Abstract

A geometric perspective involving Grammian and frame operators is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs. In particular, a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k-tensors. It is also shown that tightness of the bounds requires tight frames. The lack of tight frames of symmetric k-tensors in many cases, however, leads to consideration of sets that come as close as possible to attaining the bounds. The geometric derivation is then extended in the setting of generalized or continuous frames. The Welch bounds for finite sets and countably infinite sets become special cases of this general setting.

Original languageEnglish (US)
Pages (from-to)2455-2470
Number of pages16
JournalLinear Algebra and Its Applications
Volume437
Issue number10
DOIs
StatePublished - Nov 15 2012

Keywords

  • Frames
  • Grammian
  • Homogeneous polynomials
  • Symmetric tensors
  • Welch bounds
  • t-Designs

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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