A−Summation process in the space of locally integrable functions
DOI:
https://doi.org/10.24193/subbmath.2020.2.07Keywords:
Summation process, positive linear operators, locally integrable functions, Korovkin type theorem, modulus of continuity, rate of convergence.Abstract
In this paper, using the concept of summation process, we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from Lp,q (loc) , the space of locally integrable functions, into itself. We also study rate of convergence of these operators.
Mathematics Subject Classification (2010): 41A25, 41A36.
References
Altomare, F., Campiti, M., Korovkin Type Approximation Theory and its Applications, de Gruyter, Berlin, 1994.
Aslan, I., Duman, O., Summability on Mellin-Type nonlinear integral operators, Integral Transforms and Special Function, 30(2019), no. 6, 492-511.
Bayram, N.S¸., Strong summation process in locally integrable function spaces, Hacet. J. Math. Stat., 45(2016), no. 3, 683-694.
Bayram, N.S¸., Orhan, C., Abel convergence of the sequence of positive linear operators in Lp,q (loc), Bulletin of the Belgian Mathematical Society, Simon Stevin, 26(2019), 71-83. [5] Bernau, S.J., Theorems of Korovkin type for Lp spaces, Pacific J. Math., 53(1974), 11-19.
Costin, O., Dunne, G.V., Convergence from divergence, J. Phys. A, 51(2018), no. 4, 10 pp.
Devore, R.A., The Approximation of Continuous Functions by Positive Linear Operators, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 293, 1972.
Donner, K., Korovkin theorems in Lp spaces, J. Functional Analysis, 42(1981), 12-28. [9] Duman, O., Orhan, C., Statistical approximation in the space of locally integrable functions, Publ. Math. Debrecen, 63(2003), 134-144.
Duman, O., Orhan, C., Rates of A-statistical convergence of operators in the space of locally integrable functions, Appl. Math. Letters, (2008), 431-435.
Dzyadik, V.K., On the approximation of functions by linear positive operators and singular integrals, (Russian), Mat. Sbornik, 70(112)(1966), 508-517.
Gadjiev, A.D., On P.P. Korovkin type theorems, Math. Zametki, 20(1976).
Gadjiev, A.D., Efendiyev, R.O., I˙bikli, E., On Korovkin’s type theorem in the space of locally integrable functions, Czech. Math. J., 53(128)(2003), 45-53.
Korovkin, P.P., Linear Operators and the Theory of Approximation, Hindustan Publ. Co. Delhi, 1960.
Lorentz, G.G., A contribution to the theory of divergent sequences, Acta. Math., 80(1948), 167-190.
Orhan, C., Sakaog˘lu, I˙., Rate of convergence in Lp approximation, Periodica Mathematica Hungarica, 68(2014), 176-184.
Sakaog˘lu, I˙., Orhan, C., Strong summation process in Lp spaces, Nonlinear Analysis, 86(2013), 89-94.
Stepanets, A.I., Approximations in spaces of locally integrable functions, Ukrainian Math. J., 46(1994), no. 5, 638-670.
Stepanets, A.I., Approximations in spaces of locally integrable functions, (Russian), Akad. Nauk Ukrainy Inst. Mat. Preprint no. 18(1993), 47 pp.
Swetits, J.J., Wood, B., On degree of Lp-approximation with positive linear operators, Journal of Approximation Theory, 87(1996), 239-241.
Taberski, R., Aprroximation properties of the integral Bernstein operators and their derivatives in some classes of locally integrable functions, Funct. Approx. Comment. Math., 21(1992), 85-96.
Zygmund, A., Trigonometric Series, Cambridge University Press, 1979.
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