# Introduction

Chapter

## 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 Suppose that

*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*=Σ*c*_{ 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*= Σ*c*_{ 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)

*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)

## Keywords

Compact Subset Entire Function Half Plane Finite Type Exponential Type
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© Springer-Verlag Berlin Heidelberg 1964