The matrix method and quasi-power bases in the space of analytic functions in a disc

We denote by $A_R(0<R\leqslant\infty)$ the space of all single-valued functions analytic in the disc $|z|<R$, with the topology of compact convergence. In the paper we present a survey of the results obtained during the last twenty years from investigations (using the matrix description of continuous linear operators) of conditions for systems of analytic functions to be quasi-power bases in $A_R$. We treat applications to many classical systems of functions and to systems formed from solutions of certain differential equations.