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

摘要

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) = Sigma(n)(k=0) sin(nx - k pi)x - k pi f (k pi). 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 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.