  We obtain estimates exact in order for deviations of Zygmund sums in metrics
of spaces $L_{q}$, $1<q<\infty$, on classes of $2\pi$-periodic functions, that
admit the representation in the form of convolution of functions that belong to
unit ball of the space $L_{1}$ with fixed kernel $\Psi_{\beta}$. We show that
at certain values of the parameters that define the class $L^{\psi}_{\beta,1}$
and method of approximation, Zygmund sums provide the order of best
approximation of given classes by trigonometric polynomials in metric $L_{q}$
