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.


Octavian Agratini
Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj-Napoca, Romania


Korovkin theorem; modulus of smoothness; weighted space; rate of convergence


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O. Agratini, Linear positive operators constructed by using Beta-type bases, Hacettepe Journal of Mathematics & Statistics,  49 (2020) no. 3, pp. 1030-1038, doi: 10.15672/hujms.549015


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Hacettepe Journal of Mathematics & Statistics

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Hacettepe Üniversitesi

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[1] T. Acar, A.M. Acu and N. Manav, Approximation of functions by genuine BernsteinDurrmeyer type operators, J. Math. Ineq. 12 (4), 975–987, 2018.
[2] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, 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 Russian), 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-LeviatanBeta 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.

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