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 $L_{ q}$
Categories: math.CA