Weighted approximation: Korovkin and quantitative type theorems



Quantitative type theorems, Korovkin type theorems, weighted approximations, weighted modulus of continuity


In the present paper, we consider Korovkin and quantitative theorems, which have been treated by various authors to date, under weighted approximation. After giving the basic definitions and some of well-known spaces, we mention the main theorems and their applications to linear positive operators, which have been specially treated by the authors. Therefore, this study which can be considered as a survey study will direct the readers to literature information. Furthermore, we give a general operator including well-known operators as an application of some theorems given at the end of the paper.


T. Acar: Rate of Convergence for Ibragimov-Gadjiev-Durrmeyer Operators, Demonstr. Math., 50 (1) (2017), 119–129.

O. Agratini, A. Aral: Approximation of Some Classes of Functions by Landau Type Operators, Results Math., 76 (2021), Article ID: 12.

F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, Walter de Gruyter, Berlin (1994).

A. Aral: Approximation by Ibragimov-Gadjiyev operators in polynomial weighted space, Proc. of IMM of NAS of Azerbaijan, XIX (2003), 35–44.

A. Aral, T. Acar: On approximation properties of generalized Durrmeyer operators, Modern Mathematical Methods and High Performance Computing in Science and Technology M3HPCST, Ghaziabad (India) (2015), 1–15.

V. A. Baskakov: An example of a sequence of linear positive operators in space of continuous functions, Dokl. Akad. Nauk. SSSR, 113 (1957), 249–251 (in Russian).

V. A. Baskakov: On a sequence of linear positive operators, Research in modern constructive functions theory, Moscow (1961).

S. N. Bernstein: Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilités, Comm. Soc. Math. Kharkow (2), 13 (1912), 1–2.

J. Bustamante, L. M. de la Cruz: Positive linear operators and continuous functions on unbounded intervals, Jaen J. Approx., 1 (2) (2009), 145–173.

T. Coşkun: Some properties of linear positive operators on the spaces of weight functions, Commun. Fac. Sci. Univ. Ank. Series. A1, 47 (1998), 175–181.

T. Coşkun: Weighted approximation of continuous functions by sequences of linear positive operators, Proc. Indian Acad. Sci. (Math. Sci.), 110 (4) (2000), 357–362.

P. C. Curtis Jr.: The degree of approximation by positive convolution operators, Michigan Math. J., 12 (2) (1965), 153–160.

O. Dogru: On a certain family linear positive operators, Tr. J. Math., 21 (1997), 387–399.

O. Dogru: On the order of approximation of unbounded functions by the family of generalized linear positive operators, Commun Fac. Sci. Univ. Ankara, Series A1, 46 (1997), 173–181.

J. L. Durrmeyer: Une formule dinversion de la Transformee de Laplace: Applications a la Theorie des Moments, These de 3e Cycle, Faculte des Sciences de l’ Universite deParis, (1967).

V. K. Dzjadyk: Approximation of functions by positive linear operators and singular integrals, Russ. Mat. Sb. (N,S), 112 (70) (1966), 508–517.

A. D. Gadjiev, I. I. Ibragimov: On a sequence of linear positive operators, Soviet Math. Dokl., 11 (1970), 1092–1095.

A. D. Gadjiev: The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogous to that of P P Korovkin, Sov. Math. Dokl., 15 (5) (1974), 1433–1436 (English translated).

A. D. Gadjiev: On P. P. Korovkin type theorems, Mathem. Zametki 20 (1976), 781–786 (in Russian).

A. D. Gadjiev, R. O. Efendiyev and E. Ibikli: Generalized Bernstein Chlodowsky polynomials, Rocky Mountain J. Math., 28 (4) (1998), 1267–1277.

A. D. Gadjiev, N. Ispir: On a sequence of linear positive operators in weighted spaces, Proc. of IMM of Azerbaijan, XI(XIX) (1999), 45–56.

A. D. Gadjiev, Ö. Çakar: On uniform approximation by Bleimann, Butzer and Hahn operators on all positive semi-axis, Trans. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci., 19 (5) (1999), 21–26.

A. D. Gadjiev, E. Ibikli: On non-existence of Korovkin’s theorem in the space of Lp-locally integrable functions, Turkish J. Math., 26 (2002), 207–214.

A. D. Gadjiev, C. Orhan: Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32 (1) (2002), 129–138.

A. D. Gadjiev, R. O. Efendiyev and E. Ibikli: On Korovkin type theorem in the space of locally integrable functions, Czechoslovak Math. J., 128 (1) (2003), 45–53.

A. D. Gadjiev, A. Aral: Weighted Lp-approximation with positive linear operators on unbounded sets, Appl. Math. Lett., 20 (10) (2007), 1046–1051.

A. D. Gadjiev, A. Aral: The estimates of approximation by using a new type of weighted modulus of continuity, Comput. Math. with Appl., 54 (2007), 127–135.

A. Holhoş: Quantitative estimates for positive linear operators in weighted space, Gen. Math., 16 (4) (2008),99–110.

A. Holhoş: Uniform weighted approximation by positive linear operators, Stud. Univ. Babes Bolyai Math., 56 (3) (2011), 135–146.

L. V. Kantorovich: Sur certains developpements suivant les polynomes de la forme de S. Bernstein I, II, C. R. Acad. URRS., (1930) 563–568, 595–600.

P. P. Korovkin: Linear operators and approximation theory, Hindustan Pub. Corp., Delhi (1960).

E. V. Voronovskaya: Determination of the asymptotic form of approximation of functions by the polynomials of S.N. Bernstein, Dokl. Akad. Nauk SSSR Ser A, (1932), 79–85.




How to Cite

Aral, A. (2023). Weighted approximation: Korovkin and quantitative type theorems. Modern Mathematical Methods, 1(1), 1–21. Retrieved from https://modernmathmeth.com/index.php/pub/article/view/25