Abstract
The problem of estimating the AP-norm of univariate almost-periodic functions using a Gabor system or a wavelet system was studied by several authors, and culminated in the characterization given in a recent paper of the present author. The present article unifies and generalizes the various efforts via the study of this problem in the context of (multivariate) generalized shift-invariant (GSI) systems. The main result shows that the sought-for norm estimation of the AP functions is valid if and only if the given GSI system is an \(L_{2}({\mathbb{R}}^{d})\)-frame. Moreover, the frame bounds of the system are also the sharpest bounds in our estimation.
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Acknowledgements
The author especially thanks to Professor Amos Ron for his helpful guidance and critical advice in the manuscript. The author also thanks to the anonymous referees for their valuable comments and suggestions on the earlier version of this manuscript. This research supported by ECI Grant C61373, Central Michigan University.
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Communicated by David Walnut.
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Kim, Y.H. Representations of Almost-Periodic Functions Using Generalized Shift-Invariant Systems in \(\mathbb{R}^{d}\) . J Fourier Anal Appl 19, 857–876 (2013). https://doi.org/10.1007/s00041-013-9270-9
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DOI: https://doi.org/10.1007/s00041-013-9270-9