Decomposition systems for function spaces

By G. Kyriazis. 2003

64 citations recorded by the source corpus.

Let Θ := {θ I : e ∈ E, I ∈ D} be a decomposition system for L2(R) indexed over D, the set of dyadic cubes in R, and a finite set E, and let Θ̃ := {θ̃ I : e ∈ E, I ∈ D} be the corresponding dual functionals. That is, for every f ∈ L2(R), f = ∑ e∈E ∑ I∈D〈f, θ̃ e I〉θ I . We study sufficient conditions on Θ, Θ̃ so that they constitute a decomposition system for Triebel–Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients 〈f, θ̃ I〉, e ∈ E, I ∈ D. Typical examples of such decomposition systems are various wavelet-type unconditional bases for L2(R), and more general systems such as affine frames.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0b532335165c6f1aab33842ad2197c3675290db8

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In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Tino Ullrich
4. Yoshihiro Sawano
5. Yiyu Liang
6. Ciqiang Zhuo
7. Holger Rauhut
8. Jun Liu

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Tino Ullrich
4. Yoshihiro Sawano
5. Yiyu Liang
6. Ciqiang Zhuo
7. Holger Rauhut
8. H. Kempka

Citations in the corpus

• Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type (2010, result): “Several authors [36, 34, 44, 64, 4] used their techniques in order to generalize results in certain directions.”
• New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets (2011, result): “For some earlier results with biorthogonal wavelets on classical Besov-Triebel-Lizorkin spaces, we refer to [30].”
• Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces (2010, background): “This idea goes back to [20], [11] and [7].”
• Wavelet transforms for homogeneous mixed-norm Triebel–Lizorkin spaces (2016, background): “For homogeneous Triebel–Lizorkin spaces on Rn we may refer the reader to works of Bownik and Ho and of Kyriazis and Petrushev [8, 26–28], and for anisotropic decompositions to papers of Borup and Nielsen and of Bownik and Ho [5, 6, 8].”
• Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability (2010, background): “This idea goes back to [38], [24] and [17].”