Function spaces with dominating mixed smoothness

By Jan Vyb'iral. 2006

61 citations recorded by the source corpus.

We study several techniques whichare well known in the case of Besov and TriebelLizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal withsocalled atomic, subatomic and wavelet decompositions. All these theorems have much in common. fRoughly speaking, they say that a function belongs to some function space if, and only if, it can be decomposed into the sum of products of coefficients and corresponding building blocks, where the coefficients belong to an appropriate sequence space. These decomposition theorems estabilisha veryusefulconnection between function and sequence spaces. We use them in the study of the decay of entropy numbers of compact embeddings between two function spaces of dominating mixed smoothness reducingthis problem to the same question on the sequence space level. The considered scales cover many important specific spaces (Sobolev, Zygmund, Besov) and we get generalisations of respective assertions of Belinsky, Dinh Dung and Temlyakov.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/726fa4d491e220ecc2ac107c4dfa6644a18eb49c

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In Applied Mathematics

1. Tino Ullrich
2. Holger Rauhut
3. Wolfgang Dahmen
4. Markus Bachmayr
5. Dachun Yang
6. Wen Yuan
7. Yoshihiro Sawano
8. Yiyu Liang

In Numerical Analysis

1. Tino Ullrich
2. Wolfgang Dahmen
3. Holger Rauhut
4. Markus Bachmayr
5. Taiji Suzuki
6. M. Rosenthal

Citations in the corpus

• Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type (2010, background): “ψ1 = ψm satisfies (ML−1) for L= mand ψ0 = ϕm,ψ1 = ψm satisfy (D) and (SK) for K<m−1. Remark 5.33. (i) Atomic decompositions of spaces with dominating mixed smoothness were already given by Vyb´ıral [60]. He provides compactly supported atomic decompositions and in particular wavelet isomorphisms i”
• Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, background): “For details, we refer to [147, 116, 131]; note that the particular norm equivalence (3.”
• On tensor products of quasi-Banach spaces (2010, unknown): “For further details we refer to [24, 8].”