Iterates of q-Bernstein operators on triangular domain with all curved sides

• Autorzy: Iliyas Mohammad , Khan Asif , Arif Mohd , Mursaleen Mohammad , Lone Mudassir Rashid

Identyfikatory

DOI

10.1515/dema-2022-0173

• Języki publikacji: EN

Abstrakty

EN

In this article, Phillips-type Bernstein operators (Btm,qF)(t,s) and (Bsn,qF)(t,s) , their products, and Boolean sum based on q-integer have been studied on a triangle with all curved sides. Furthermore, convergence of iterates of these operators have been analyzed using the weakly Picard operators technique and the contraction principle.

Słowa kluczowe

EN

quantum calculus; Phillips q-Bernstein operators; product operators; Boolean sum operators; weakly Picard operators; contraction principle

PL

obliczenia kwantowe; operatory Bernsteina; operatory Picarda

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2022
• Tom: Vol. 55, nr 1
• Strony: 891--899
• Opis fizyczny: Bibliogr. 39 poz.

Twórcy

autor

Iliyas Mohammad

• Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

autor

Khan Asif

• Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

autor

Arif Mohd

• Department of Mathematical Sciences, Baba Ghulam Shah Badshah University, Rajouri-185234, India

autor

Mursaleen Mohammad

• Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
• Department of Medical Research, China Medical University Hospital, China Medical University (Taiwan), Taichung, Taiwan

autor

Lone Mudassir Rashid

• Department of Mathematical Sciences, Baba Ghulam Shah Badshah University, Rajouri-185234, India

Bibliografia

• [1] A. Sard, Linear approximation, Vol. 9 of Mathematics Surveys, American Mathematical Society, 1963.
• [2] K. Weierstrass, Über die analytische darstellbarkit sogenannter willkürlicher functionen einer reelen varänderlichen sitzungsberichtedr, Koniglish Preussicschen Akademie der Wissenschcaften zu Berlin, Vol. 633–639, 1885, pp. 789–805.
• [3] A. Lupas, A q-analogue of the Berstein operater, Seminar on Numerical and Statistiacal, Calculus, University of Cluj-Napoca, Vol. 9, 1987, pp. 85–92.
• [4] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1885), 511–518.
• [5] G. M. Phillips, A generalization of the Berstein polynomials based on the q-integers, Anziam J. 42 (2000), no. 1, 79–86.
• [6] R. E. Barnhill and L. Mansfield, Sard kernel theorems on triangular and rectangu-lar domains with extensions and applications to finite element error, Technical Report 11, Department of Mathematics, Brunel University, 1972.
• [7] R. E. Barnhill, Surfaces in Computer aided geometric design: survey with new results, Comput. Aided Geom. Design 2 (1985), 1–17.
• [8] K. Khan and D. K. Lobiyal, Bézier curves based on Lupas ( )p q, -analogue of Bernstein functions in CAGD, J. Comput. Appl. Math. 317 (2017), 458–477.
• [9] A. Rababah and S. Manna, Iterative process for G2-multi degree reduction of Bézier curves, Appl. Math. Comput. 217 (2011), 8126–8133.
• [10] A. Khan, M. S. Mansoori, K. Khan, and M. Mursaleen, Phillips-type q-Bernstien operators on triangles, J. Funct. Spaces 2021 (2021), 13, DOI: 10.1155/2021/6637893.
• [11] A. Khan, M. S. Mansoori, K. Khan, and M. Mursaleen Lupaş-type Bernstein operators on triangles based on quantum analogue, Alexandria Eng. J. 60 (2021), no. 6, 5909–5919, DOI: https://doi.org/10.1016/j.aej.2021.04.038.
• [12] I. A. Rus, Iterates of Bernstein operators via contraction principle, J. Math. Anal. Appl. 292 (2004), no. 1, 259–261.
• [13] R. E. Barnhill, G. Birkhoff, and W. L. Gordon, Smooth interpolation in triangles, J. Approx. Theory 8 (1973), 114–128.
• [14] R. E. Barnhill and L. Mansfield, Error bounds for smooth interpolation in trian-gles, J. Approx. Theory 11 (1974), 306–318.
• [15] D. D. Stancu, Evaluation of the remainder term in approximation formulas by Bernstein polynomials, Math. Comp. 17 (1963), 270–278.
• [16] D. D. Stancu, The remainder of certain linear approximation formulas in two vari-ables, SIAM Numer. Anal. Ser. B 1 (1964), 137–163.
• [17] I. Gavrea and M. Ivan, On the iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010), no. 2, 366–367.
• [18] I. A. Rus, Fixed points and interpolation point set of positive linear operator on ( )C D¯ , Studia Uniersitatis Babes-Bolyai 55 (2010), no. 4, 243–248.
• [19] S. Karlin and Z. Ziegler, Iteration of positive approximation operators, J. Approx. Theory 3 (1970), 310–339.
• [20] T. Acar and A. Aral, On pointwise convergence of q-Berstein operators and their q-derivatives, Numer. Funct. Anal. Optim. 36 (2014), no. 3, 287–304.
• [21] T. Acar, A. Aral, and S. A. Mohiuddien, Approximation by bivariate ( )p q, -Bernstein-Kantorovich operators, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018), no. 2, 655–662.
• [22] M. Ayman-Mursaleen and S. Serra-Capizzano, Statistical convergence via q-calculus and a Korovkinas type approximation theorem, Axioms 11 (2022), no. 2, 70, DOI: https://doi.org/10.3390/axioms11020070.
• [23] M. Ayman-Mursaleen, A. Kiliçman, and Md. Nasiruzzaman, Approximation by q-Bernstein-Stancu-Kantorovich operators with shifted knots of real parameters, FILOMAT 36 (2022), no. 4, 1179–1194.
• [24] Q. Cai, A. Kiliçman, and M. Ayman-Mursaleen, Approximation properties and q-statistical convergence of Stancu-type generalized Baskakov-Szász operators, J. Funct. Spaces 2022 (2022), 2286500.
• [25] R. T. Farouki and V. T. Rajan, Algorithms for polynomials in Bernstein form, Comput. Aided Geom. Design 5 (1988), no. 1, 1–26.
• [26] M. Mursaleen, K. J. Ansari, and A. Khan, Approximation properties and error estimation of q-Berstein shifted operators, Numer. Algorithms 84 (2020), no. 1, 207–227.
• [27] R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511–520.
• [28] S. Ostrovska, On the Lupaš q-analogue of the Bernstein operator, Rocky Mountain J. Math. 36 (2006), no. 5, 1615–1629.
• [29] D. Occorsio, M. G. Russo, and W. Themistoclakis, Some numerical applications of generalized Bernstein operators, Constr. Math. Anal. 4 (2021), no. 2, 186–214.
• [30] K. Bozkurt, F. Ozsarac, and A. Aral, Bivariate Bernstein polynomials that reproduce exponential functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 70 (2021), no. 1, 541–554.
• [31] M. C. Montano and V. Leonessa, A sequence of Kantorovich-type operators on mobile intervals, Constr. Math. Anal. 2 (2019), no. 3, 130–143.
• [32] T. Acar, A. Aral, and S. A. Mohiuddine, On Kantorovich modification of (p,q)-Bernstein operators, Iran. J. Sci. Technol. Trans. A Sci. 42 (2018), no. 3, 1459–1464.
• [33] R. Paltanea, Durrmeyer type operators on a simplex, Constr. Math. Anal. 4 (2021), no. 2, 215–228.
• [34] K. Victor and C. Pokman, Quantum Calculus, Springer-Verlag, New York, 2002.
• [35] T. Cătinas, Iterates of Bernstien type operators on a triangle with all curved sides, Abstr. Appl. Anal. 7 (2014).
• [36] I. A. Rus, Picard operators and applications, Sci. Math. Jpn. 58 (2003), no. 1, 191–219.
• [37] T. Acar, A. Aral and I. Rasa, Iterated Boolean sums Bernstien type operators, Numer. Funct. Anal. Optim. 41 (2020), no. 12, 1515–1527.
• [38] I. Gavrea and M. Ivan, On the iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010), no. 2, 366–368.
• [39] I. Gavrea and M. Ivan, On the iterates of positive linear operators, J. Approx. Theory 163 (2011), no. 9, 1076–1079.

Uwagi

PL

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).