# Introduction

• Ralph P. BoasJr.
• R. Creighton Buck
Part of the Ergebnisse der Mathematik und Ihrer Grenzgebiete book series (MATHE2, volume 19)

## Abstract

The place of our work in the theory of polynomial expansions will be seen best if we begin with some general remarks. Let 𝕻 be the complex linear space of all polynomials, with the topology of uniform convergence on all compact subsets of a simply-connected region Ω. The completion of 𝕻 is then the space 𝕬(Ω) of all functions f which are analytic in Ω. Let σ = {p n } be a sequence of polynomials which forms a basis for 𝕻: that is, any p∈𝕻 has a unique representation as a finite sum p = Σc n p n . It is customary to call such a σ a basic set of polynomials. Then every f∈𝕬(Ω) is the limit of a sequence of finite sums of the form $$\sum\limits_{n} {{a_{{k,n}}}{p_{n}}}$$. Of course this by no means implies that there are numbers c n such that f = Σc n p n with a convergent or even a summable series. One way of attaching a series to a given function is as follows. Since σ is a basis, in particular there is a row-finite infinite matrix, unique among all such matrices, such that
$${z^{k}} = \sum\limits_{{n = 0}}^{\infty } {{\pi _{{k,n}}}{p_{n}}\left( z \right)} ,\quad k = 0,1,2, \ldots$$
(1.1)
.

Hull Diene