Frames of translates

1998/11/24 by Peter G. Casazza, Casazza, Peter G., Ole Christensen +3
Mathematics · #46B20 #46C05 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B20 #msc:46C05

paper · pdf · doi:10.48550/arxiv.math/9811144

23 pages

arxiv created 1998/11/24 · arxiv updated 2009/11/30

Abstract

We give necessary and sufficient conditions for a subfamily of regularly spaced translates of a function to form a frame (resp. a Riesz basis) for its span. One consequence is that ifthetranslates are taken only from a subset of the natural numbers, then this family is a frame if and only if it is a Riesz basis. We also consider arbitrary sequences of translates and show that for sparse sets, having an upper frame bound is equivalent to the family being a frame sequence. Finally, we use the fractional Hausdorff dimension to identify classes of exact frame sequences.

Related

• Linear combinations of frame generators in systems of translates
• Properties of translates of frames on Vilenkin group
• Frames and Oversampling Formulas for Band Limited Functions
• An operator based approach to irregular frames of translates
• Frames containing a Riesz basis and preservation of this property under perturbation