On Conic Equations Under Bernstein Operators

Yıl 2024, , 161 - 169, 24.03.2024

Tuncay Tunç , Ghofran Alhazzori

https://doi.org/10.17798/bitlisfen.1364241

Öz

One of the most important problems in approximation theory in mathematical analysis is the determination of sequences of polynomials that converge to functions and have the same geometric properties. This type of approximation is called the shape-preserving approximation. These types of problems are usually handled depending on the convexity of the functions, the degree of smoothness depending on the order of differentiability, or whether it satisfies a functional equation. The problem addressed in this paper belongs to the third class. A quadratic bivariate algebraic equation denotes geometrically some well-known shapes such as circles, ellipses, hyperbolas and parabolas. Such equations are known as conic equations. In this study, it is investigated whether conic equations transform into a conic equation under bivariate Bernstein polynomials, and if so, which conic equation it transforms into.

Anahtar Kelimeler

Bivariate Bernstein polynomials, Conic equations, Shape-preserving approximation, Korovkin type theorem.

Kaynakça

• [1] J. Pal, “Approksimation of konvekse Funktioner ved konvekse Polynomier,” Mat. Tidsskrift, B, pp. 60–65, 1925.
• [2] T. Popoviciu, “About the Best Polynomial Aprroximation of Continious Functions,” Mathematical Monography, Sect. Mat. Univ. Cluj, III, 1937.
• [3] L. Lupaş, “A property of the S. N. Bernstein Operator.” Mathematica (Cluj), vol. 9, no. 32, pp. 299-301, 1967.
• [4] D. Leviatan, “Shape preserving approximation by polynomials and splines, in: Approximation Theory and Function Series,” Bolyai Soc. Math. Stud. vol. 5, no. 1, pp. 63–84, 1996.
• [5] D. Leviatan, “Shape preserving approximation by polynomials,” J. Comput. Appl. Math. vol. 121, no. 1–2, pp. 73–94, 2000.
• [6] L. M. Kocic and G. M. Milovanovic, “Shape preserving approximation by polynomials and splines,” Comput. Math. Appl., vol. 33, vo. 11, pp. 59–97, 1997.
• [7] Y. K. Hu and X. M. Yu, “Copositive polynomial approximation revisited,” Applied Mathematics Reviews (G.A. Anastassiou ed.), vol. I, pp. 157–174, 2000.
• [8] S. G. Gal, Shape Preserving Approximation By Real and Complex Polynomials. 1th ed.; Springer Science and Business Media. 2008.
• [9] T. H. Hildebrandt and I. J. Schoenberg, “On Linear Functional Operations and the Moment Problem for a Finite Interval in One or Several Dimensions.” Annals of Mathematics, vol. 34, no. 2, pp. 317-328, 1933.
• [10] E. H. Kingsley, “Bernstein polynomials for functions of two variables of class C(k).” Proceedings of the American Mathematical Society, vol. 12, no. 1, pp. 64-71, 1951.
• [11] T. Popoviciu, “Sur Quelques Proprietes Des Fonctions D’une Ou De Deux Variables Reelles”. Mathematica (Cluj). vol. 8, no. 1, pp. 1-85, 1934.
• [12] M. Uzun, and T. Tunc, “B-Convexity and B-Concavity Preserving Property Of Two-Dimensional Bernstein Operators.” 7th Int. IFS and Contemporary Mathematics Conf., May, 25-29, 2021, Türkiye, 1-7.
• [13] S.N. Bernstein, “Demonstration Du Theoreme De Weierstrass Fondee Sur Le Calcul De Probabilities,” Comm. Soc. Math. Kharkov, vol. 13, no.2, pp. 1-2, 1912.
• [14] D. D. Stancu, “A method for obtaining polynomials of Bernstein type of two variables.” The American Mathematical Monthly, vol. 70, no. 3, pp. 260-264, 1963.