Approximately dual frames of nonstationary Gabor frames for l2(Z) and reconstruction errors

Abstract

Nonstationary Gabor (NSG) frames for L 2 (R) allow for flexible sampling and varying window functions and have found applications in adaptive signal analysis. Since any numerical implementation of NSG frames is based on a discrete model, the first author of this paper introduced discrete-time NSG frames and investigated the existence and construction of NSG frames for l 2 (Z) recently. In this paper, we consider the reconstruction of any functions in l 2 (Z) from coefficients obtained from NSG frames. Since perfect reconstruction is in general not feasible from NSG frame coefficients, we resort to approximately dual frames and estimate the reconstruction errors. We provide different approximately dual frames of NSG frames under some conditions and give the corresponding reconstruction error bounds. In particular, for a class of NSG frames that are related to painless NSG frames, we show that the approximately dual frames carry the NSG structure. Finally, a constructive example is given to illustrate the theoretical results. KEYWORDSadaptive representations, approximately dual frames, nonstationary Gabor frames, reconstruction MSC CLASSIFICATION 42C15; 42C40Math Meth Appl Sci. 2020;43:6643-6662.wileyonlinelibrary.com/journal/mma

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