Uniform approximation of periodical functions by trigonometric sums of a special type

The approximation properties of the trigonometric sums U_{n,p}^\psi of a special type are investigated on the classes C^\psi_{\beta, \infty} of (\psi,\beta)-differentiable (in the sense of Stepanets) periodical functions. The solution of Kolmogorov-Nikol’skii problem in a sufficiently general case is found as a result of consistency between the parameters of approximating sums and approximated classes. It is shown that, in some important cases the sums under consideration provide higher order of approximation in the uniform metric on the classes C^\psi_{\beta, \infty} than Fourier sums, Zygmund sums and de la Valle Poussin sums do. The range of parameters within the limits of it the sums U_{n,p}^\psi supply the order of the best uniform approximation on the classes C^\psi_{\beta, \infty} is indicated.