Title

Dilations And Completions For Gabor Systems

Keywords

Affine systems; Dual frame pair dilation; Frames; Gabor frames; Projective unitary representations; Time-frequency lattices; Von Neumann algebras

Abstract

Let Λ=K×L be a full rank time-frequency lattice in a, d ×a, d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L 2(a, d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419-433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel-Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ) ( j=1N G(g j ,Λ)) for L 2(a, d ). We show that this is true whenever v(Λ) N. In particular, when v(Λ) 1, any Bessel-Gabor system is a subset of a tight Gabor frame G(g,Λ) G(h,Λ) for L 2(a, d ). Related results for affine systems are also discussed. © 2008 Birkhäuser Boston.

Publication Date

4-1-2009

Publication Title

Journal of Fourier Analysis and Applications

Volume

15

Issue

2

Number of Pages

201-217

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00041-008-9028-y

Socpus ID

67349209306 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/67349209306

This document is currently not available here.

Share

COinS