Abstract
The summability process introduced by Bell (Proc Am Math Soc 38: 548–552, 1973) is a more general and also weaker method than ordinary convergence. Recent studies have demonstrated that using this convergence in classical approximation theory provides many advantages. In this paper, we study the summability process to approximate a function and its derivatives by means of a wider class of linear operators than a family of positive linear operators. Our results improve not only Baskakov’s idea in (Mat Zametki 13: 785–794, 1973) but also the Korovkin theory based on positive linear operators. In order to verify this we display a specific sequence of approximating operators by plotting their graphs.
Similar content being viewed by others
References
F. Altomare, M. Campiti, Korovkin-type Approximation Theory and Its Applications, de Gruyter Studies in Mathematics, vol. 17 (Walter de Gruyter & Co., Berlin, 1994)
G.A. Anastassiou, O. Duman, A Baskakov type generalization of statistical Korovkin theory. J. Math. Anal. Appl. 340, 476–486 (2008)
G.A. Anastassiou, O. Duman, Towards Intelligent Modeling: Statistical Approximation Theory, Intelligent Systems Reference Library, vol. 14 (Springer, Berlin, 2011)
Ö.G. Atlihan, C. Orhan, Summation process of positive linear operators. Comput. Math. Appl. 56, 1188–1195 (2008)
V.A. Baskakov, Generalization of certain theorems of P. P. Korovkin on positive operators. Mat. Zametki 13, 785–794 (1973)
H.T. Bell, Order summability and almost convergence. Proc. Am. Math. Soc. 38, 548–552 (1973)
K. Demirci, \(A\)-statistical core of a sequence. Demonstr. Math. 33, 343–353 (2000)
H. Fast, Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
A.R. Freedman, J.J. Sember, Densities and summability. Pac. J. Math. 95, 293–305 (1981)
J.A. Fridy, C. Orhan, Statistical limit superior and limit inferior. Proc. Am. Math. Soc. 125, 3625–3631 (1997)
A.D. Gadjiev, C. Orhan, Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32, 129–138 (2002)
W.B. Jurkat, A. Peyerimhoff, Fourier effectiveness and order summability. J. Approx. Theory 4, 231–244 (1971)
W.B. Jurkat, A. Peyerimhoff, Inclusion theorems and order summability. J. Approx. Theory 4, 245–262 (1971)
P.P. Korovkin, Linear Operators and Approximation theory (Hindustan Publishing Corp, Delhi, 1960)
G.G. Lorentz, A contribution to the theory of divergent sequences. Acta Math. 80, 167–190 (1948)
G.G. Lorentz, Bernstein Polynomials, 2nd edn. (Chelsea Publishing Co., New York, 1986)
R.N. Mohapatra, Quantitative results on almost convergence of a sequence of positive linear operators. J. Approx. Theory 20, 239–250 (1977)
C. Orhan, I. Sakaoglu, Rate of convergence in \(L_{p}\) approximation. Period. Math. Hungar. 68, 176–184 (2014)
H.L. Royden, Real Analysis, 3rd edn. (Macmillan Publishing Company, New York, 1988)
J.J. Swetits, On summability and positive linear operators. J. Approx. Theory 25, 186–188 (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aslan, İ., Duman, O. A summability process on Baskakov-type approximation. Period Math Hung 72, 186–199 (2016). https://doi.org/10.1007/s10998-016-0120-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-016-0120-9