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Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence

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Abstract

We investigate the approximation of d-variate periodic functions in Sobolev spaces of dominating mixed (fractional) smoothness \(s>0\) on the d-dimensional torus, where the approximation error is measured in the \(L_2\)-norm. In other words, we study the approximation numbers \(a_n\) of the Sobolev embeddings \(H^s_\mathrm{mix}(\mathbb {T}^d)\hookrightarrow L_2(\mathbb {T}^d)\), with particular emphasis on the dependence on the dimension d. For any fixed smoothness \(s>0\), we find two-sided estimates for the approximation numbers as a function in n and d. We observe super-exponential decay of the constants in d, if n, the number of linear samples of f, is large. In addition, motivated by numerical implementation issues, we also focus on the error decay that can be achieved by approximations using only a few linear samples (small n). We present some surprising results for the so-called “preasymptotic” decay and point out connections to the recently introduced notion of quasi-polynomial tractability of approximation problems.

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Acknowledgments

The authors would like to thank the organizers of the Dagstuhl seminar 12391 “Algorithms and Complexity for Continuous Problems”, 2012, and the organizers of the HCM workshop “Discrepancy, Numerical Integration, and Hyperbolic Cross Approximation” where this work was initiated and pursued, for providing a pleasant and fruitful working atmosphere. They would also like to thank Dinh Dũng for commenting on earlier versions of this manuscript and pointing out the relation to [6]. Last but not least, the authors would like to thank an anonymous referee for asking for the details in Lemma 4.13.

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Authors and Affiliations

  1. Universität Leipzig, Augustusplatz 10, 04109, Leipzig, Germany

    Thomas Kühn

  2. Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07737, Jena, Germany

    Winfried Sickel

  3. Hausdorff-Center for Mathematics, Endenicher Allee 62, 53115, Bonn, Germany

    Tino Ullrich

Authors
  1. Thomas Kühn
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  2. Winfried Sickel
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  3. Tino Ullrich
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Correspondence to Thomas Kühn.

Additional information

Communicated by Wolfgang Dahmen.

Dedicated to Vladimir N. Temlyakov on the occasion of his 60th birthday.

Thomas Kühn is supported in part by the Spanish Ministerio de Economía y Competitividad (MTM2013-42220-P).

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Kühn, T., Sickel, W. & Ullrich, T. Approximation of Mixed Order Sobolev Functions on the d-Torus: Asymptotics, Preasymptotics, and d-Dependence. Constr Approx 42, 353–398 (2015). https://doi.org/10.1007/s00365-015-9299-x

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  • DOI: https://doi.org/10.1007/s00365-015-9299-x

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