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

2012 ◽  
Vol 36 (1) ◽  
pp. 1-51 ◽  
Author(s):  
Markus Hansen ◽  
Winfried Sickel
Keyword(s):  
Besov Spaces ◽  
Compact Embeddings ◽  
Term Approximation ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

scholarly journals Pointwise multipliers for Besov spaces of dominating mixed smoothness - II

2017 ◽  
Vol 60 (11) ◽  
pp. 2241-2262 ◽  
Author(s):  
Van Kien Nguyen ◽  
Winfried Sickel
Keyword(s):  
Besov Spaces ◽  
Pointwise Multipliers ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

scholarly journals Trace operators on fractals, entropy and approximation numbers

2011 ◽  
Vol 18 (3) ◽  
pp. 549-575
Author(s):  
Cornelia Schneider
Keyword(s):  
Besov Spaces ◽  
Trace Operator ◽  
Approximation Numbers ◽  
Atomic Decompositions ◽  
Sharp Estimates ◽  
Compact Embeddings ◽  
Trace Space

Abstract First we compute the trace space of Besov spaces – characterized via atomic decompositions – on fractals Γ, for parameters 0 < p < ∞, 0 < q ≤ min(1, p) and s = (n – d)/p. New Besov spaces on fractals are defined via traces for 0 < p, q ≤ ∞, s ≥ (n – d)/p and some embedding assertions are established. We conclude by studying the compactness of the trace operator TrΓ by giving sharp estimates for entropy and approximation numbers of compact embeddings between Besov spaces. Our results on Besov spaces remain valid considering the classical spaces defined via differences. The trace results are used to study traces in Triebel–Lizorkin spaces as well.

The Jawerth–Franke Embedding of Spaces with Dominating Mixed Smoothness

2009 ◽  
Vol 16 (4) ◽  
pp. 667-682
Author(s):  
Markus Hansen ◽  
Jan Vybíral
Keyword(s):  
Function Spaces ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

Abstract We give a proof of the Jawerth embedding for function spaces with dominating mixed smoothness of Besov and Triebel–Lizorkin type where 0 < 𝑝0 < 𝑝1 ≤ ∞ and 0 < 𝑞0,𝑞1 ≤ ∞ and with If 𝑝1 < ∞, we prove also the Franke embedding Our main tools are discretization by a wavelet isomorphism and multivariate rearrangements.

Frame Characterizations of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type and their Applications

2002 ◽  
Vol 9 (3) ◽  
pp. 567-590
Author(s):  
Dachun Yang
Keyword(s):  
Besov Spaces ◽  
Interpolation Method ◽  
Real Interpolation ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Entropy Numbers ◽  
Compact Embeddings

Abstract The author first establishes the frame characterizations of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type. As applications, the author then obtains some estimates of entropy numbers for the compact embeddings between Besov spaces or between Triebel–Lizorkin spaces. Moreover, some real interpolation theorems on these spaces are also established by using these frame characterizations and the abstract interpolation method.

scholarly journals Function Values Are Enough for $$L_2$$-Approximation

2020 ◽  
Author(s):  
David Krieg ◽  
Mario Ullrich
Keyword(s):  
Hilbert Space ◽  
Fourier Coefficients ◽  
Approximation Number ◽  
Approximation Numbers ◽  
Approximation Of Functions ◽  
Worst Case ◽  
Linear Information ◽  
Grid Algorithm ◽  
Dominating Mixed Smoothness ◽  
Mixed Smoothness

AbstractWe study the $$L_2$$ L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number $$e_n$$ e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number $$a_n$$ a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that $$\begin{aligned} e_n \,\lesssim \, \sqrt{\frac{1}{k_n} \sum _{j\ge k_n} a_j^2}, \end{aligned}$$ e n ≲ 1 k n ∑ j ≥ k n a j 2 , where $$k_n \asymp n/\log (n)$$ k n ≍ n / log ( n ) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces $$H^s_\mathrm{mix}(\mathbb {T}^d)$$ H mix s ( T d ) with dominating mixed smoothness $$s>1/2$$ s > 1 / 2 and dimension $$d\in \mathbb {N}$$ d ∈ N , and we obtain $$\begin{aligned} e_n \,\lesssim \, n^{-s} \log ^{sd}(n). \end{aligned}$$ e n ≲ n - s log sd ( n ) . For $$d>2s+1$$ d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak’s (sparse grid) algorithm is optimal.

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