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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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Abstract: Consider 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.
Publication Date: 19-Jun-2015
Citation: Harman, 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/s00365-015-9296-0
DOI: doi:10.1007/s00365-015-9296-0
ISSN: 0176-4276
EISSN: 1432-0940
Pages: 303 - 311
Language: en
Type of Material: Journal Article
Journal/Proceeding Title: Constructive Approximation
Version: Author's manuscript

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