Robertson Type Theorems for Frames

2001/08/14 by Eric Weber, Weber, Eric
Computer Science · Engineering · Mathematics · #42C15 #43A70 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #math.FA #msc:42C15 #msc:43A70

paper · pdf · doi:10.48550/arxiv.math/0108097

10 pages

arxiv created 2001/08/14 · openalex publication_date 2001/08/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend Robertson's theorem to apply to frames generated by the action of a discrete, countable abelian unitary group. Within this setup we use Stone's theorem and the theory of spectral multiplicity to analyze wandering frame collections. Motivated by wavelet theory, we explicitly apply our results to the action of the integers given by translations on \ltwo. This yields a new functional analytic method of constructing a wavelet from a multiresolution analysis.

Related

• Geometric Aspects of Frame Representations of Abelian Groups
• The unitary extension principle on locally compact abelian groups
• Frames generated by actions of countable discrete groups
• A Constructive Approach to the Finite Wavelet Frames over Prime Fields
• Linear independence of translates implies linear independence of affine Parseval frames on LCA groups