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Complexity of oscillatory integrals on the real line

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Abstract

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space \(H^{s} (\mathbb {R})\) and from the space \(C^{s}(\mathbb {R})\) with an arbitrary integer s ≥ 1. We find tight upper and lower bounds for the worst case error of optimal algorithms that use n function values. More specifically, we study integrals of the form

$$ I_{k}^{\varrho} (f) = {\int}_{\mathbb{R}} f(x) \,\mathrm{e}^{-i\,kx} \varrho(x) \, \mathrm{d} x\ \ \ \text{for}\ \ f\in H^{s}(\mathbb{R})\ \ \text{or}\ \ f\in C^{s}(\mathbb{R}) $$
(1)

with \(k\in {\mathbb {R}}\) and a smooth density function ρ such as \( \rho (x) = \frac {1}{\sqrt {2 \pi }} \exp (-x^{2}/2)\). The optimal error bounds are \({\Theta }((n+\max (1,|k|))^{-s})\) with the factors in the Θ notation dependent only on s and ϱ.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Erich Novak

  2. Institut für Analysis, Johannes Kepler Universität, Linz, Austria

    Mario Ullrich

  3. Department of Computer Science, Columbia University, New York, NY, 10027, USA

    Henryk Woźniakowski

  4. Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02097, Warszawa, Poland

    Henryk Woźniakowski

  5. School of Computer Science and Technology, Anhui University, Hefei, 230601, People’s Republic of China

    Shun Zhang

Authors
  1. Erich Novak
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  2. Mario Ullrich
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  3. Henryk Woźniakowski
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  4. Shun Zhang
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Corresponding author

Correspondence to Shun Zhang.

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Communicated by: Gitta Kutyniok

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Novak, E., Ullrich, M., Woźniakowski, H. et al. Complexity of oscillatory integrals on the real line. Adv Comput Math 43, 537–553 (2017). https://doi.org/10.1007/s10444-016-9496-6

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  • DOI: https://doi.org/10.1007/s10444-016-9496-6

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