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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 $L_{q}$

Approximation of the classes of convolutions of periodic functions by Zygmund sums in integral metrics