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 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))

.