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Relaxed quaternionic Gabor expansions at critical density
Author(s) -
Hartmann Stefan
Publication year - 2016
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4087
Subject(s) - mathematics , quaternion , exponential function , gaussian , lattice (music) , space (punctuation) , pure mathematics , point (geometry) , representation (politics) , critical point (mathematics) , series (stratigraphy) , mathematical analysis , geometry , paleontology , linguistics , philosophy , physics , quantum mechanics , politics , political science , acoustics , law , biology
Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion‐valued signal f in the Wiener space can be expanded into a unique ℓ 2 series on a lattice at critical density 1, provided one more point is added in the middle of a cell. We call that a relaxed Gabor expansion . Copyright © 2016 John Wiley & Sons, Ltd.