Preservation properties of the Baskakov–Kantorovich operators

Zhang, Chungou; Zhu, Zhihui

doi:10.1016/j.camwa.2009.01.027

Cited by 22 publications

(9 citation statements)

References 8 publications

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“…Goyal and Agrawal [8] introduced the Bézier variant of the generalized Baskakov-Kantorovich operators, established a direct approximation theorem with the aid of the Ditzian-Totik modulus of smoothness, and studied the rate of convergence for the functions having the derivatives of bounded variation for these operators. Zhang and Zhu [29] studied preservation properties, such as monotonicity, convexity, and smoothness, as well as those under the average, of the Baskakov-Kantorovich operators. Gadjev [7] studied the approximation of functions by the Baskakov-Kantorovich operator in the space L p [0, ∞) and obtained the double inequality…”

Section: Motivations and Main Resultsmentioning

confidence: 99%

Equivalent theorem of approximation by linear combination of weighted Baskakov–Kantorovich operators in Orlicz spaces

Han

Guo

2019

J Inequal Appl

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Section: Motivations and Main Resultsmentioning

confidence: 99%

Equivalent theorem of approximation by linear combination of weighted Baskakov–Kantorovich operators in Orlicz spaces

Han

Guo

2019

J Inequal Appl

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“…Mihesan constructed the generalization of the Baskakov operators and the convergence rate of the generalization obtained by Wafi and Khatoon . Moreover, the preservation properties of the Baskakov‐Kantorovich operators are studied by Chungou and Zhihui . A novel collection of similar studies can be found in the book by Gupta and Tachev …”

Section: Introductionmentioning

confidence: 98%

Kantorovich‐type generalization of parametric Baskakov operators

İlarslan

Erbay

Aral

2019

Math Methods in App Sciences

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In this manuscript, we define a Kantorovich generalization of the nonnegative parametric Baskakov operators. After that, the weighted uniform convergence of the generalized operators is proved. Also, we present Voronovskaja-type asymptotic approximation theorem then establish weighted approximation properties for parametric Kantorovich operator. Numerical results show that depending on the value of the parameter, we obtain a better approximation. KEYWORDS-Kantorovich operator, Voronovskaja-type theorem, weighted approximation MSC CLASSIFICATION 41A36; 41A25; 26A15 INTRODUCTIONThe primary focus of the approximation theory is to approximate real-valued continuous functions by a simpler class of functions such as algebraic polynomials. Such topics have attracted the attention of many mathematicians. Kantorovich 1 presented a class of positive linear operators on bounded the interval [0, 1] for Lebesgue integrable functions defined on the interval [0, 1]. This class of operators has been studied by many researchers. Butzer 2 studied Voronovskaja-type results for the Kantorovich polynomials. Abel 3 gave an approximation as well as the convergence rate of these polynomials. On the other hand, Baskakov 4 introduced a sequence of positive linear operators, called Baskakov operators, on unbounded the interval [0, ∞) for suitable functions defined on the interval [0, ∞). Later, the Baskakov operators have been studied by many researchers. Pethe 5 studied approximation properties of Baskakov operators. Gupta 6 studied the rate of convergence of Baskakov-type operators. Mihesan 7 constructed the generalization of the Baskakov operators and the convergence rate of the generalization obtained by Wafi and Khatoon. 8 Moreover, the preservation properties of the Baskakov-Kantorovich operators are studied by Chungou and Zhihui. 9 A novel collection of similar studies can be found in the book by Gupta and Tachev. 10 On the other hand, q-analogous to the Baskakov operators were introduced by Aral and Gupta. 11 The same authors introduced another q-analogous to the Baskakov operators and studied the convergence rate in weighted norm and some shape preserving properties. 12 Recently, Aral and Erbay 13 proposed -Baskakov operators and studied their approximation properties. They showed that even though convergence is independent of the parameter , the approximation errors depend on .The aim of the manuscript is to define a Kantorovich generalization of the nonnegative parametric Baskakov operators introduced by Aral and Erbay. 13 We name it -Baskakov-Kantorovich operator. Later, we calculate the moments and central moments of the new operator. The weighted uniform convergence of the generalized operators is proved.

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“…Here we would also like to mention Refs. [39,41], in which the authors investigate properties of preserving shape such as convexity, star-shape, semi-additivity and those under the average for the Szász-Kantorovich operators and Baskakov-Kantorovich operators, respectively.…”

Section: Introductionmentioning

confidence: 99%

On preservation of binomial operators

2021

Self Cite

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Binomial operators are the most important extension to Bernstein operators, defined by $$ \bigl(L^{Q}_{n} f\bigr) (x)=\frac{1}{b_{n}(1)} \sum ^{n}_{k=0}\binom { n}{k } b_{k}(x)b_{n-k}(1-x)f\biggl( \frac{k}{n}\biggr),\quad f\in C[0, 1], $$ ( L n Q f ) ( x ) = 1 b n ( 1 ) ∑ k = 0 n ( n k ) b k ( x ) b n − k ( 1 − x ) f ( k n ) , f ∈ C [ 0 , 1 ] , where $\{b_{n}\}$ { b n } is a sequence of binomial polynomials associated to a delta operator Q. In this paper, we discuss the binomial operators $\{L^{Q}_{n} f\}$ { L n Q f } preservation such as smoothness and semi-additivity by the aid of binary representation of the operators, and present several illustrative examples. The results obtained in this paper generalize what are known as the corresponding Bernstein operators.

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Preservation properties of the Baskakov–Kantorovich operators

Cited by 22 publications

References 8 publications

Equivalent theorem of approximation by linear combination of weighted Baskakov–Kantorovich operators in Orlicz spaces

Equivalent theorem of approximation by linear combination of weighted Baskakov–Kantorovich operators in Orlicz spaces

Kantorovich‐type generalization of parametric Baskakov operators

On preservation of binomial operators

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