Summation Process of Positive Linear Operators in Two-Dimensional Weighted Spaces
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Sevda Akdağ
Abstract
In this paper, using an A-summation process we give a theorem of the Korovkin type for double sequences of positive linear operators acting from a weighted space Cp1 into Bp2. We also give a quantitative version of our main theorem with the help of the weighted modulus of continuity.
References
[1] ATLIHAN, Ö. G.-ORHAN, C.: Summation process of positive linear operators, Comput. Math. Appl. 56 (2008), 1188-1195.10.1016/j.camwa.2008.02.020Search in Google Scholar
[2] BAŞARIR, M.: On strong almost convergence of double sequences, Period. Math. Hungar. 30 (1995), 99-103.10.1007/BF01876616Search in Google Scholar
[3] BOJANIC, R.-CHENG, F.: Estimates for the rate of approximation of functions of bounded variation by Hermite-Fejer polynomials, Proc. Conf. Can. Math. Soc. 3 (1983), 5-17.Search in Google Scholar
[4] BOJANIC, R.-KHAN, M. K.: Summability of Hermite-Fejer interpolation for functions of bounded variation, J. Natur. Sci. Math. 32 (1992), 5-10.Search in Google Scholar
[5] CAO, F.-LIU, Y.: Approximation theorems by positive linear operators in weighted spaces, Positivity 15 (2011), 87-103.10.1007/s11117-009-0043-2Search in Google Scholar
[6] DEVORE, R. A.: The Approximation of Continuous Functions by Positive Linear Operators. Lecture Notes in Math. 293, Springer-Verlag, Berlin, 1972.10.1007/BFb0059493Search in Google Scholar
[7] DUMAN, O.-ORHAN, C.: Statistical approximation by positive linear operators, Studia Math. 161 (2004), 187-197.10.4064/sm161-2-6Search in Google Scholar
[8] DUMAN, O.-ORHAN, C.: Rates of A-statistical convergence of positive linear operators, Appl. Math. Lett. 18 (2005), 1339-1344.10.1016/j.aml.2005.02.029Search in Google Scholar
[9] GADŽIEV, A. D.: Theorems of the type of P. P. Korovkin’s theorems, Mat. Zametki. 20 (1976), 781-786 (Presented at the International Conference on the Theory of Approximation of Functions, Kaluga, 1975) (Russian).Search in Google Scholar
[10] GADŽIEV, A. D.: Weighted approximation of continuous functions by positive linear operators on the whole real axis, Izv. Akad. Nauk Azerb. SSR Ser. Fiz.-Tehn. Mat. Nauk (1975), 41-45 (Russian).Search in Google Scholar
[11] HARDY, G. H.: Divergent Series, Oxford Univ. Press, London, 1949.Search in Google Scholar
[12] HAMILTON, H. J.: Transformations of multiple sequences, Duke Math. J. 2 (1936), 29-60.10.1215/S0012-7094-36-00204-1Search in Google Scholar
[13] KARAKUŞ, S.-DEM˙IRC˙I, K.: A-summation process and Korovkin-type approximation theorem for double sequences of positive linear operators, Math. Slovaca 62 (2012), 281-292.10.2478/s12175-012-0009-xSearch in Google Scholar
[14] KOROVKIN, P. P.: Linear Operators and Theory of Approximation, Hindustan Publ. Co, Delhi, 1960.Search in Google Scholar
[15] MÓRICZ, F.-RHOADES, B. E.: Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Cambridge Philos. Soc. 104 (1988), 283-294.10.1017/S0305004100065464Search in Google Scholar
[16] MÓRICZ, F.: Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2004), 82-89.10.1007/s00013-003-0506-9Search in Google Scholar
[17] MURSALEEN-EDELY, O. H. H.: Statistical convergence of double sequences, J.Math. Anal. Appl. 288 (2003), 223-231.10.1016/j.jmaa.2003.08.004Search in Google Scholar
[18] MURSALEEN-SAVAŞ, E.: Almost regular matrices for double sequences, Studia Sci. Math. Hungar. 40 (2003), 205-212.Search in Google Scholar
[19] PATTERSON, R. F.-SAVAŞ, E.: Uniformly summable double sequences, Studia Sci. Math. Hungar. 44 (2007), 147-158.Search in Google Scholar
[20] PRINGSHEIM, A.: Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann. 53 (1900), 289-321.10.1007/BF01448977Search in Google Scholar
[21] ROBISON, G. M.: Divergent double sequences and series, Trans. Amer. Math. Soc. 28 (1926), 50-73.10.1090/S0002-9947-1926-1501332-5Search in Google Scholar
[22] SAVAŞ, E.-RHOADES, B. E.: Double summability factor theorems and applications, Math. Inequal. Appl. 10 (2007), 125-149. Search in Google Scholar
Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Radius, Diameter and the Degree Sequence of a Graph
- On Primary Ideals in Posets
- Characterization of the Set of Regular Elements in Ordered Semigroups
- Tame Automorphisms with Multidegrees in the Form of Arithmetic Progressions
- A Result Concerning Additive Mappings in Semiprime Rings
- Characterizing Jordan Derivations of Matrix Rings Through Zero Products
- Existence Results for Impulsive Nonlinear Fractional Differential Equations With Nonlocal Boundary Conditions
- The Radon-Nikodym Property and the Limit Average Range
- A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting
- On Booth Lemniscate and Hadamard Product of Analytic Functions
- Recursion Formulas for Srivastava Hypergeometric Functions
- Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments
- Singular Degenerate Differential Operators and Applications
- The Interior Euler-Lagrange Operator in Field Theory
- On Selections of Set-Valued Maps Satisfying Some Inclusions in a Single Variable
- The Family F of Permutations of ℕ
- Summation Process of Positive Linear Operators in Two-Dimensional Weighted Spaces
- On Iλ-Statistical Convergence in Locally Solid Riesz Spaces
- Some Norm one Functions of the Volterra Operator
- Some Results on Absolute Retractivity of the Fixed Points Set of KS-Multifunctions
- Convexity in the Khalimsky Plane
- Natural Boundary Conditions in Geometric Calculus of Variations
- Exponential Inequalities for Bounded Random Variables
- Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences
- Second Order Riemannian Mechanics
- Further Remarks on an Order for Quantum Observables
