On series with respect to the Faber–Schauder system and with respect to the bases of the space $C[0,1]$

In this paper we investigate the sequence of coefficients of the expansions of continuous functions with respect to normed and quasinormed bases of the space $C[0,1]$. In this connection one makes use in an essential way of the properties of the Faber–Schauder system of functions. Some adjoining questions are investigated.