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Spectral Concentration of Positive Functions on Compact Groups

  • Gorjan Alagic
  • Alexander RussellEmail author
Article

Abstract

The problem of understanding the Fourier-analytic structure of the cone of positive functions on a group has a long history. In this article, we develop the first quantitative spectral concentration results for such functions over arbitrary compact groups. Specifically, we describe a family of finite, positive quadrature rules for the Fourier coefficients of band-limited functions on compact groups. We apply these quadrature rules to establish a spectral concentration result for positive functions: given appropriately nested band limits \(\mathcal {A}\subset \mathcal {B} \subset\widehat{G}\), we prove a lower bound on the fraction of L 2-mass that any \(\mathcal {B}\)-band-limited positive function has in \(\mathcal {A}\). Our bounds are explicit and depend only on elementary properties of \(\mathcal {A}\) and \(\mathcal {B}\); they are the first such bounds that apply to arbitrary compact groups. They apply to finite groups as a special case, where the quadrature rule is given by the Fourier transform on the smallest quotient whose dual contains the Fourier support of the function.

Keywords

Approximation Positive functions Band-limited functions Spectral analysis 

Mathematics Subject Classification

41A17 43A77 

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute for Quantum Computing & Department of Combinatorics and OptimizationU. WaterlooWaterlooCanada
  2. 2.Department of Computer Science and Engineering & Department of MathematicsU. ConnecticutStorrsUSA

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