Auswahl der wissenschaftlichen Literatur zum Thema „ISMAIL-MAY OPERATORS“

We introduce here a modification of the Ismail-May operators, preserving affine function and estimate the order of approximation with the help of classical approach viz. the second order modulus of continuity, and the Peetre?s K-functional. Further, we provide the convergence estimates for the differences of Ismail-May operators and its Kantorovich variants. In the end, the convergence of the operators have been depicted through illustrative graphics.

In the present paper, we introduce a modification of Ismail-May operators having weights of Baskakov basis functions. We estimate weighted Korovkin's theorem and difference estimates between two operators and establish a Voronovskaja type asymptotic formula in simultaneous approximation.

In this article, we deal with the approximation properties of Ismail-May operators [16] based on a non-negative real parameter ?. We provide some graphs and error estimation table for a numerical example depicting the convergence of our proposed operators. We further define the B?zier variant of these operators and give a direct approximation theorem using Ditizan-Totik modulus of smoothness and a Voronovoskaya type asymptotic theorem. We also study the error in approximation of functions having derivatives of bounded variation. Lastly, we introduce the bivariate generalization of Ismail May operators and estimate its rate of convergence for functions of Lipschitz class.

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth.

AbstractIn the theory of approximation, linear operators play an important role. The exponential-type operators were introduced four decades ago, since then no new exponential-type operator was introduced by researchers, although several generalizations of existing exponential-type operators were proposed and studied. Very recently, the concept of semi-exponential operators was introduced and few semi-exponential operators were captured from the exponential-type operators. It is more difficult to obtain semi-exponential operators than the corresponding exponential-type operators. In this paper, we extend the studies and define semi-exponential Bernstein, semi-exponential Baskakov operators, semi-exponential Ismail–May operators related to $$2x^{3/2}$$ 2 x 3 / 2 or $$x^{3}$$ x 3 . Furthermore, we present a new derivation for the semi-exponential Post–Widder operators. In some examples, open problems are indicated.

AbstractIn the year 1978, Ismail and May studied operators of exponential type and proposed some new operators which are connected with a certain characteristic function $$p\left( x\right) $$ p x . Several of these operators were not separately studied by researchers due to its unusual behavior. The topic of the present paper is the local rate of approximation of a sequence of exponential type operators $$R_{n}$$ R n belonging to $$p\left( x\right) =x\left( 1+x\right) ^{2}$$ p x = x 1 + x 2 . As the main result we derive a complete asymptotic expansion for the sequence $$\left( R_{n}f\right) \left( x\right) $$ R n f x as n tends to infinity.

This thesis is mainly a study of convergence estimates of various approximation opera tors. Approximation theory is indeed an old topic in mathematical analysis that remains an appealing field of study with several applications. The findings presented here are related to the approximation of specific classes of linear positive operators. The introduc tory chapter is a collection of relevant definitions and literature of concepts that are used throughout this thesis. The second chapter is based on approximation of certain exponential type opera tors. The first section of this chapter presents the study of convergence estimates of Kan torovich variant of Ismail-May operators. Further, a two variable generalisation of the proposed operators is also discussed. The second section is dedicated to a modification of Ismail-May exponential type operators which preserve functions of exponential growth. The modified operators in general are not of exponential type. In chapter three, we present a Durrmeyer type construction involving a class of or thogonal polynomials called Apostol-Genocchi polynomials and Palt ˇ anea operators with ˇ real parameters α, λ and ρ. We establish approximation estimates such as a global approx imation theorem and rate of approximation in terms of usual, r−th and weighted modulus of continuity. We further study asymptotic formulae such as Voronovskaya theorem and quantitative Voronovskaya theorem. The rate of convergence of the proposed operators for the functions whose derivatives are of bounded variation is also presented. Inspired by the King’s approach, chapter four deals with the preservation of func tions of the form t s , s ∈ N ∪ {0}. Followed by some useful lemmas, we determine the rate of convergence of the proposed operators in terms of usual modulus of continuity and Peetre’s K- functional. Further, the degree of approximation is also established for the function of bounded variation. We also illustrate via figures and tables that the proposed modification provides better approximation for preservation of test function e3. In chapter five, we consider a Kantorovich variant of the operators proposed by Gupta and Holhos (68) using arbitrary sequences which preserves the exponential func tions of the form a −x . It is shown that the order of approximation can be made better xi xii Abstract with appropriate choice of sequences with certain conditions. We therefore provide nec essary moments and central moments and some useful lemmas. Further, we present a quantitative asymptotic formula and estimate the error in approximation. Graphical rep resentations are provided in the end with different choices of sequences satisfying the given conditions. The last chapter summarizes the thesis with a brief conclusion and also discusses the future prospects of this thesis.