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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)

Keywords

Compact Subset Entire Function Half Plane Finite Type Exponential Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1964

Authors and Affiliations

  • Ralph P. BoasJr.
    • 1
  • R. Creighton Buck
    • 2
  1. 1.EvanstonUSA
  2. 2.MadisonUSA

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