Note on B-splines, wavelet scaling functions, and Gabor frames

Let g be a continuous, compactly supported function on such that the integer translates of g constitute a partition of unity. We show that the Gabor system (g,a,b), with window g and time-shift and frequency-shift parameters a,b>0 has no lower frame bound larger than 0 if b=2,3,... and a>0. In...

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Bibliographic Details
Published in:IEEE transactions on information theory 2003-12, Vol.49 (12), p.3318-3320
Main Authors: Grochenig, K., Janssen, A.J.E.M., Kaiblinger, N., Pfander, G.E.
Format: Article
Language:English
Subjects:
Continuous wavelet transforms
Frames
Frequency
Information processing
Information theory
Integers
Laboratories
Mathematical models
Mathematics
Partitions
Pictures
Spline
Triangles
Unity
Wavelet
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Summary:Let g be a continuous, compactly supported function on such that the integer translates of g constitute a partition of unity. We show that the Gabor system (g,a,b), with window g and time-shift and frequency-shift parameters a,b>0 has no lower frame bound larger than 0 if b=2,3,... and a>0. In particular, (g,a,b) is not a Gabor frame if g is a continuous, compactly supported wavelet scaling function and if b=2,3,... and a>0. We give an example for our result for the case that g=B/sub 1/, the triangle function supported by [-1,1], by showing pictures of the canonical dual corresponding to (g,a,b) where ab=1/4 and b crosses the lines N=2,3,.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2003.820022