%0 Journal Article
%T An Interplay between Gabor and Wilson Frames
%A S. K. Kaushik
%A Suman Panwar
%J Journal of Function Spaces
%D 2013
%R 10.1155/2013/610917
%X Wilson frames as a generalization of Wilson bases have been defined and studied. We give necessary condition for a Wilson system to be a Wilson frame. Also, sufficient conditions for a Wilson system to be a Wilson Bessel sequence are obtained. Under the assumption that the window functions and for odd and even indices of are the same, we obtain sufficient conditions for a Wilson system to be a Wilson frame (Wilson Bessel sequence). Finally, under the same conditions, a characterization of Wilson frame in terms of Zak transform is given. 1. Introduction In 1946, Gabor [1] proposed a decomposition of a signal in terms of elementary signals, which displays simultaneously the local time and frequency content of the signal, as opposed to the classical Fourier transform which displays only the global frequency content for the entire signal. On the basis of this development, in 1952, Duffin and Schaeffer [2] introduced frames for Hilbert spaces to study some deep problems in nonharmonic Fourier series. In fact, they abstracted the fundamental notion of Gabor for studying signal processing. Janssen [3] showed that while being complete in , the set suggested by Gabor is not a Riesz basis. This apparent failure of Gabor system was then rectified by resorting to the concept of frames. Since then, the theory of Gabor systems has been intimately related to the theory of frames, and many problems in frame theory find their origins in Gabor analysis. For example, the localized frames were first considered in the realm of Gabor frames [4¨C7]. Gabor frames have found wide applications in signal and image processing. In view of Balian-Low theorem [8], Gabor frame for (which is a Riesz basis) has bad localization properties in time or frequency. Thus, a system to replace Gabor systems which does not have bad localization properties in time and frequency was required. For more literature on Gabor frames one may refer to [8¨C12]. Wilson et al. [13, 14] suggested a system of functions which are localized around the positive and negative frequency of the same order. The idea of Wilson was used by Daubechies et al. [15] to construct orthonormal ˇ°Wilson basesˇ± which consist of functions given by with a smooth well-localized window function . For such bases the disadvantage described in the Balian-Low theorem is completely removed. Independently from the work of Daubechies, Jaffard, and Journe, orthonormal local trigonometric bases consisting of the functions , , were introduced by Malvar [16]. Some generalizations of Malvar bases exist in [17, 18]. A drawback of Malvar's
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