scholarly journals Entropy Numbers of Diagonal Operators of Logarithmic Type

2001 ◽  
Vol 8 (2) ◽  
pp. 307-318
Author(s):  
Thomas Kühn
Keyword(s):  
Asymptotic Behaviour ◽  
Entropy Numbers ◽  
Diagonal Operators ◽  
Logarithmic Type

Abstract We determine the asymptotic behaviour (as 𝑘 → ∞, up to multiplicative constants not depending on k) of the entropy numbers 𝑒𝑘 (D σ : l p → l q ), 1 ≤ p ≤ q ≤ ∞, of diagonal operators generated by logarithmically decreasing sequences σ = (σ n ). This complements earlier results by Carl [J. Approx. Theory 32: 135–150, 1981] who investigated the case of power-like decay of the diagonal.

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

Compactness of embeddings of function spaces on quasi-bounded domains and the distribution of eigenvalues of related elliptic operators

2013 ◽  
Vol 56 (3) ◽  
pp. 829-851 ◽  
Author(s):  
Hans-Gerd Leopold ◽  
Leszek Skrzypczak
Keyword(s):  
Asymptotic Behaviour ◽  
Spectral Properties ◽  
Necessary Conditions ◽  
Function Spaces ◽  
Elliptic Operators ◽  
Bounded Domains ◽  
Entropy Numbers ◽  
Sobolev Embeddings ◽  
Distribution Of Eigenvalues ◽  
Sufficient And Necessary Conditions

AbstractWe prove sufficient and necessary conditions for compactness of the Sobolev embeddings of Besov and Triebel–Lizorkin spaces defined on bounded and unbounded uniformly E-porous domains. The asymptotic behaviour of the corresponding entropy numbers is calculated. Some applications to the spectral properties of elliptic operators are described.

Entropy numbers of embeddings of weighted Besov spaces III. Weights of logarithmic type

2006 ◽  
Vol 255 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Thomas Kühn ◽  
Hans-Gerd Leopold ◽  
Winfried Sickel ◽  
Leszek Skrzypczak
Keyword(s):  
Besov Spaces ◽  
Entropy Numbers ◽  
Weighted Besov Spaces ◽  
Logarithmic Type

scholarly journals !ENTROPY NUMBERS OF EMBEDDINGS OF WEIGHTED BESOV SPACES. II

2006 ◽  
Vol 49 (2) ◽  
pp. 331-359 ◽  
Author(s):  
Thomas Kühn ◽  
Hans-Gerd Leopold ◽  
Winfried Sickel ◽  
Leszek Skrzypczak
Keyword(s):  
Asymptotic Behaviour ◽  
Large Class ◽  
Besov Space ◽  
Besov Spaces ◽  
Compact Embedding ◽  
Entropy Numbers ◽  
Weighted Besov Spaces ◽  
Weighted Besov Space

AbstractWe investigate the asymptotic behaviour of the entropy numbers of the compact embedding $B^{s_1}_{p_1,q_1}(\mathbb{R}^d,w_1)\hookrightarrow B^{s_2}_{p_2,q_2}(\mathbb{R}^d,w_2)$. Here $B^s_{p,q}(\mathbb{R}^d,w)$ denotes a weighted Besov space. We present a general approach which allows us to work with a large class of weights.

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