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Tractability of Approximation for Some Weighted Spaces of Hybrid Smoothness

  • Arthur G. WerschulzEmail author
Chapter

Abstract

A great deal of work has studied the tractability of approximating (in the L2-norm) functions belonging to weighted unanchored Sobolev spaces of dominating mixed smoothness of order 1 over the unit d-cube. In this paper, we generalize these results. Let r and s be non-negative integers, with r ≤ s. We consider the approximation of complex-valued functions over the torus \(\mathbb {T}^d=[0,2\pi ]^d\) from weighted spaces \(H^{s,1}_\varGamma (\mathbb {T}^d)\) of hybrid smoothness, measuring error in the \(H^r(\mathbb {T}^d)\)-norm. Here we have isotropic smoothness of order s, the derivatives of order s having dominating mixed smoothness of order 1. If r = s = 0, then \(H^{0,1}(\mathbb {T}^d)\) is a well-known weighted unachored Sobolev space of dominating smoothness of order 1, whereas we have a generalization for other values of r and s. Besides its independent interest, this problem arises (with r = 1) in Galerkin methods for solving second-order elliptic problems. Suppose that continuous linear information is admissible. We show that this new approximation problem is topologically equivalent to the problem of approximating \(H^{s-r,1}_\varGamma (\mathbb {T}^d)\) in the \(L_2(\mathbb {T}^d)\)-norm, the equivalence being independent of d. It then follows that our new problem attains a given level of tractability if and only if approximating \(H^{s-r,1}_\varGamma (\mathbb {T}^d)\) in the \(L_2(\mathbb {T}^d)\)-norm has the same level of tractability. We further compare the tractability of our problem to that of \(L_2(\mathbb {T}^d)\)-approximation for \(H^{0,1}_\varGamma (\mathbb {T}^d)\). We then analyze the tractability of our problem for various families of weights.

Notes

Acknowledgements

I am happy to thank Erich Novak and Henryk Woźniakowski for their helpful and insightful remarks. Moreover, the referees made suggestions that improved the paper, for which I also extend my thanks.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer and Information ScienceFordham UniversityNew YorkUSA
  2. 2.Department of Computer ScienceColumbia UniversityNew YorkUSA

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