Dunkl generalization of Szász operators via q-calculus

İçöz, Gürhan; Çekіm, Bayram

doi:10.1186/s13660-015-0809-y

Cited by 40 publications

(36 citation statements)

References 25 publications

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“…We also prove several approximation results and successfully extend the results of [19]. Several other related results are also discussed.…”

Section: Introductionsupporting

confidence: 80%

“…Motivated essentially by Içöz and Çekim’s [19] recent investigation of Dunkl generalization of Szász-Mirakjan operators via q -calculus, we show that our modified operators have better error estimation than those in [19]. We also prove several approximation results and successfully extend the results of [19].…”

Section: Introductionsupporting

confidence: 68%

“…Here we have defined a Dunkl generalization of these modified operators. This type of modification enables better error estimation on the interval if compared to the classical Dunkl-Szász operators [19]. We obtained some approximation results via the well-known Korovkin-type theorem.…”

Section: Resultsmentioning

confidence: 99%

“…In [19], Içöz and Çekim gave the Dunkl generalization of Szász operators via q -calculus as follows: for , , and .…”

Section: Introductionmentioning

confidence: 99%

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On modified Dunkl generalization of Szász operators via q-calculus

2017

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The purpose of this paper is to introduce a modification of q-Dunkl generalization of exponential functions. These types of operators enable better error estimation on the interval than the classical ones. We obtain some approximation results via a well-known Korovkin-type theorem and a weighted Korovkin-type theorem. Further, we obtain the rate of convergence of the operators for functions belonging to the Lipschitz class.

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“…We also prove several approximation results and successfully extend the results of [19]. Several other related results are also discussed.…”

Section: Introductionsupporting

confidence: 80%

Section: Introductionsupporting

confidence: 68%

Section: Resultsmentioning

confidence: 99%

“…In [19], Içöz and Çekim gave the Dunkl generalization of Szász operators via q -calculus as follows: for , , and .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On modified Dunkl generalization of Szász operators via q-calculus

2017

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“…They gave definitions of q -Dunkl analogues of exponential functions, recursion relations and notations for and . For and , Gürhan Içöz gave a Dunkl generalization of Szász operators via q -calculus [12] as …”

Section: Introductionmentioning

confidence: 99%

A Dunkl type generalization of Szász operators via post-quantum calculus

2018

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The object of this paper to construct Dunkl type Szász operators via post-quantum calculus. We obtain some approximation results for these new operators and compute convergence of the operators by using the modulus of continuity. Furthermore, we obtain the rate of convergence of these operators for functions belonging to the Lipschitz class. We also study the bivariate version of these operators.

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Some approximation results involving the q‐Szász–Mirakjan–Kantorovich type operators via Dunkl's generalization

Srivástava

Mursaleen

Alotaibi

et al. 2017

Math Methods in App Sciences

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Communicated by W. SprößigThe purpose of this paper is to introduce a family of q-Szász-Mirakjan-Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well-known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K-functional, and the second-order modulus of continuity. Furthermore, we obtain the approximation results for bivariate q-Szász-Mirakjan-Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Keywords: basic (or q-) integers; basic (or q-) hypergeometric functions; basic (or q-) exponential functions; q-Dunkl's analogue; Szász operator; q-Szász-Mirakjan-Kantorovich operator; rate of convergence; modulus of continuity; Peetre's K-functionalDuring the past two decades or so, applications of basic (or q-) calculus emerged as a new area in the field of approximation theory. Lupaş [3] was the first who (in the year 1987) introduced a q-analogue of the well-known Bernstein polynomials in (1.1) and investigated their approximating and shape-preserving properties. In the year 1997, Phillips [4] considered another q-analogue of the classical Bernstein polynomials. Later on, many authors introduced q-generalizations of various operators and investigated several approximation and other interesting properties in each case (see, for example, [5-13] and [14]).We begin our present sequel to some of the aforementioned investigations by a number of basic definitions and concept details of the q-calculus, which are used in this paper.In this section, we derive the Korovkin type and weighted Korovkin type approximation properties for the q-operator Q K n,q .f ; x/ defined by (2.1). Korovkin-type theorems furnish simple and useful tools for ascertaining whether a given sequence of positive linear operators, acting on some given function space, is an approximation process.

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Dunkl generalization of Szász operators via q-calculus

Cited by 40 publications

References 25 publications

On modified Dunkl generalization of Szász operators via q-calculus

On modified Dunkl generalization of Szász operators via q-calculus

A Dunkl type generalization of Szász operators via post-quantum calculus

Some approximation results involving the q‐Szász–Mirakjan–Kantorovich type operators via Dunkl's generalization

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