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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 747268, 8 pages
http://dx.doi.org/10.1155/2013/747268
Research Article

Regularity of Dual Gabor Windows

1Department of Applied Mathematics and Computer Science, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
2Department of Mathematical Sciences, KAIST, 373-1 Guseong-Dong, Yuseong-Gu, Daejeon 305-701, Republic of Korea
3Department of Mathematics, Yeungnam University, 214-1 Dae-Dong, Gyeongsan-Si, Gyeongsangbuk-Do 712-749, Republic of Korea

Received 29 May 2013; Accepted 25 August 2013

Academic Editor: Anna Mercaldo

Copyright © 2013 Ole Christensen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a construction of dual windows associated with Gabor frames with compactly supported windows. The size of the support of the dual windows is comparable to that of the given window. Under certain conditions, we prove that there exist dual windows with higher regularity than the canonical dual window. On the other hand, there are cases where no differentiable dual window exists, even in the overcomplete case. As a special case of our results, we show that there exists a common smooth dual window for an interesting class of Gabor frames. In particular, for any value of there is a smooth function which simultaneously is a dual window for all B-spline generated Gabor frames for B-splines of order with a fixed and sufficiently small value of .