Lupas type Bernstein operators on square with two curved sides
Resumo
The motive of this paper is to construct Lupa\c{s} type Bernstein operators $\big(B^{x_{1}}_{r,q}F\big)(x_{1},x_{2})$, $\big(B^{x_{2}}_{s,q}F\big)(x_{1},x_{2})$, their products $\big(P_{rs,q}F\big)(x_{1},x_{2})$ and $\big(Q_{sr,q}F\big)(x_{1},x_{2})$ and their Boolean sums
$\big(S_{rs,q}F\big)(x_{1},x_{2})$ and $\big(T_{sr,q}F)(x_{1},x_{2}) $ on square $D_{h}$ with two curved side. Which interpolate a given function on
the some edges and at the vertices of square. The remainders from the approximation formulas are computed using Peano's theorem.
Downloads
Referências
F. A. M. Ali, S. A. A. Karim, A. Saaban, M. K. Hasan, A. Ghaffar, K. S. Nisar, and D. Baleanu, Construction of cubic timmer triangular patches and its application in scattered data interpolation, Mathematics, 8 (2), 159.
G. E. Andrews, R. Askey and R. Roy, Special functions, Encyclopedia Math. Appl., vol. 71, Cambridge Univ. Press, Cambridge, 1999.
R. E. Barnhill, G. Birkhoff and W. J. Gordon, Smooth interpolation in triangle, J. Approx. Theory, 8(1973), 114-128.
R. E. Barnhill and J. A. Gregory, Polynomial interpolation to boundary data on triangles, Math. Comp., 29(131)(1975), 726-735.
R. E. Barnhill and L. Mansfield, Error bounds for smooth interpolation in trian- gles, J. Approx. Theory, 11(1974), 306-318.
S. N. Bernstein, constructive proof of Weierstrass approximation theorem, Comm. Kharkov Math. Soc. (1912).
P. Blaga and G. Coman, Bernstein-type operators on triangles, Rev. Anal. Num´er. Th´eor. Approx., 38(1)(2009), 11-23.
K. B¨ohmer and Gh. Coman, Blending interpolation schemes on triangle with error bounds, Lecture Notes in Mathematics, 571, Springer Verlag, Berlin, Heidelberg, New York, (1977), 14-37.
Q. B. Cai, W. T. Cheng, Convergence of λ-Bernstein operators based on (p, q)-integers, J. Ineq. App.(2020) 2020:35.
Q. B. Cai, B. Y. Lian and G. Zhou, Approximation properties of λ-Bernstein operators, J. Ineq. App.(2018) 2018:61.
P. Blaga, T. Catinas and G. Coman, Bernstein-type operators on a triangle with one curved side, Mediterr. J. Math., 9(4)(2011), 1-13.
M. Mursaleen, K. J. Ansari and A. Khan, Approximation properties and error estimation of q-Bernstein shifted operators, Numer. Algor. 84(2020), 207-227.
M. Mursaleen and A. Khan, Generalized q-Bernstein-Schurer operators and some approximation theorems, J. Funct. Spaces Volume 2013, Article ID 719834, 7 pages http://dx.doi.org/10.1155/2013/719834.
G. M. Nielson, D. H. Thomas and J. A. Wixom, Interpolation in triangles, Bull.Austral. Math. Soc., 20(1)(1979), 115-130.
G. M. Phillips, A generalization of the Bernstein polynomials based on the q-integers, The ANZIAM Journal, 42 (2000), 79-86.
G. M. Phillips, Bernstein polynomials based on the q-integers, Annals Numer. Math. 4 (1997), 511-518.
L. L. Schumaker, Fitting surfaces to scattered data, Approximation Theory II, G. G. Lorentz, C. K. Chui, L. L. Schumaker, eds., Academic Press, (1976), 203-268.
Asif Khan, M.S. Mansoori, Khalid Khan and M.Mursaleen, Phillips-type q-Bernstein on triangles, J. Funct. Spaces, 2021, 22-22. Article number: 6637893.
Asif Khan and M.S. Mansoori and Khalid Khan and M. Mursaleen, Lupas type Bernstein operators on triangles based on quantum analogue, Alex. Eng. J., 60 (6), 5909-5919, 2021, doi.org/10.1016/j.aej.2021.04.038.
D. D. Stancu, Approximation of bivariate functions by means of some Bernstein-type operators , Multivariate Approximation, (Sympos., Univ. Durham, Durham, 1977), Academic Press, London-New York, (1978), 189-208.
D. D. Stancu, Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Comp., 17(1963), 270-278.
K. Victor, C. Pokman, Quantum Calculus, Springer-Verlag, New York, 2002.
P. Blaga, T. Catinas and G. Coman, Bernstein-type operators on square with one and two curved sides, Studia Univ. Babes Bolyai, Mathematica, Volume LV, Number 3, September 2010.
N. Braha, T. Mansour, M. Mursaleen, T. Acar, Convergence of λ-Bernstein operators via power series summability method, J. Appl. Math. Comput., 65, 125-146 (2021).
A. Lupas, A q-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, University of Cluj-Napoca, vol. 9, pp. 85-92, (1987).
M J D Powell, Approximation Theory and Methods, 1981 (CUP, reprinted 1988).
Khalid Khan, D. K. Lobiyal, B´ezier curves based on Lupas (p, q)-analogue of Bernstein functions in CAGD, J. Comput. Appl. Math., 317, 2017, 458-477.
Copyright (c) 2025 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
