Abstract
We consider the problem of completely characterizing when a system of integer translates in a finitely generated shift-invariant subspace of L 2(ℝd) is stable in the sense that rectangular partial sums for the system are norm convergent. We prove that a system of integer translates is stable in L 2(ℝd) precisely when its associated Gram matrix satisfies a suitable Muckenhoupt A 2 condition.
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Communicated by Michael Frazier.
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Nielsen, M. On Stability of Finitely Generated Shift-Invariant Systems. J Fourier Anal Appl 16, 901–920 (2010). https://doi.org/10.1007/s00041-009-9096-7
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DOI: https://doi.org/10.1007/s00041-009-9096-7
Keywords
- Shift-invariant space
- Schauder basis
- Integer translates
- Vector Hunt-Muckenhoupt-Wheeden theorem
- Muckenhoupt condition