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Bounded Power Series on the Real Line

Davide Sclosa

Abstract

We investigate power series that converge to a bounded function on the real line. First, we establish relations between coefficients of a power series and boundedness of the resulting function; in particular, we show that boundedness can be prevented by certain Turán inequalities and, in the case of real coefficients, by certain sign patterns. Second, we show that the set of bounded power series naturally supports three topologies and that these topologies are inequivalent and incomplete. In each case, we determine the topological completion. Third, we study the algebra of bounded power series, revealing the key role of the backward shift operator.

Bounded Power Series on the Real Line

Abstract

We investigate power series that converge to a bounded function on the real line. First, we establish relations between coefficients of a power series and boundedness of the resulting function; in particular, we show that boundedness can be prevented by certain Turán inequalities and, in the case of real coefficients, by certain sign patterns. Second, we show that the set of bounded power series naturally supports three topologies and that these topologies are inequivalent and incomplete. In each case, we determine the topological completion. Third, we study the algebra of bounded power series, revealing the key role of the backward shift operator.
Paper Structure (4 sections, 13 theorems, 37 equations)

This paper contains 4 sections, 13 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Boundedness and Coefficients
  3. The Topology of Bounded Power Series
  4. The Algebra of Bounded Power Series

Key Result

Theorem 1.1

Let $f(x)=\sum_{n\geq 0} a_n x^n$ be a real power series, bounded on the real line, and satisfying eq:convexity for some $N\geq 1$ and $0<\chi< 1$. Let and let $L(n)$ denote the supremum of the integers $L\geq 0$ for which the tail $(a_k)_{k\geq n}$ contains $L$ consecutive elements with the same sign. Then

Theorems & Definitions (32)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Corollary 1.3
  • proof
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 22 more