Meng, Jiana; Zhao, Lianxia On the best convergence order of a new class of triangular summation operators. (English) Zbl 1156.41309 J. Shanghai Univ. 10, No. 5, 399-401 (2006). Summary: A new class of triangular summation operators based on the equidistant nodes is constructed. It is proved that this class of operators converges uniformly to arbitrary continuous functions with the period \(2\pi\) on the whole axis. Furthermore, the best approximation order and the highest convergence order are obtained. In contrast to certain operators constructed by Bernstein and O. Kis in the previous works, the convergence properties of the new operator constructed in this paper are superior. MSC: 41A25 Rate of convergence, degree of approximation 41A55 Approximate quadratures Keywords:triangular summation operator; uniform convergence; best approximation order; highest convergence order PDFBibTeX XMLCite \textit{J. Meng} and \textit{L. Zhao}, J. Shanghai Univ. 10, No. 5, 399--401 (2006; Zbl 1156.41309) Full Text: DOI References: [1] Bernstein S N. On a class of modifying Lagrange interpolation formula [J]. Acad Nauk Bernstein Collected Works, 1954, 2: 130–140. [2] Shen Xie-chang. Interpolation polynomial (one)–La-grange interpolation [J]. Advances in Mathematics, 1983, 12(3): 193–214 (in Chinese). [3] Zhang Yu-lei, He Jia-xing. Triangle interpolation polynomials of S. N. Bernstein type[J]. Mathematica Numerical Sinica, 1997, 19(2): 154–158 (in Chinese). [4] Yuan Xue-gang, Wang Min. On new study of S.N. Bernstein problem[J]. Acta Mathematica Scintica, 2000, 20(2): 256–260 (in Chinese). · Zbl 0973.42002 [5] Yuan Xue-gang, Wang De-hui. Approximation to continuous functions by a kind of interpolation polynomials [J]. Northeastern Mathematic Journal, 2001, 17(1): 39–44. · Zbl 1015.41012 [6] Meng Jia-na. On new study of Bernstein interpolation process of the third type[J]. Journal of Jilin Univerity (Science Edition), 2003, 41(2): 140–143 (in Chinese). · Zbl 1033.41001 [7] Xie Ting-fan, Zhou Song-ping. A remark on the approximation of Lagrange interpolation polynomials based on the Chebyshev nodes [J]. Journal of Mathematic Research and Exposition, 2003, 23(1): 177–181 (in Chinese). · Zbl 1019.41002 [8] Kis O. On certain interpolation processes of S. N. Bernstein [J]. Acta Mathematica Hungarica, 1973, 24: 353–361. · Zbl 0269.41001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.