Linear positive operators constructed by using Beta-type bases
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.
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- [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] 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] 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] 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] 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] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence,
Rocky Mountain J. Math. 32, 129–138, 2002.
- [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] G.C. Jain, Approximation of functions by a new class of linear operators, J. Aust.
Math. Soc. 13 (3), 271–276, 1972.
- [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] G.G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, Inc., New
York, 1966.
- [11] G.G. Lorentz, Bernstein Polynomials, 2nd Ed., Chelsea Publ. Comp., New York, NY,
1986.
- [12] G. Mastroianni, Su un operatore lineare e positivo, Rend. Acc. Sc. Fis. Mat. Napoli,
46, 161,-176, 1979.
- [13] C.A. Micchelli, Saturation classes and iterates of operators, Dissertation, Stanford,
1969.
- [14] M. Mursaleen and M. Nasiruzzaman, Approximation of modified Jakimovski-Leviatan-
Beta type operators, Constr. Math. Anal. 1 (2), 88–98, 2018.
- [15] P.C. Sikkema, On some linear positive operators, Indag. Math. 32, 327–337, 1970.
- [16] S. Tarabie, On Jain-Beta linear operators, Appl. Math. Inform. Sci. 6 (2), 213–216,
2012.