Optimal simultaneous approximation via \(\mathcal{A}\)-summability. (English) Zbl 1470.41011

Summary: We present optimal convergence results for the \(m\)th derivative of a function by sequences of linear operators. The usual convergence is replaced by \(\mathcal{A}\)-summability, with \(\mathcal{A}\) being a sequence of infinite matrices with nonnegative real entries, and the operators are assumed to be \(m\)-convex. Saturation results for nonconvergent but almost convergent sequences of operators are stated as corollaries.


41A35 Approximation by operators (in particular, by integral operators)
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[1] Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Mathematica, 80, 1, 167-190 (1948) · Zbl 0031.29501
[2] Altomare, F.; Campiti, M., Korovkin-Type Approximation Theory and Its Applications. Korovkin-Type Approximation Theory and Its Applications, de Gruyter Studies in Mathematics, 17 (1994), Berlin, Germany: Walter de Gruyter, Berlin, Germany · Zbl 0924.41001
[3] King, J. P.; Swetits, J. J., Positive linear operators and summability, Australian Mathematical Society A, 11, 281-290 (1970) · Zbl 0199.45101
[4] Mohapatra, R. N., Quantitative results on almost convergence of a sequence of positive linear operators, Journal of Approximation Theory, 20, 3, 239-250 (1977) · Zbl 0351.41010
[5] Swetits, J. J., On summability and positive linear operators, Journal of Approximation Theory, 25, 2, 186-188 (1979) · Zbl 0422.41019
[6] Bell, H. T., Order summability and almost convergence, Proceedings of the American Mathematical Society, 38, 548-552 (1973) · Zbl 0259.40003
[7] Aguilera, F.; Cárdenas-Morales, D.; Garrancho, P.; Hernández, J. M., Quantitative results in conservative approximation and summability, Automation Computers Applied Mathematics, 17, 2, 201-208 (2008)
[8] Garrancho, P.; Cárdenas-Morales, D.; Aguilera, F., On asymptotic formulae via summability, Mathematics and Computers in Simulation, 81, 10, 2174-2180 (2011) · Zbl 1227.40006
[9] Cárdenas-Morales, D.; Garrancho, P., Local saturation of conservative operators, Acta Mathematica Hungarica, 100, 1-2, 83-95 (2003) · Zbl 1058.41017
[10] Garrancho, P.; Cárdenas-Morales, D., A converse of asymptotic formulae in simultaneous approximation, Applied Mathematics and Computation, 217, 6, 2676-2683 (2010) · Zbl 1205.41016
[11] Lorentz, G. G.; Schumaker, L. L., Saturation of positive operators, Journal of Approximation Theory, 5, 4, 413-424 (1972) · Zbl 0233.41007
[12] Berens, H., Pointwise saturation of positive operators, Journal of Approximation Theory, 6, 2, 135-146 (1972) · Zbl 0262.41017
[13] Karlin, S. J.; Studden, W. J., Tchebycheff Systems (1966), New York, NY, USA: Wiley-Interscience, New York, NY, USA · Zbl 0153.38902
[14] Pesin, I. N., Classical and Modern Integration Theories (1951), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0206.06401
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