(ωx) Can Approximate Almost Every Finite Set of Samples

Gilbert H. Harman, Sanjeev R. Kulkarni, Hariharan Narayanan

Research output: Contribution to journalArticlepeer-review

Abstract

Consider a set of points (x1,y1),…,(xn,yn) with distinct 0≤xi≤1 and with -1<yi<1. The question of whether the function y=sin(ωx) can approximate these points arbitrarily closely for a suitable choice of ω is considered. It is shown that such approximation is possible if and only if the set (Formula presented.) is linearly independent over the rationals. Furthermore, a constructive sufficient condition for such approximation is provided. The results provide a sort of counterpoint to the classical sampling theorem for bandlimited signals. They also provide a stronger statement than the well-known result that the collection of functions (Formula presented.)has infinite pseudo-dimension.

Original languageEnglish (US)
Pages (from-to)303-311
Number of pages9
JournalConstructive Approximation
Volume42
Issue number2
DOIs
StatePublished - Oct 4 2015

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

Keywords

  • Approximation
  • Diophantine approximation
  • Pseudo dimension
  • Sampling theorem
  • Sinusoids
  • VC dimension

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