On approximation of functions from the class $L^{\psi}_{\beta, 1}$ by the Abel-Poisson integrals in the integral metric

In the paper, we investigate an asymptotic behavior of sharp upper bounds in integral metric deviations Abel-Poisson integrals from functions class $L^{\psi}_{\beta, 1}$. The are solutions partial differential equations elliptic type with corresponding boundary conditions, and they play important role applied problems. approximative properties on different classes differentiable were studied a number papers. Nevertheless, problem respective approximation $L^{\psi}_{\beta,1}$ space $L$ remained unsolved. We managed to obtain estimates for values $(\psi, \beta)$-differentiable unit ball by integrals. some cases, also write down equalities these quantities, that is solve Kolmogorov-Nikol'skii metric.