# Introduction

• Ralph P. BoasJr.
• R. Creighton Buck
Part of the Ergebnisse der Mathematik und Ihrer Grenzgebiete book series (MATHE1, 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 pc 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 a is a basis, in particular there is a row-finite infinite matrix, unique among all such matrices, such tha
$${z^k} = \sum\limits_{n = 0}^\infty {{\pi _{k,n}}{p_n}(z),k = 0,1,2....}$$
(1.1)
Suppose that Ω contains the origin, let f be analytic at the origin, and write
$$f(z) = \sum\limits_{k = 0}^\infty {{f^{(k)}}(0){z^k}/k!}$$
(2.1)

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