We establish interpolation analogs of Lebesgue-type inequalities on the sets \(C^{\psi }_{\beta }L_{1}\) of \(2\pi \) -periodic functions f defined as convolutions of the generating kernel \(\Psi _{\beta }(t) = \displaystyle \sum \nolimits _{k = 1}^{\infty }\psi (k)\cos \bigg (kt-\dfrac{\beta \pi }{2}\bigg ),\) \(\psi (k)\ge 0,\) \(\displaystyle \sum \nolimits _{k = 1}^{\infty }\psi (k)<\infty ,\) \(\beta \in \mathbb {R},\) with functions \(\varphi \) from \(L_{1}.\) In these inequalities, for each \(x\in \mathbb {R},\) the moduli of deviations \(|f(x)- \tilde{S}_{n-1}(f;x)|\) of the interpolation Lagrange polynomials \( \tilde{S}_{n-1}(f;\cdot )\) are estimated via the best approximations of functions \(\varphi \) by trigonometric polynomials in \(L_{1}\) -metrics. If the sequences \(\psi (k)\) decrease to zero faster than any power function, then the obtained inequalities are asymptotically exact in numerous important cases. In these cases, we also establish asymptotic equalities for the exact upper bounds of pointwise approximations by interpolation trigonometric polynomials in the classes of convolutions of the generating kernel \(\Psi _{\beta }\) with functions \(\varphi \) that belong to the unit ball in the space \(L_{1}.\)
