Geodesic
Entropy Numbers in Weighted Function Spaces. The Case of Intermediate Weights
Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 170-179.

Voir la notice de l'article provenant de la source Math-Net.Ru

The exact asymptotic behavior of the entropy numbers of compact embeddings of weighted Besov spaces is known in many cases, in particular for power-type weights and logarithmic weights. Here we consider intermediate weights that are strictly between these two scales; a typical example is $w(x)=\exp\bigl(\sqrt {\log (1+|x|)}\,\bigr)$. For such weights we prove almost optimal estimates of the entropy numbers $e_k\bigl (\mathrm{id}:B^{s_1}_{p_1 q_1}(\mathbb R^d,w)\to B^{s_2}_{p_2 q_2}(\mathbb R^d)\bigr)$. 
@article{TRSPY_2006_255_a12,
     author = {T. K\"uhn},
     title = {Entropy {Numbers} in {Weighted} {Function} {Spaces.} {The} {Case} of {Intermediate} {Weights}},
     journal = {Informatics and Automation},
     pages = {170--179},
     publisher = {mathdoc},
     volume = {255},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a12/}
}
TY  - JOUR
AU  - T. Kühn
TI  - Entropy Numbers in Weighted Function Spaces. The Case of Intermediate Weights
JO  - Informatics and Automation
PY  - 2006
SP  - 170
EP  - 179
VL  - 255
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a12/
LA  - en
ID  - TRSPY_2006_255_a12
ER  - 
%0 Journal Article
%A T. Kühn
%T Entropy Numbers in Weighted Function Spaces. The Case of Intermediate Weights
%J Informatics and Automation
%D 2006
%P 170-179
%V 255
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a12/
%G en
%F TRSPY_2006_255_a12
T. Kühn. Entropy Numbers in Weighted Function Spaces. The Case of Intermediate Weights. Informatics and Automation, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 170-179. http://geodesic.mathdoc.fr/item/TRSPY_2006_255_a12/

[1] Carl B., Stephani I., Entropy, compactness and the approximation of operators, Cambridge Univ. Press, Cambridge, 1990 | MR

[2] Edmunds D.E., Triebel H., Function spaces, entropy numbers, differential operators, Cambridge, 1996 | MR

[3] Haroske D., “Embeddings of some weighted function spaces on $\mathbb R^n$; entropy and approximation numbers. A survey of some recent results”, An. Univ. Craiova. Ser. Mat. Inform., 24 (1997), 1–44 | MR | Zbl

[4] Haroske D.D., Triebel H., “Wavelet bases and entropy numbers in weighted function spaces”, Math. Nachr., 278 (2005), 108–132 | DOI | MR | Zbl

[5] König H., Eigenvalue distribution of compact operators, Birkhäuser, Basel, 1986 | MR

[6] Kühn T., “A lower estimate for entropy numbers”, J. Approx. Theory, 110 (2001), 120–124 | DOI | MR | Zbl

[7] Kühn T., “Entropy numbers of general diagonal operators”, Rev. Mat. Complut., 18 (2005), 479–491 | MR | Zbl

[8] Kühn T., Leopold H.-G., Sickel W., Skrzypczak L., “Entropy numbers of embeddings of weighted Besov spaces”, Constr. Approx., 23 (2006), 61–77 | DOI | MR | Zbl

[9] Kühn T., Leopold H.-G., Sickel W., Skrzypczak L., “Entropy numbers of embeddings of weighted Besov spaces. II”, Proc. Edinburgh Math. Soc., 49 (2006), 331–359 | DOI | MR | Zbl

[10] Kühn T., Leopold H.-G., Sickel W., Skrzypczak L., “Entropy numbers of embeddings of weighted Besov spaces. III: Weights of logarithmic type”, Math. Ztschr., 255 (2007), 1–15 | DOI | MR | Zbl

[11] Peetre J., New thoughts on Besov spaces, Duke Univ. Press, Durham, 1976 | MR | Zbl

[12] Pietsch A., Operator ideals, VEB Dtsch. Verl. Wissensch., Berlin, 1978 ; North-Holland, Amsterdam; New York, 1980 | MR | Zbl | Zbl

[13] Schütt C., “Entropy numbers of diagonal operators between symmetric Banach spaces”, J. Approx. Theory, 40 (1984), 121–128 | DOI | MR | Zbl

[14] Triebel H., Theory of function spaces, Birkhäuser, Basel, 1983 | MR | Zbl