Abstract
In this paper, we characterize finitely generated shift-invariant subspaces of \(L^2(G)\), where G is a locally compact abelian group. In particular, we give a formula for the coefficients in the known representation of the Fourier transform of the elements of finitely generated shift-invariant subspaces. Also, certain orthogonalization procedure for generators which is reminiscent of the Gram–Schmidt orthogonalization process is given.
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Notes
A first version of the main results of this paper, restricted to the case of the real line, has been posted on ArXiv [14] by one of the authors.
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Tabatabaie, S.M., Kazarian, K., Kamyabi Gol, R.A. et al. The Structure of Finitely Generated Shift-Invariant Subspaces on Locally Compact Abelian Groups. Mediterr. J. Math. 18, 27 (2021). https://doi.org/10.1007/s00009-020-01677-2
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DOI: https://doi.org/10.1007/s00009-020-01677-2