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sin(ωx) Can Approximate Almost Every Finite Set of Samples

Author(s): Harman, Gilbert H; Kulkarni, Sanjeev R; Narayanan, Hariharan

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dc.contributor.authorHarman, Gilbert H-
dc.contributor.authorKulkarni, Sanjeev R-
dc.contributor.authorNarayanan, Hariharan-
dc.identifier.citationHarman, Gilbert H, Kulkarni, Sanjeev R, Narayanan, Hariharan. (2015). $$\sin (\omega x)$$ sin ( ω x ) Can Approximate Almost Every Finite Set of Samples. Constructive Approximation, 42 (2), 303 - 311. doi:10.1007/s00365-015-9296-0en_US
dc.description.abstractConsider a set of points (x1, y1), . . . , (xn, yn) with distinct 0 ≤ xi ≤ 1 and with −1 < yi < 1. The question of whether the function y = sin(ωx) can approximate these points arbitrarily closely for a suitable choice of ω is considered. It is shown that such approximation is possible if and only if the set {x1,..., xn} is linearly independent over the rationals. Furthermore, a constructive sufficient condition for such approximation is provided. The results provide a sort of counterpoint to the classical sampling theorem for bandlimited signals. They also provide a stronger statement than the well-known result that the collection of functions {sin(ωx) : ω < ∞} has infinite pseudo-dimension.en_US
dc.format.extent303 - 311en_US
dc.relation.ispartofConstructive Approximationen_US
dc.rightsAuthor's manuscripten_US
dc.titlesin(ωx) Can Approximate Almost Every Finite Set of Samplesen_US
dc.typeJournal Articleen_US

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