Fourier Interpolation and Time-Frequency Localization
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AbstractWe prove that under very mild conditions for any interpolation formula $$f(x) = \sum _{\lambda \in \Lambda } f(\lambda )a_\lambda (x) + \sum _{\mu \in M} {\hat{f}}(\mu )b_{\mu }(x)$$ f ( x ) = ∑ λ ∈ Λ f ( λ ) a λ ( x ) + ∑ μ ∈ M f ^ ( μ ) b μ ( x ) we have a lower bound for the counting functions $$n_\Lambda (R_1) + n_{M}(R_2) \ge 4R_1R_2 - C\log ^{2}(4R_1R_2)$$ n Λ ( R 1 ) + n M ( R 2 ) ≥ 4 R 1 R 2 - C log 2 ( 4 R 1 R 2 ) which very closely matches recent interpolation formulas of Radchenko and Viazovska and of Bondarenko, Radchenko and Seip.
2003 ◽
Vol 13
(2)
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pp. 133-140
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2021 ◽
Vol 10
(4)
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pp. 2033-2044
2016 ◽
Vol 63
(8)
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pp. 1718-1727
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2007 ◽
Vol 29
(2)
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pp. 73-82
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2013 ◽
Vol 385-386
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pp. 1389-1393
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2012 ◽
Vol 452-453
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pp. 782-788
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2020 ◽
Vol 18
(06)
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pp. 2050050