Oscillation properties of differential operators and convolution operators, and some applications

A criterion is obtained for the validity of Rolle's theorem on the interval for linear differential operators. The exact order is found of decrease of the Fourier coefficients of the convolution kernels guaranteeing the variation diminishing property. Applications are given to several problems of approximation theory (such as quadrature formulas, recovering, widths, trigonometric interpolation, etc.).
Bibliography: 24 titles.