Embeddings and Entropy Numbers for General Weighted Sequence Spaces: The Non-Limiting Case

2000 ◽  
Vol 7 (4) ◽  
pp. 731-743 ◽  
Author(s):  
Hans-Gerd Leopold
Keyword(s):  
Sequence Spaces ◽  
Entropy Numbers ◽  
Compact Embeddings ◽  
Weighted Sequence ◽  
Limiting Case

Abstract The paper deals with embeddings for sequence spaces with general weights. Our main results clarify in a rather final way the compactness of embeddings between these spaces including estimates for the entropy numbers of such compact embeddings in a generalized nonlimiting case.

ENTROPY NUMBERS OF DIAGONAL OPERATORS BETWEEN VECTOR-VALUED SEQUENCE SPACES

2001 ◽  
Vol 64 (3) ◽  
pp. 739-754 ◽  
Author(s):  
THOMAS KÜHN ◽  
TOMAS P. SCHONBEK
Keyword(s):  
Lower Bounds ◽  
Sequence Spaces ◽  
Upper And Lower Bounds ◽  
Entropy Numbers ◽  
Diagonal Operators ◽  
Weighted Sequence ◽  
Banach Sequence Spaces ◽  
Vector Valued ◽  
Banach Sequence

Upper and lower bounds are established for the entropy numbers of certain diagonal operators between Banach sequence spaces. These diagonal operators are isomorphisms between the spaces considered in the paper and weighted sequence spaces considered by Leopold so that the entropy numbers in question coincide with those considered by Leopold. The results in the paper improve the previous results in at least two ways. The estimates in the paper are ‘almost’ sharp in the sense that the upper and lower estimates differ only by logarithmic factors for a much wider range of parameters. Moreover, all the upper estimates are improvements on the previous ones, the improvement being quite significant in some cases.

scholarly journals Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases

2011 ◽  
Vol 9 (2) ◽  
pp. 129-178 ◽  
Author(s):  
Dorothee D. Haroske ◽  
Leszek Skrzypczak
Keyword(s):  
Singular Point ◽  
Polynomial Growth ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Weighted Spaces ◽  
Muckenhoupt Weights ◽  
Asymptotic Estimates ◽  
Entropy Numbers ◽  
Wavelet Bases ◽  
Compact Embeddings

We study compact embeddings for weighted spaces of Besov and Triebel-Lizorkin type where the weight belongs to some MuckenhouptApclass. This extends our previous results [25] to more general weights of logarithmically disturbed polynomial growth, both near some singular point and at infinity. We obtain sharp asymptotic estimates for the entropy numbers of this embedding. Essential tools are a discretisation in terms of wavelet bases, as well as a refined study of associated embeddings in sequence spaces and interpolation arguments in endpoint situations.

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.

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.

Entropy numbers of operators acting between vector-valued sequence spaces

2012 ◽  
Vol 286 (5-6) ◽  
pp. 614-630 ◽  
Author(s):  
David E. Edmunds ◽  
Yuri Netrusov
Keyword(s):  
Sequence Spaces ◽  
Entropy Numbers ◽  
Vector Valued

On inclusion properties of discrete Morrey spaces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hendra Gunawan ◽  
Denny Ivanal Hakim ◽  
Mochammad Idris
Keyword(s):  
Weak Type ◽  
Morrey Spaces ◽  
Sequence Spaces ◽  
Necessary Condition ◽  
Inclusion Properties ◽  
Compact Embeddings ◽  
Inclusion Relations ◽  
Pitt’S Theorem

Abstract We discuss a necessary condition for inclusion relations of weak type discrete Morrey spaces which can be seen as an extension of the results in [H. Gunawan, E. Kikianty and C. Schwanke, Discrete Morrey spaces and their inclusion properties, Math. Nachr. 291 2018, 8–9, 1283–1296] and [D. D. Haroske and L. Skrzypczak, Morrey sequence spaces: Pitt’s theorem and compact embeddings, Constr. Approx. 51 2020, 3, 505–535]. We also prove a proper inclusion from weak type discrete Morrey spaces into discrete Morrey spaces. In addition, we give a necessary condition for this inclusion. Some connections between the inclusion properties of discrete Morrey spaces and those of Morrey spaces are also discussed.

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