Order estimates of the uniform approximations by Zygmund sums on the classes of convolutions of periodic functions

The Zygmund sums of a function f is an element of L-1 are trigonometric polynomials of the form Z(n-1)(s)(f; t) := (a(0)/2+ Sigma(n-1)(k=1) (1 -(k/n)(s)) (a(k)(f)cos kt+b(k)(f)sin kt),s > 0, where a(k) (f) and b(k) (f) are the Fourier co-efficients of f. We establish the exact-order estimates of uniform approximations by the Zygmund sums Z(n-1)(s) of 2 pi-periodic continuous functions from the classes C-beta,p(psi). These classes are defined by the convolutions of functions from the unit ball in the space L-p, 1 <= p < infinity, with generating fixed kernels psi(beta)(t) similar to Sigma(infinity)(k=1) (k)cos kt + beta pi/2, psi(beta) is an element of L-p ', beta is an element of R, 1/p + 1/p ' =1. We additionally assume 2 that the product psi(k)k(s+1/p) is generally monotonically increasing with the rate of some power function, and, besides, for 1 < p < infinity it holds that Sigma(infinity)(k=n) psi(p)'(k)k(p)'-2 < infinity, and for p = 1 the following condition Sigma(infinity)(k=n) psi(k) < infinity is true. It is shown, that under these conditions Zygmund sums Z(n-1)(s) and Feje ' r sums sigma(n-1) = Z(n-1)(1) realize the order of the best uniform approximations by trigonometric polynomials of these classes, namely for 1 < p < infinity E-n(C-beta,p(psi))(C) (sic) E (E-beta,E-p (psi);Z(n-1)(s))(C) (sic) (Sigma(infinity)(k=n) psi(p)'(k)k(p)'(-2))(1/p)', 1/p + 1/p '= 1, and for p = 1 I-n(C-beta,1(psi))(C) (sic) E (C-beta,1(psi);Z(n-1)(s))(C) (sic) {Sigma(infinity)(k=1)psi(k), cos beta pi/2 not equal 0, psi(n)n, cos beta pi/2 not equal 0, where E-n(C-beta,p(psi))(C) := sup inf parallel to f(.) - tn-1(.)parallel to(C,) f is an element of C-beta,p(psi) (tn-1 is an element of T2n-1) and T2n-1 is the subspace of trigonometric polynomials t(n-1) of order n - 1 with real coefficients, E(C-beta,p(psi); Z(n-1)(s))(C) := sup inf parallel to f(.) - Z(n-1)(s) (f;.)parallel to(C,) f is an element of C-beta,p(psi)