Abstract
The aim of the present paper is to point out basic results concerning the approximation of functions by using linear positive operators. We indicate the main research directions of this field and some of the most remarkable results obtained in the last half-century. Our presentation will bring to light classical and recent results in Korovkin-type approximation theory, obviously just as much as it can be done in a fewpages.
Authors
Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
positive linear operator; Korovkin-type approximation theory; statistical convergence.
Paper coordinates
O. Agratini, On approximation of functions by positive linear operators, Stud. Cercet. Stiint, Ser. Mat, Proceedings of ICMI 45 (2006), 17-28.
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Stud. Cercet. Stiint, Ser. Mat, Proceedings of ICMI
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[1] O. Agratini, On statistical approximation in spaces of continuous functions, Math. Inequalities and Applications, 9(2006), in print.
[2] F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, de Gruyter Studies in Mathematics, Vol.17, Walter de Gruyter & Co., Berlin-New York, 1994.
[3] G. Anastassiou and S. Gal, Approximation Theory, Birkhäuser, Boston, 2000.
[4] D. Andrica and C. Mustăţa, An abstract Korovkin type theorem and applications, Studia Univ. Babeş-Bolyai, Mathematica, 34(1989), 44-51.
[5] S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Soc. Math. Kharkow (2), 13(1912), 1-2.
[6] H. Bohman, On approximation of continuous and of analytic functions, Ark. Mat., 2(1952-54), 43-56.
[7] Z. Ditzian and V. Totik, Moduli of smoothness, Springer-Verlag, New York Inc., 1987.
[8] O. Duman, Statistical approximation for periodic functions, Demonstratio Mathematica, 36(2003), f.4, 873-878.
[9] W. Feller, An Introduction to Probability Theory and its Applications, Vol.II, Wiley, 1966.
[10] A.D. Gadjiev and C. Orhan, Some approximation theorems via statistical convergence, Rocky Mountain Journal of Mathematics, 32(2002), f.1, 129-138.
[11] H.H. Gonska, On approximation in spaces of continuous functions, Bull. Austral. Math. Soc., 28(1983), 411-432.
[12] J.P. King, Probabilistic interpretation of some positive linear operators, Rev. Roum. Math. Pures Appl., 25(1980), f.1, 77-82.
[13] P.P. Korovkin, On convergence of linear positive operators in the space of continuous functions (Russian), Dokl. Akad. Nauk SSSR, 90(1953), 961-964.
[14] P.P. Korovkin, Linear Operators and Approximation Theory, translated from the Russian ed.1959, Russian Monographs and Texts 28 on Advanced Mathematics and Physics, Vol.III, Gordon and Breach Publishers, Inc., New York, Hindustan Publishing Corp. (India), Delhi, 1960.
[15] G.G. Lorentz, Bernstein Polynomials, Toronto: Univ. of Toronto Press, 1953.
[16] A. Pinkus, Weierstrass and Approximation Theory, Journal of Approximation Theory, 107(2000), 1-66.
[17] T. Popoviciu, Asupra demonstraţiei teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare, Lucrările Sesiunii Gen. Şt. Acad. Române, 2-12 iunie 1950, Editura Academiei Republicii Populare Române, 1951, pp. 1664-1667. [Translated in English by Daniela Kacsó: On the proof of Weierstrass’ theorem using interpolation polynomials, East Journal on Approximations, 4(1998), f.1, 107-110.]
[18] O. Shisha and B. Mond, The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. USA, 60(1968), 1196-1200.
[19] D.D. Stancu, Use of probabilistic methods in the theory of uniform approximation of continuous functions, Rev. Roum. Math. Pures et Appl., 14(1969), f.5, 673-691.