Tests for pointwise and uniform convergence of sinc approximations of continuous functions on a~closed interval

A. Yu. Trynin

Sbornik. Mathematics, Tome 198 (2007) no. 10, pp. 1517-1534

Résumé

The result obtained in this paper allows one to identify the approximate convergence at a point (or its absence) of the values of the Whittaker operators: $$ L_n(f,x)=\sum_{k=0}^{n}\frac{\sin(nx-k\pi)}{nx-k\pi}\,f\biggl(\frac{k\pi}{n}\biggr). $$ The only requirement on the function $f$ to be approximated is its continuity on $[0,\pi]$. The information about $f$ can be reduced to its values at the nodes $k\pi/n$ lying in a neighbourhood of the point at which the approximation properties are actually under consideration. A test for the uniform convergence of these operators on compact subsets of $(0,\pi)$ is also obtained for continuous functions, which is similar to Privalov's criterion of the convergence of the Lagrange–Chebyshev interpolation polynomials and trigonometric polynomials. Bibliography: 32 titles.