On a generalization of the Stancu-Schurer operator of higher order
DOI:
https://doi.org/10.33993/jnaat-1713Keywords:
Stancu-Schurer operator, generalized Stancu operator of higher order, convex combination, linear positive operatorAbstract
In this paper, we introduce a generalization of the higher-order Stancu–Schurer operator. Starting from a particular form of the operator recently introduced by the authors, we extend the convex combination of two terms appearing in its expression to a convex combination of $m$ functions, where $m\in\mathbb{N}$ with $m\geq 1$. For these generalized operators, we study classical properties such as linearity, positivity, monotonicity, moments, and certain convexity properties. We conclude the paper with some remarks on a nonlinear extension with data-dependent weights.
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G. Başcanbaz-Tunca, A. Erençin and F. Taşdelen, Some properties of Bernstein type Cheney and Sharma operators, Gen. Math., 24 (2016) nos.1-2, pp. 17-25.
S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc., 13 (1912/1913) no. 1, pp. 1-2.
T. Bostanci and G. Başcanbaz-Tunca, A Stancu type extension of Cheney and Sharma operator, J. Numer. Anal. Approx. Theory, 47 (2018) no. 2, pp. 124-134. https://doi.org/10.33993/jnaat472-1133
T. Cătinaş, Extension of Some Cheney-Sharma Type Operators to a Triangle With One Curved Side, Miskolc Math. Notes 21 (2020), pp. 101-111. https://dx.doi.org/10.18514/MMN.2020.2686
T. Cătinaş, Cheney–Sharma Type Operators on a Triangle with Straight Sides, Symmetry 14 (2022), no. 11, 2446. https://doi.org/10.3390/sym14112446
T. Cătinaş, A Stancu type extension of the Campiti-Metafune operator, J. Numer. Anal. Approx. Theory, 54 (2025) no. 2, pp. 229-236. DOI: 10.33993/jnaat542-1638
T. Cătinaş and I. Buda, An extension of the Cheney-Sharma operator of the first kind, J. Numer. Anal. Approx. Theory, 52 (2023) no. 2, pp. 172-181. https://doi.org/10.33993/jnaat522-1373
T. Cătinaş and D. Otrocol, Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15 (2013), No. 7, pp. 1240-1246.
T. Cătinaş and D. Otrocol, Iterates of Cheney-Sharma type operators on a triangle with curved side, J. Comput. Anal. Appl. 28 (2020), No. 4, pp. 737-744.
N. Çetin, A new generalization of complex Stancu operators, Math. Methods Appl. Sci. 42 (2019), pp. 5582-5594.
N. Çetin, A new complex generalized Bernstein-Schurer operator, Carpathian J. Math., 37 (2021) no. 1, pp. 81-89.
N. Çetin and N.M. Mutlu, Complex generalized Stancu-Schurer operators, Math. Slovaca, 74 (2024) no. 5, pp. 1215-1232.
E.W. Cheney and A. Sharma, On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 2 (1964), pp. 77-84.
S.G. Gal, Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics, vol. 8, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009.
E.S. Grigoriciuc, A Stancu type extension of the Cheney-Sharma Chlodovsky operators, J. Numer. Anal. Approx. Theory, 53 (2024) no. 1, pp. 103-117. DOI: 10.33993/jnaat531-1406
E.S. Grigoriciuc and A. Malina, A Stancu–Schurer type extension of higher order of the Cheney–Sharma operators, Math. Slovaca, 76 (2026) no. 1, pp. 225-244. DOI: 10.1515/ms-2025-1155
D. Miclăuş, The generalization of some results for Schurer and Schurer-Stancu operators, Rev. Anal. Numér. Théor. Approx., 40 (2011) no. 1, pp. 52-63.
F. Schurer, Linear positive operators in approximation theory, Math. Inst. Technol. Univ. Delft Rep., 1962.
D. Söylemez and F. Taşdelen, On Cheney-Sharma Chlodovsky operators, Bull. Math. Anal. Appl., 11 (2019) no. 1, pp. 36-43.
D.D. Stancu, Quadrature formulas constructed by using certain linear positive operators, in Numerical Integration. ISNM 57: International Series of Numerical Mathematics, vol. 57, G. Hämmerlin, Ed., Basel, Birkhäuser, 1981, pp. 241-251.
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