On estimating the rate of best trigonometric approximation by a modulus of smoothness
The best trigonometric approximation in \(L_{p},1\leq p\leq \infty \), is characterized by a modulus of smoothness of order \(r\in N\), which is equivalent to zero if the function is a trigonometric polynomial of a given degree. The characterization is similar to the one given by the classical modulus of smoothness. The modulus possesses properties similar to those of the classical one. The Jackson-type estimate from the paper was established for the Hilbert space \(L_{2}\) by \textit{A. G. Babenko}, \textit{N. I. Chernykh} and \textit{V. T. Shevaldin} [Math.\ Notes 65, No. 6, 777-781 (1999; Zbl 0960.42001)]. Estimates for the best constant on the right side were also given, for \(p=\infty ,r=2,\) by \textit{V. T. Shevaldin} [Proceedings of the Steklov Institute of Mathematics 2001, Suppl. 1, S206--S213 (2001; Zbl 1117.41016)].
\(K\)-functional, trigonometric B-spline
scientific article; zbMATH DE number 5990616
scientific article