On the expansion of analytic functions in series of polynomials and in series of other analytic functions

Abstract

On several occasionst the present writer has studied the expansion of functions of a complex variable in series of more or less arbitrary functions. The results obtained are connected with Birkhoff's generalization of Taylor's series,t but yield additional results when interpreted, after conformal transformation of the plane, as the expansion of arbitrary functions in series of polynomials. The object of the present note is to prove the following theorem, which is easily established by the methods already used, but seems to go in some directions beyond results proved by other writers? concerning polynomials associated with a region bounded by an arbitrary Jordan curve: