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Lattice Algorithms for Multivariate L ∞ Approximation in the Worst-Case Setting

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Constructive Approximation Aims and scope

A Correction to this article was published on 21 January 2020

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Abstract

We approximate d-variate functions from weighted Korobov spaces with the error of approximation defined in the L ∞ sense. We study lattice algorithms and consider the worst-case setting in which the error is defined by its worst-case behavior over the unit ball of the space of functions. A lattice algorithm is specified by a generating (integer) vector. We propose three choices of such vectors, each corresponding to a different search criterion in the component-by-component construction. We present worst-case error bounds that go to zero polynomially with n −1, where n is the number of function values used by the lattice algorithm. Under some assumptions on the weights of the function space, the worst-case error bounds are also polynomial in d, in which case we have (polynomial) tractability, or even independent of d, in which case we have strong (polynomial) tractability. We discuss the exponents of n −1 and stress that we do not know if these exponents can be improved.

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Change history

  • 21 January 2020

    We correct the expression for the worst-case error derived in [Kuo, Wasilkowski, Wo��niakowski, Construct. Approx. 30 (2009), 475���493] and explain that the main theorem of the paper holds with enlarged constants.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

    Frances Y. Kuo

  2. Department of Computer Science, University of Kentucky, Lexington, KY, 40506, USA

    Grzegorz W. Wasilkowski

  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, 02-097, Warszawa, Poland

    Henryk Woźniakowski

Authors
  1. Frances Y. Kuo
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  2. Grzegorz W. Wasilkowski
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  3. Henryk Woźniakowski
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Corresponding author

Correspondence to Frances Y. Kuo.

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Communicated by Ian Sloan.

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Kuo, F.Y., Wasilkowski, G.W. & Woźniakowski, H. Lattice Algorithms for Multivariate L ∞ Approximation in the Worst-Case Setting. Constr Approx 30, 475–493 (2009). https://doi.org/10.1007/s00365-009-9075-x

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  • DOI: https://doi.org/10.1007/s00365-009-9075-x

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