Best m-Term Approximation and Sobolev–Besov Spaces of Dominating Mixed Smoothness—the Case of Compact Embeddings

Abstract

We shall investigate the asymptotic behavior of the widths of best m-term approximation with respect to tensor products of Sobolev as well as Besov spaces in case of compact embeddings. Furthermore, we compare best m-term approximation with optimal linear approximation and entropy numbers.

Isotropic and dominating mixed Besov spaces: A comparison

“…For further results in this directions supporting the importance of spaces with p, q < 1, we refer to Jawerth and Milman [11], [12]. Similar descriptions of S [8], [9]. For us this motivates the investigation of our problem also for p, q < 1.…”

Optimal space–time adaptive wavelet methods for degenerate parabolic PDEs

“…The main difficulties arising for a characterization are the tensor product structure of the basis and the weighted spaces in time. The first issue has been addressed in the recent work [26]. We also refer to [17] for general results.…”

Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions

“…Another bound is obtained in the next proposition by exploiting results on hyperbolic cross approximation [12] (see also [7]). Related conversions from hyperbolic cross approximations to tensor formats have also been considered in [9, §7.6] and [20].…”

“…Proof. We rely on results on m-term approximation from [12]. We consider the tensor product wavelet system {φ j } j∈I from [12, Section 3.2], where I ⊂ N d × Z d and where for (l, k) ∈ I, φ l,k (x) = ϕ l 1 ,k…”

“…which is an hyperbolic cross with cardinality #I L ∼ L d−1 2 L (see [12,Remark 5.7]). Then from [12,Proposition 5.6] and the fact that f H s mix ∼ (c j (f )) j∈I s s 2,2 f , we have that the approximation…”