Some Applications of New Modifiedq-Integral Type Operators

Abstract

We introduce a new sequence of -integral operators. We show that it is a weighted approximation process in the polynomial space of continuous functions defined on unit interval. Weighted statistical approximation theorem, Korovkin type theorems for fuzzy continuous functions, and an estimate for the rate of convergence for these operators. IntroductionThe study of -Calculus is a generalization of any subjects, such as hyper geometric series, complex analysis, and particle physics. Currently it continues being an important subject of study. It has been shown that positive linear operators constructed by -numbers are quite effective as far as the rate of convergence is concerned and we can have some unexpected results, which are not observed for classical case. In the last decade, some new generalizations of wellknown positive linear operators, based on -integers, were introduced and studied by several authors. For example, -Meyer Konig and Zeller operators were studied by Trif [1], Dogru and Duman [2], Aral and Gupta [3], and so forth. In 20011, Aral and Gupta [4,5] introduced a -generalization of the classical Baskakov operators. In 2012, Sharma [6,7] introduced the -Durrmeyer type operators. Orkcu and Dogru [8] introduced Kantorovich type generalization ofSzasz-Mirakjan operators and discussed their -statistical approximation properties. In this paper motivated by Sharma we introduced a -analogue of the -Durrmeyer operators and we study better rate of convergence and statistical approximation properties.We mention some important definitions of -Calculus.Definition 1. For any fixed real numbers > 0 and ∈ , the -integers are defined byIn this way for real number one may writeDefinition 2. The -factorial is defined byDefinition 3. For any number ∈ (0, ), the -binomial coefficient is defined byAral and Gupta [4] introduced a -generalization of the classical Baskakov operators. For all ∈ [0, ∞), ∈ (0, 1), and each positive integer , the operators are defined as whereand they established some approximation results on it. Sharma [6] introduced the following -Durrmeyer type operators defined as follows: forwhereand they established some approximation results on it. In this paper motivated by Sharma [6,7,[10][11][12] we introduce a -analogue of the -Baskakov-Durrmeyer type operators defined as follows: for ∈ , ,where we set , ( ; ) = ( + −1 ) (Kasana et al.[13] obtained a sequence of modified Szâsz type operators for integrable function on [0, ∞) defined aswhere and belong to [0, ∞) and is fixed.In this paper, motivated by Kasana and Sharma, we introduce a -analogue of the -Baskakov-Durrmeyer type operators defined as follows: for ∈ , ,where and belong to , and is fixed. The aim of this paper is to study some approximation properties of a new generalization of operators based onintegers. We estimate moments for these operators. Also, we study statistical convergence and Korovkin type theorems for fuzzy continuous functions. Finally, we give better error estimations for operators (10) and (12). Estimation of MomentsWe ...

Some new applications of modified q-integral operators

“…and established some approximation results on it. Very recently we motivated by H. Sharma [4],and N. Deo [7], [8], [9] and introduced a q-analogue of the Baskakov-Durrmeyer type operators [11] defined as:…”

“…Very recently we published a paper based on q-Baskakov-Durrmeyer type operators [11]. In the present paper motivated by H. Sharma we investigate some new applications of q-analogue of the Baskakov-Durrmeyer type and we study better rate of convergence.…”

Some new applications of integrally modified q-Szasz-Mirakyan operators

“…Very recently we published a paper based on q-Durrmeyer type operators [12] . In this paper motivated by H. Sharma we introduced a q-analogue of the Szasz-Mirakyan Operators and we study better rate of convergence.…”