Approximation by certain linear operators preserving $x^2$

We investigate certain positive linear operators $L_n$ preserving the functions $e_k(x) = x^k , k = 0, 1$, and modified operators $L^∗_n$ which preserve $e_0$ and $e_2$ . We show that the error of approximation of f by $L^∗_n(f)$ is smaller than for $L_n(f)$.

PDF

___

[1] Abel U.: Asymptotic approximation with Stancu beta operators, Rev. Anal. Num´er. Th´eor. Approx., 27(1), 5-13 (1998).
[2] Agratini O.: Linear operators that preserve some test functions, Int. J. Math. Math. Sci. Art. ID 94136 11 (2006).
[3] Agratini O.: On the iterates of a class of summation-type linear positive operators, Comput. Math. Appl. 55, 1178-1180 (2008).
[4] Baskakov V. A.: An example of sequence of linear positive operators in the space continuous functions, Dokl. Akad. Nauk SSSR, 113, 249-251 (1957).
[5] Becker M.: Global approximation theorems for Sz´asz-Mirakyan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27(1), 127-142 (1978).
[6] De Vore R. A. and Lorentz G. G.: Constructive Approximation, Springer-Verlag, Berlin, New York, 1993.
[7] Ditzian Z. and Totik V.: Moduli of Smoothness, Springer-Verlag, New-York, 1987.
[8] Duman O. and ¨Ozarslan M. A.: Sz´asz-Mirakyan type operators providing a better error estimation, Applied Math. Letters 20(12), 1184-1188 (2007).
[9] Duman O., ¨Ozarslan M. A. and Aktu˘glu H.: Better error estimation for Sz´asz-Mirakjan-Beta operators, J. Comput. Anal. Appl. 10, 53-59 (2008).
[10] King I. P.: Positive linear operators which preserve x2 , Acta Math. Hungar., 99(3), 203-208 (2003).
[11] ¨Ozarslan M. A. and Duman O.: MKZ type operators providing a better estimation on [1/2,1), Canadian Math. Bull. 50, 434-439 (2007).
[12] Rempulska L. and Skorupka M.: On strong approximation applied to Post-Widder operators, Anal. Theor. Applic., 22(2), 172-182 (2006).
[13] Rempulska L. and Skorupka M.: Approximation properties of modified Stancu beta operators, Rev. Anal. Num´er. Th´eor. Approx., 35(2), 189-197 (2006).
[14] Stancu D.D.: On the beta approximating operators of second kind, Rev. Anal. Num´er. Th´eor. Approx., 24(1-2), 231-239 (1995).
[15] Sz´asz O.: Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards Sect. B, 45(1), 239-245 (1950).