Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan pp 1215-1242 | Cite as

# Tractability of Approximation for Some Weighted Spaces of Hybrid Smoothness

## Abstract

A great deal of work has studied the tractability of approximating (in the *L*_{2}-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.

## References

- 1.Bellman, R.E.: Dynamic Programming. Princeton University Press, Princeton, NJ (1957)zbMATHGoogle Scholar
- 2.Borthwick, D.: Introduction to Partial Differential Equations. Universitext. Springer International Publishing, Cham (2017)zbMATHGoogle Scholar
- 3.Dahlke, S., Novak, E., Sickel, W.: Optimal approximation of elliptic problems by linear and nonlinear mappings. I. J. Complex.
**22**(1), 29–49 (2006)MathSciNetCrossRefGoogle Scholar - 4.Griebel, M., Knapek, S.: Optimized tensor-product approximation spaces. Constr. Approx.
**16**(4), 525–540 (2000)MathSciNetCrossRefGoogle Scholar - 5.Kühn, T., Sickel, W., Ullrich, T.: Approximation numbers of Sobolev embeddings—sharp constants and tractability. J. Complex.
**30**(2), 95–116 (2014)MathSciNetCrossRefGoogle Scholar - 6.Kühn, T., Sickel, W., Ullrich, T.: Approximation of mixed order Sobolev functions on the
*d*-torus: asymptotics, preasymptotics, and*d*-dependence. Constr. Approx.**42**(3), 353–398 (2015)MathSciNetCrossRefGoogle Scholar - 7.Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume I: Linear Information. EMS Tracts in Mathematics, vol. 6. European Mathematical Society (EMS), Zürich (2008)Google Scholar
- 8.Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume II: Standard Information For Functionals. EMS Tracts in Mathematics, vol. 12. European Mathematical Society (EMS), Zürich (2010)Google Scholar
- 9.Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems. Volume III: Standard Information For Operators. EMS Tracts in Mathematics, vol. 18. European Mathematical Society (EMS), Zürich (2012)Google Scholar
- 10.Siedlecki, P., Weimar, M.: Notes on (
*s*,*t*)-weak tractability: a refined classification of problems with (sub)exponential information complexity. J. Approx. Theory**200**, 227–258 (2015)MathSciNetCrossRefGoogle Scholar - 11.Werschulz, A.G., Woźniakowski, H.: Tractability of multivariate approximation over a weighted unanchored Sobolev space. Constr. Approx.
**30**(3), 395–421 (2009)MathSciNetCrossRefGoogle Scholar - 12.Werschulz, A.G., Woźniakowski, H.: Tight tractability results for a model second-order Neumann problem. Found. Comput. Math.
**15**(4), 899–929 (2015)MathSciNetCrossRefGoogle Scholar - 13.Werschulz, A.G., Woźniakowski, H.: A new characterization of (
*s*,*t*)-weak intractability. J. Complex.**38**, 68–79 (2017)CrossRefGoogle Scholar