Pseudo Wavelet Frames
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Pseudo Wavelet Frames T. C. Easwaran Nambudiri1 · K. Parthasarathy2 Received: 22 December 2018 / Revised: 13 October 2019 / Accepted: 16 October 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We seek to alleviate the unpleasantness of the fact that the class of wavelet frames is not closed under canonical duals by introducing and studying a new class of frames that we call pseudo wavelet frames, which includes all wavelet frames and is ‘self-dual’ in terms of canonical duals. The class also has the ‘extension property’ and is invariant under the Fourier transform. Keywords Bessel sequence · Frame · Canonical dual · Frame operator · Gabor frame · Wavelet frame · Pseudo wavelet frame Mathematics Subject Classification (2010) 42C15
1 Introduction Frames are less rigid and more flexible alternatives for orthonormal bases in Hilbert spaces. In terms of an orthonormal basis, every element of the space can be expanded uniquely as a series. However, frame expansions, by design, are nonunique. In the context of time-frequency analysis, there are mainly two types of such systems: the Gabor and the wavelet frames. Gabor frame theory originates from the confluence of time-frequency analysis and theory of Hilbert space frames ([8]). The work of Daubechies-GrossmannMeyer [6] marked its emergence as an important research area and, in the same work, they observed parallel techniques for the development of wavelet frames in L2 (R). Gabor frames are formed by modulating and translating a single function whereas for wavelet frames, dilations and translations of a single function are considered. T. C. Easwaran Nambudiri
[email protected] K. Parthasarathy [email protected] 1
Department of Mathematics, Government Brennen College, Dharmadam, Thalassery, Kerala, 670106, India
2
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, 600005, India
T.C. Easwaran Nambudiri, K. Parthasarathy
In spite of these similarities in their construction, Gabor and wavelet systems have some serious dissimilarities. The major difference is that the canonical dual of a Gabor frame is again a Gabor frame, while that of a wavelet frame may not be a wavelet frame (see [1, 3–5, 13] for Daubechies’ example; see also [11]). This aspect of wavelet frames is considered in [11], where constructions of several families of wavelet frames with wavelet frame canonical duals are given. Here, we look at the same phenomenon from a different perspective, seeking a larger, self-dual class. More precisely, the purpose of the present article is to seek a class of frames in L2 (R) that contains all wavelet frames and is closed under taking canonical duals. Towards this end, we introduce and study a family of frames in L2 (R) that we call pseudo wavelet frames. This class has the sought-after property on canonical duals. The role of the frame operator is very significant in frame theory since the canonical dual frame required for the fr
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