On One- and Two-Dimensional α–Stancu–Schurer–Kantorovich Operators and Their Approximation Properties

Heshamuddin, Md.; Rao, Nadeem; Lamichhane, Bishnu P.; Kılıçman, Adem; Mursaleen, Mohammad Ayman

doi:10.3390/math10183227

Cited by 17 publications

(4 citation statements)

References 28 publications

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“…The magnificent polynomials of Bernstein were likewise used in many different mathematical fields for the purpose of solving partial differential equations numerically, computer-aided geometric design (CAGD), computer graphics, and so on. For more information, readers are referred to the literature [30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

Berwal,

Mohiuddine,

Kajla

et al. 2024

Math Methods in App Sciences

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In this research paper, we construct a new sequence of Riemann–Liouville type fractional ‐Bernstein–Kantorovich operators. We prove a Korovkin type approximation theorem and discuss the rate of convergence with the first order modulus of continuity of these operators. Further, we study Voronovskaja type theorem, quantitative Voronovskaya type theorem, Chebyshev–Grüss inequality and Grüss–Voronovskaya type theorem.

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Section: Introductionmentioning

confidence: 99%

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

Berwal,

Mohiuddine,

Kajla

et al. 2024

Math Methods in App Sciences

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“…Mohiuddine et al [6], Acu et al [7], İçöz and Çekim [8,9], and Kajla and Micláus [10,11] constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in diferent functional spaces in terms of several generating functions. Some other researchers developed many other useful operators [6,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] in the same feld. In the recent past, for g ∈ [0, 1], m ∈ N and α ∈ [−1, 1], Chen et al [31] constructed a sequence of new linear positive operators as…”

Section: Introductionmentioning

confidence: 99%

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

Ayman-Mursaleen,

Rao,

Rani

et al. 2023

Journal of Mathematics

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The objective of this paper is to construct univariate and bivariate blending type α-Schurer–Kantorovich operators depending on two parameters α∈0,1 and ρ>0 to approximate a class of measurable functions on 0,1+q,q>0. We present some auxiliary results and obtain the rate of convergence of these operators. Next, we study the global and local approximation properties in terms of first- and second-order modulus of smoothness, weight functions, and by Peetre’s K-functional in different function spaces. Moreover, we present some study on numerical and graphical analysis for our operators.

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“…It is well known that for every continuous function g the Bernstein polynomials Bernstein (1912) converge uniformly to g(x) for all x 2 ½0; 1. The Bernstein polynomials are defined by The Sza ´sz Sza ´sz (1950) and Baskakov Baskakov (1957) operators were constructed to approximate the continuous functions defined on the unbounded interval ½0; 1Þ: The Baskakov operators are define by (see also Al-Abied et al (2021), Cai and Aslan (2021), Cai et al (2022), Kajla et al (2021), Khan et al (2022), Kilicman et al (2020), Heshamuddin et al (2022), Mohiuddine et al (2020Mohiuddine et al ( , 2021, Mohiuddine and O ¨zger (2020), Ayman Mursaleen et al (2022), Rao et al (2021)…”

Section: Introductionmentioning

confidence: 99%

Construction of q-Baskakov Operators by Wavelets and Approximation Properties

Nasiruzzaman

Kılıçman

Mursaleen

2022

Iran J Sci Technol Trans Sci

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We use wavelets to define the Kantorovich variant of q-Baskakov type operators, and for $$1\le p< \infty$$ 1 ≤ p < ∞ , we study the $$L_{p}$$ L p -approximation. Let $$\xi$$ ξ be any positive constant and $$\Psi _{k}(x)$$ Ψ k ( x ) be any continuous derivative function such that $$\int _{\mathbb {R}}x^{s}\Psi _{k}(x)\mathrm {d}_{q}x=0$$ ∫ R x s Ψ k ( x ) d q x = 0 where $$0\le s \le k,\;k\in \mathbb {N}$$ 0 ≤ s ≤ k , k ∈ N , $$0<q<1$$ 0 < q < 1 $$.$$ . For all $$\Psi \in L_{\infty }(\mathbb {R})$$ Ψ ∈ L ∞ ( R ) suppose the following conditions hold: (i) a finite positive $$\xi$$ ξ exits with the property $$\sup \Psi \subset [0,\xi ],$$ sup Ψ ⊂ [ 0 , ξ ] , (ii) its first k moments vanish: For $$1\le s \le k,\;k\in \mathbb {N}$$ 1 ≤ s ≤ k , k ∈ N , we have $$\int _{\mathbb {R}}t^{s}\Psi (t)\mathrm {d}_{q}t=0$$ ∫ R t s Ψ ( t ) d q t = 0 and $$\int _{\mathbb {R} }\Psi (t)\mathrm {d}_{q}t=1$$ ∫ R Ψ ( t ) d q t = 1 . Then in the sense of Haar basis for $$0<q<1,$$ 0 < q < 1 , the $$q-$$ q - analogue of Baskakov–Kantorovich type wavelets operators are defined by $$\begin{aligned} \left( \mathcal {S}_{r,s,q}\;g\right) (x)=[r]_{q}\sum_{s =0}^{\infty }q^{s -1}B_{r,s,q}(x)\int _{\mathbb {R}}g\left( t\right) \Psi \left( q^{s-1}[r]_{q}t-[s ]_{q}\right) \mathrm {d}_{q}t. \end{aligned}$$ S r , s , q g ( x ) = [ r ] q ∑ s = 0 ∞ q s - 1 B r , s , q ( x ) ∫ R g t Ψ q s - 1 [ r ] q t - [ s ] q d q t .

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On One- and Two-Dimensional α–Stancu–Schurer–Kantorovich Operators and Their Approximation Properties

Cited by 17 publications

References 28 publications

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

Approximation by Riemann–Liouville type fractional α$$ \alpha $$‐Bernstein–Kantorovich operators

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

Construction of q-Baskakov Operators by Wavelets and Approximation Properties

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