Fibonacci, Van der Corput and Riesz-Nágy

Lluís Bibiloni, Jaume Paradís, Pelegrí Viader

Research output: Contribution to journalArticleResearchpeer-review

5 Citations (Scopus)

Abstract

What do the three names in the title have in common? The purpose of this paper is to relate them in a new and, hopefully, interesting way. Starting with the Fibonacci numeration system - also known as Zeckendorff's system - we will pose ourselves the problem of extending it in a natural way to represent all real numbers in (0, 1). We will see that this natural extension leads to what is known as the φ{symbol}-system restricted to the unit interval. The resulting complete system of numeration replicates the uniqueness of the binary system which, in our opinion, is responsible for the possibility of defining the Van der Corput sequence in (0, 1), a very special sequence which besides being uniformly distributed has one of the lowest discrepancy, a measure of the goodness of the uniformity. Lastly, combining the Fibonacci system and the binary in a very special way we will obtain a singular function, more specifically, the inverse of one of the family of Riesz-Nágy. © 2009 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)401-414
JournalJournal of Mathematical Analysis and Applications
Volume362
Issue number2
DOIs
Publication statusPublished - 15 Feb 2010

Keywords

  • Binary systems
  • Fibonacci numeration system
  • Singular function
  • Van der Corput sequence

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