On the convergence of generalizations of the sinc approximations on the Privalov–Chanturia class

We establish a condition described in terms of the left- or right modulus of continuity and the negative or positive modulus of variation of a function $f$ respectively, which is sufficient for uniform approximation of $f$ by the values of the function interpolation operators constructed from the solutions of the Cauchy problem with a linear differential expression of the second order inside an interval. These operators are some generalization of the classical sinc approximations used in the Whittaker–Kotelnikov–Shannon Sampling Theorem. We also show that this condition is sufficient for uniform convergence over the entire segment of one modification of the function interpolation operator, which allows us to eliminate the Gibbs phenomenon near the ends of the segment.