Approximation of generalized Poisson integrals by interpolation trigonometric polynomials

In this paper we establish asymptotically best possible interpolation Lebesgue-type inequalities for $2\pi$-periodic functions $f$, which are representable as generalized Poisson integrals of the functions $\phi$ from the space $L_p$, $1\le p\le \mathrm{\infty }$. In these inequalities the deviation of the interpolation Lagrange polynomials $\mathrm{\mid }f(x)-{\tilde{S}}_{n-1}(f;x)\mathrm{\mid }$ for every $x\in \mathbb{R}$ is expressed via the best approximations $E_n(\phi {)}_{L_p}$ of the functions $\phi$ be trigonometric polynomials in $L_p$-metrics. We also find asymptotic equalities for the exact upper bounds of points approximations by interpolation trigonometric polynomials on the classes $C_{\beta ,p}^{\alpha ,r}$ of generalized Poisson integrals of the functions, which belong to the unit balls of the spaces $L_p$, $1\le p\le \mathrm{\infty }$.