Abstract
A dimension invariance property for finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation of duals in a very large dimensional space is now easily accessible by merely working in a space of some much smaller dimension. A number of uniform and non-uniform schemes are studied. To satisfy the support conditions, a method of finding valid alternate dual functions with small support via a known parametric dual frame formula is discussed. Oftentimes it is convenient to have truncated approximate duals that satisfy the support conditions. Stability studies of the dimension invariance principle via such approximate duals are also presented.
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Communicated by Qiyu Sun.
The first author is supported in part by grants NSF DMS 1008183, DTRA/NSF DMS 1042701, and AFOSR FA9550-11-1-0245. The second author is supported in part by NSF grants DMS-0709384 and DMS-1010058, and by AFSOR grant FA9550-11-1-0245.
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Cahill, J., Li, S. Dimension invariance of finite frames of translates and Gabor frames. Adv Comput Math 37, 505–520 (2012). https://doi.org/10.1007/s10444-011-9222-3
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DOI: https://doi.org/10.1007/s10444-011-9222-3
Keywords
- Frames
- Dual frames
- Frame expansions
- Frame representations
- Gabor frames
- Dimension invariance
- Multi-variable vector space
- Multi-Gabor expansions