Introduction

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 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)