Approximation of Generalized Poisson Integrals by Interpolating Trigonometric Polynomials

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities for 2π-periodic functions f that can be represented in the form of generalized Poisson integrals of functions φ from the space L_p, 1 ≤ p ≤ ∞. In these inequalities, the moduli of deviations of the interpolation Lagrange polynomials \(\left|f\left(x\right)-{\widetilde{S}}_{n-1}\left(f;x\right)\right|\) for every x ∈ ℝ are expressed via the best approximations \({E}_{n}{\left(\varphi \right)}_{{L}_{p}}\) of the functions φ by trigonometric polynomials in the L_p-metrics. We also deduce asymptotic equalities for the exact upper bounds of pointwise approximations of the generalized Poisson integrals of functions that belong to the unit balls in the spaces L_p, 1 ≤ p ≤ ∞, by interpolating trigonometric polynomials on the classes \({C}_{\beta ,p}^{\alpha ,r}\).