operators

In this paper we give the strong converse inequality of type B for Meyer-König and Zeller operators.     

The generalization of bivariate MKZ operators by multiple generating functions

In the present paper, we study approximation properties of multiple generating functions type bivariate Meyer-König and Zeller (MKZ) operators with the help of Volkov type theorem. We compute the order of convergence of these operators by means of modulus of continuity and the elements of modified Lipschitz class. Finally, we give application to partial differential equations.     

Approximation by generalized MKZ-operators in polynomial weighted spaces

We prove some approximation properties of generalized Meyer-Konig and Zeller operators for differentiable functions in polynomial weighted spaces. The results extend some results proved in [1-3,7-16].     

The generalization of Meyer-König and Zeller operators by generating functions

In this paper, theorems are proved concerned with some approximation properties of generating functions type Meyer-König and Zeller operators with the help of a Korovkin type theorem. Secondly, we compute the rates of convergence of these operators by means of the modulus of continuity, Peetre's K-functional and the elements of the modified Lipschitz class. Also we introduce the rth order generalization of these operators and we obtain approximation properties of them. In the last part, we give some applications to the differential equations.     

Properties of convergence for the q-Meyer-König and Zeller operators

In this paper, we discuss properties of convergence for the q-Meyer-König and Zeller operators Mn,q. Based on an explicit expression for Mn,q(t²,x) in terms of q-hypergeometric series, we show that for qn∈(0,1], the sequence (Mn,qn(f))_n?1 converges to f uniformly on [0,1] for each f∈C[0,1] if and only if limn→∞qn=1. For fixed q∈(0,1), we prove that the sequence (Mn,q(f)) converges for each f∈C[0,1] and obtain the estimates for the rate of convergence of (Mn,q(f)) by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions. We also give explicit formulas of Voronovskaya type for the q-Meyer-König and Zeller operators for fixed 0<q<1. If 0<q<1, f∈C¹[0,1], we show that the rate of convergence for the Meyer-König and Zeller operators is o(qn) if and only if

     

Parameter convexity and concavity of generalized trigonometric functions

We study the convexity properties of the generalized trigonometric functions viewed as functions of the parameter. We show that p→_sinp?(y)$p\to {\sin }_p?(y)$ and p→_cosp?(y)$p\to {\cos }_p?(y)$ are log-concave on the appropriate intervals while p→_tanp?(y)$p\to {\tan }_p?(y)$ is log-convex. We also prove similar facts about the generalized hyperbolic functions. In particular, our results settle a major part of the conjecture recently put forward in [4].     

On uniform approximation by some classical Bernstein-type operators

We investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval. The discussed examples include the Szász operators, the Szász-Durrmeyer operators, the gamma operators, the Baskakov operators, and the Meyer-König and Zeller operators. We show that some results of Totik remain valid for unbounded functions, at the same time that we give simple rates of convergence in terms of the usual modulus of continuity. We also show by a counterexample that the result for Meyer-König and Zeller operators does not extend to Cheney and Sharma operators.     

The generalized Roper-Suffridge extension operator for some biholomorphic mappings

Suppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C. Let , where 1?p₁?2 (or 0<p₁?2), pj?1, j=2,…,n, are real numbers. In this paper, we prove that

     

More on convexity and smoothness of operators

Let X and Y be Banach spaces and T:Y→X be a bounded operator. In this note, we show first some operator versions of the dual relation between q-convexity and p-smoothness of Banach spaces case. Making use of them, we prove then the main result of this note that the two notions of uniform q-convexity and uniform p-smoothness of an operator T introduced by J. Wenzel are actually equivalent to that the corresponding T-modulus δT of convexity and the T-modulus ρT of smoothness introduced by G. Pisier are of power type q and of power type p, respectively. This is also an operator version of a combination of a Hoffman's theorem and a Figiel-Pisier's theorem. As their application, we show finally that a recent theorem of J. Borwein, A.J. Guirao, P. Hajek and J. Vanderwerff about q-convexity of Banach spaces is again valid for q-convexity of operators.     

K-moment problem for recursive sequences of order ∞: Application to the zeros of analytic functions

In this paper we solve the K-moment problem for a family of sequences defined by linear recursive relations of order ∞ indexed by Z. Extension of some known properties are studied and others results are established. Moreover, we paraphrase a fundamental application on the existence of zeros of an analytic function, in terms of the support of the representing measure of the former sequences.     

Schrödinger operators with matrix potentials. Transition from the absolutely continuous to the singular spectrum

It is proven that the absolutely continuous spectrum of matrix Schrödinger operators coincides (with the multiplicity taken into account) with the spectrum of the unperturbed operator if the (matrix) potential is square integrable. The same result is also proven for some classes of slower decaying potentials if they are smooth.     

Completely monotonic functions of positive order and asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function

We introduce completely monotonic functions of order r>0 and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any positive integer order.     

Factoring and recognition of read-once functions using cographs and normality and the readability of functions associated with partial k-trees

An approach for factoring general boolean functions was described in Golumbic and Mintz [Factoring logic functions using graph partitioning, in: Proceedings of IEEE/ACM International Conference on Computer Aided Design, November 1999, pp. 195-198] and Mintz and Golumbic [Factoring Boolean functions using graph partitioning, Discrete Appl. Math. 149 (2005) 131-153] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring read-once functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality.For non-read-once functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial k-tree, then F is a read 2^k function and a read 2^k formula for F can be obtained in polynomial time.     

Derivative polynomials and closed-form higher derivative formulae

In a recent paper, Adamchik [1] expressed in a closed-form symbolic derivatives of four functions belonging to the class of functions whose derivatives are polynomials in terms of the same functions. In this sequel, simple closed-form higher derivative formulae which involve the Carlitz-Scoville higher order tangent and secant numbers are derived for eight trigonometric and hyperbolic functions.     

On the precise growth, covering, and distortion theorems for normalized biholomorphic mappings

In this paper, we consider a normalized biholomorphic mapping f(x) defined on the unit ball in a complex Banach space, where the origin 0 is a zero of order k+1 of f(x)−x. The precise growth and covering theorem for f(x) is obtained when f(x) is a starlike mapping or a starlike mapping of order α. Especially, the precise growth and covering theorem for f(x) is also established when f(x) is a quasi-convex mapping. Moreover, the precise distortion theorem for f(x) is given when f(x) is a convex mapping. Our result includes many known results.     

On the complete monotonicity of the waiting time density in some GI/G/k systems

In this note the complete monotonicity of the waiting time density in GI/G/k queues is proved under the assumption that the service time density is completely monotone. This is an extension of Keilson's [3] result for M/G/1 queues. We also provide another proof of the result that complete monotonicity is preserved by geometric compounding.