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   -> Volume 6, Issue 8

Preprint: Wavelet and Gabor frames: The whole series
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"Amos Ron" (

PostPosted: Mon Jul 21, 1997 1:58 pm    
Subject: Preprint: Wavelet and Gabor frames: The whole series
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#3 Preprint: Wavelet and Gabor frames: The whole series

Fiberization of wavelet, Gabor, and other systems: the entire series

Zuowei Shen and I (Amos Ron) had just finished our fifth (and last?) paper
is this series. All are available as .ps and .ps.Z from
file names are provided. There is also a file at that location.
So here is the list:

paper1: file:
title: Frames and stable bases for shift-invariant subspaces
of $L_2(R^d)$.
journal: Canadian J. Math. 47 (1995)
content: The theory of fiberization. Very technical. Read
if you intend to use fiberization yourself.

paper2: file:
title: Weyl-Heisenberg frames and Riesz bases in $L_2(R^d)$.
journal: Duke Math. Journal (1997)
content: applications of fiberization to Gabor systems: the
duality principle of such systems, gramian analysis
and zak transform analysis. do not read if your
interest is squarely in wavelet systems.

paper3 file:
title: Affine systems: the analysis of the analysis operator
journal: J. Functional Analysis (1997)
content: the theory of affine frames: quasi-affine fiberization,
complete characterization, affine product, the
rectangular extension principle, compactly supported
spline tight frames. read only if you are interested
in wavelets.

paper4 file:
title: Affine systems II: dual systems
journal: new (the paper, not the journal)
content: complements paper3 with a study of the
system-dual system setup, for affine frames and
affine Riesz basis.

paper5 file:
title: Compactly supported tight affine spline frames
in $L_2(R^d)$.
journal: Math. Comp. (1997)
content: in the title. for example, we derive a tight frame
from the Powell-Zwart element (bivariate C^1
piecewise quadratic on a 4-direction mesh, whose
shifts are neither Riesz basis nor a frame). the
algorithm derives a compactly supported tight frame
from any refinable box spline.

bonus file:, co-author: Charly Gr"ochenig
title: Tight compactly supported wavelet frames of
arbitarily high smoothness.
jounral: Proceedings of AMS (1997)
content: it extends the paper5 construction to arbitrary dilation
matrices. the wavelets, though, are not piecewise-polynomials.
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