Representation of measurable functions by series in the Faber–Schauder system, and universal series

In this work criteria are established for various types of universal series in the Faber–Schauder system of functions. By means of these criteria the maximal speed of decrease is established for coefficients of universal series in this system, and existence is proved for continuous functions with guaranteed (and best possible) smoothness in terms of moduli of continuity whose basis expansions in the Faber–Schauder system are universal in some sense or other. Convergence almost everywhere as well as convergence in integral “metrics” is considered.
Bibliography: 11 titles.