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The rate of convergence for linear shape-preserving algorithms. (English) Zbl 1331.41033

Summary: We prove some results which give explicit methods for determining an upper bound for the rate of approximation by means of operators preserving a cone. Then we obtain some quantitative results on the rate of convergence for some sequences of linear shape-preserving operators.

MSC:

41A36 Approximation by positive operators
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[1] [1] Barnabas B., Coroianu L., Gal Sorin G., Approximation and shape preserving properties of the Bernstein operator of maxproduct kind, Int. J. of Math. and Math., 2009, Article ID 590589, 1-26; · Zbl 1188.41016
[2] Boytsov D. I., Sidorov S. P., Linear approximation method preserving k-monotonicity, Siberian electronic mathematical reports, 2015, 12, 21-27; · Zbl 1347.41021
[3] Cárdenas-Morales D., Garrancho P., Rasa I., Bernstein-type operators which preserve polynomials, Comput. Math. Appl., 2011, 62, 158-163; · Zbl 1228.41019
[4] Cárdenas-Morales D., Muñoz-Delgado F. J., Improving certain Bernstein-type approximation processes, Mathematics and Computers in Simulation, 2008, 77, 170-178; · Zbl 1142.41302
[5] Cárdenas-Morales D., Muñoz-Delgado F. J., Garrancho P., Shape preserving approximation by Bernstein-type operators which fix polynomials, Applied Mathematics and Computation, 2006, 182, 1615-1622; · Zbl 1136.65018
[6] Floater M. S., On the convergence of derivatives of Bernstein approximation, J. Approx. Theory, 2005, 134, 130-135; · Zbl 1068.41010
[7] Gal Sorin G., Shape-Preserving Approximation by Real and Complex Polynomials, Springer, 2008; · Zbl 1154.41002
[8] Gonska H. H., Quantitative Korovkin type theorems on simultaneous approximation, Mathematische Zeitschrift, 1984, 186 (3), 419-433; · Zbl 0523.41013
[9] Knoop H.-B., Pottinger P., Ein satz vom Korovkin-typ fur Ck raume, Math. Z., 1976, 148, 23-32; · Zbl 0322.41016
[10] Kopotun K. A., Leviatan D., Prymak A., Shevchuk I. A., Uniform and pointwise shape preserving approximation by algebraic polynomials, Surveys in Approximation Theory, 2011, 6, 24-74; · Zbl 1296.41001
[11] Kopotun K., Shadrin A., On k-monotone approximation by free knot splines, SIAM J. Math. Anal., 2003, 34, 901-924; · Zbl 1031.41007
[12] Korovkin P. P., On the order of approximation of functions by linear positive operators, Dokl. Akad. Nauk SSSR, 1957, 114 (6), 1158-1161 (in Russian); · Zbl 0084.06104
[13] Kvasov B. I., Methods of shape preserving spline approximation, Singapore: World Scientific Publ. Co. Pte. Ltd., 2000; · Zbl 0960.41001
[14] Muñoz-Delgado F. J., Cárdenas-Morales D., Almost convexity and quantitative Korovkin type results, Appl.Math. Lett., 1998, 94 (4), 105-108; · Zbl 0942.41013
[15] Muñoz-Delgado F. J., Ramírez-González V., Cárdenas-Morales D., Qualitative Korovkin-type results on conservative approximation, J. Approx. Theory, 1998, 94, 144-159; · Zbl 0911.41015
[16] Pál J., Approksimation of konvekse funktioner ved konvekse polynomier, Mat. Tidsskrift, 1925, B, 60-65; · JFM 51.0210.02
[17] Popoviciu T., About the Best Polynomial Approximation of Continuous Functions. Mathematical Monography. Sect. Mat. Univ. Cluj., 1937, fasc. III, (in Romanian); · JFM 63.0959.03
[18] Pˇaltˇanea R., A generalization of Kantorovich operators and a shape-preserving property of Bernstein operators, Bulletin of the Transilvania University of Brasov, Series III: Mathematics, Informatics, Physics, 2012, 5 (54), 65-68; · Zbl 1324.41032
[19] Shisha O., Mond B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 1968, 60, 1196- 1200; · Zbl 0164.07102
[20] Sidorov S. P., Negative property of shape preserving finite-dimensional linear operators, Appl.Math. Lett., 2003, 16 (2), 257- 261; · Zbl 1062.41018
[21] Sidorov S. P., Linear relative n-widths for linear operators preserving an intersection of cones, Int. J. of Math. and Math., 2014, Article ID 409219, 1-7; · Zbl 1310.41013
[22] Sidorov S.P., On the order of approximation by linear shape-preserving operators of finite rank, East Journal on Approximations, 2001, 7 (1), 1-8; · Zbl 1085.41508
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