Abstract
In this paper, we propose a general strategy for rapidly computing sparse Legendre expansions. The resulting methods yield a new class of fast algorithms capable of approximating a given function f : [−1, 1] → ℝ with a near-optimal linear combination of s Legendre polynomials of degree ≤ N in just \((s \log N)^{\mathcal {O}(1)}\)-time. When s ≪ N, these algorithms exhibit sublinear runtime complexities in N, as opposed to traditional Ω(NlogN)-time methods for computing all of the first N Legendre coefficients of f. Theoretical as well as numerical results demonstrate the effectiveness of the proposed methods.
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M.A. Iwen was supported in part by NSF DMS-1416752.
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Hu, X., Iwen, M. & Kim, H. Rapidly computing sparse Legendre expansions via sparse Fourier transforms. Numer Algor 74, 1029–1059 (2017). https://doi.org/10.1007/s11075-016-0184-x
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DOI: https://doi.org/10.1007/s11075-016-0184-x