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We establish interpolation analogues of Lebesgue type inequalities on thesets of $C^{\psi}_{\beta}L_{1}$ $2\pi$-periodic functions $f$, which arerepresentable as convolutions of generating kernel $\Psi_{\beta}(t) =\sum\limits_{k=1}^{\infty}\psi(k)\cos \big(kt-\frac{\beta\pi}{2}\big)$,$\psi(k)\geq 0$, $\sum\limits_{k=1}^{\infty}\psi(k)<\infty$,$\beta\in\mathbb{R}$, with functions $\varphi$ from $L_{1}$ . In obtainedinequalities for each $x\in\mathbb{R}$ the modules of deviations $|f(x)-\tilde{S}_{n-1}(f;x)|$ of interpolation Lagrange polynomials $\tilde{S}_{n-1}(f;\cdot)$ are estimated via best approximations$E_{n}(\varphi)_{L_{1}}$ of functions $\varphi$ by trigonometric polynomials in$L_{1}$-metrics. When the sequences $\psi(k)$ decrease to zero faster than anypower function, the obtained inequalities in many important cases areasymptotically exact. In such cases we also establish the asymptotic equalitiesfor exact upper boundaries of pointwise approximations by interpolationtrigonometric polynomials on the classes of convolutions of generating kernel$\Psi_{\beta}$ with functions $\varphi$, which belong to the unit ball from thespace $L_{1}$.

Estimates of approximations by interpolation trigonometric polynomials on the classes of convolutions of high smoothness