Uniform convergence of Lagrange–Sturm–Liouville processes on one functional class

Abstract

We establish the uniform convergence inside an arbitrary interval ${ (a, b) \subset [0, \pi] }$ for the values of the Lagrange–Sturm–Liouville operators for functions in a class defined by one-side moduli of continuity and oscillations. Outside this interval, the sequence of values of the Lagrange–Sturm–Liouville operators may diverge. The conditions describing this functional class contain a restriction only on the rate and magnitude of the increasing (or decreasing) of the continuous function. Each element of the proposed class can decrease (or, respectively, increase) arbitrarily fast. Popular sets of functions satisfying the Dini–Lipschitz condition or the Krylov criterion are proper subsets of this class, even if, under their conditions, the classical modulus of continuity and the variation are replaced by the one-sided ones. We obtain sharp upper bounds for functions and Lebesgue constants of the Lagrange–Sturm–Liouville processes. We establish sufficient conditions of the uniform convergence of the Lagrange–Sturm–Liouville processes in terms of the maximal absolute value of the sum and the maximal sum of the absolute values of the weighted first order differences. We prove the equiboundedness of the sequence of fundamental functions of Lagrange–Sturm–Liouville processes. Three new operators are proposed, which are modifications of the Lagrange–Sturm-Liouville operator and they allow one to approximate uniformly an arbitrary continuous function vanishing at the ends on the segment $ [0, \pi] $. All the results of the work remain valid if the one-sided moduli of continuity and oscillations are replaced by the classical ones.