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Construction of \(J^{\text{th}}\)-stage discrete periodic wave packet frames. (English) Zbl 1394.42026

Summary: In this paper, our main goal is to introduce the construction of the \(J^{\text{th}}\)-stage periodic wave packet frames in a discrete setting. We first establish a general characterization of wave packet systems to be Parseval frames in \(l^2(\mathbb Z_N)\) using discrete Fourier transforms, and provide a sufficient condition for the system to be a first-stage discrete periodic wave packet frame for \(l^2(\mathbb Z_N)\). Then, we construct a class of \(J^{\text{th}}\)-stage discrete periodic wave packet frames by iterating the filter sequence, and establish the associated decomposition and reconstruction algorithms for these wave packet frames, which include the corresponding results of wavelet analysis and Gabor theory as special cases. Finally, we give an illustrative example to demonstrate the validity of the proposed scheme.

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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