Linear positive operators constructed by using Beta-type bases

Abstract

Starting from a discrete linear approximation process that has the ability to turn polynomials into polynomials of the same degree, we introduce an integral generalization by using Beta-type bases. Some properties of this new sequence of operators are investigated in unweighted and weighted spaces of functions defined on unbounded interval. In our construction particular cases are outlined.

Keywords

• Korovkin theorem,modulus of smoothness,weighted space,rate of convergence

References

1. [1] T. Acar, A.M. Acu and N. Manav, Approximation of functions by genuine Bernstein- Durrmeyer type operators, J. Math. Ineq. 12 (4), 975–987, 2018.
2. [2] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applica- tions, de Gruyter Series Studies in Mathematics, Vol. 17, Walter de Gruyter & Co., Berlin, New York, 1994.
3. [3] I. Chlodovsky, Sur le développement des fonctions définies dans un interval infini en séries de polynômes de N.S. Bernstein, Compositio Math. 4, 380–393, 1937.
4. [4] A.D. Gadjiev, Theorems of Korovkin type, Mat. Zametki, 20 (5), 781–786 (in Rus- sian), 1976; Mathematical Notes, 20 (5), 995-998 (English translation), 1976.
5. [5] A.D. Gadjiev and A. Aral, The estimates of approximation by using a new type of weighted modulus of continuity, Comput. Math. Appl. 54, 127–135, 2007.
6. [6] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain J. Math. 32, 129–138, 2002.
7. [7] V. Gupta and M.A. Noor, Convergence of derivatives for certain mixed Szász-Beta operators, J. Math. Anal. Appl. 321, 1–9, 2006.
8. [8] G.C. Jain, Approximation of functions by a new class of linear operators, J. Aust. Math. Soc. 13 (3), 271–276, 1972.

9. [9] A.S. Kumar and T. Acar, Approximation by generalized Baskakov-Durrmeyer-Stancu type operators, Rend. Circ. Mat. Palermo, Series 2, 65 (3), 411–424, 2016.
10. [10] G.G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, Inc., New York, 1966.
11. [11] G.G. Lorentz, Bernstein Polynomials, 2nd Ed., Chelsea Publ. Comp., New York, NY, 1986.
12. [12] G. Mastroianni, Su un operatore lineare e positivo, Rend. Acc. Sc. Fis. Mat. Napoli, 46, 161,-176, 1979.
13. [13] C.A. Micchelli, Saturation classes and iterates of operators, Dissertation, Stanford, 1969.
14. [14] M. Mursaleen and M. Nasiruzzaman, Approximation of modified Jakimovski-Leviatan- Beta type operators, Constr. Math. Anal. 1 (2), 88–98, 2018.
15. [15] P.C. Sikkema, On some linear positive operators, Indag. Math. 32, 327–337, 1970.
16. [16] S. Tarabie, On Jain-Beta linear operators, Appl. Math. Inform. Sci. 6 (2), 213–216, 2012.