On approximation numbers of the canonical embedding ℓ p m → ℓ q m

1988 ◽  
pp. 145-149
Author(s):  
Yuly Makovoz
Keyword(s):  
Canonical Embedding ◽  
Approximation Numbers

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

scholarly journals Operators constructed by means of basic sequences and nuclear matrices

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ahmed Morsy ◽  
Nashat Faried ◽  
Samy A. Harisa ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Schauder Basis ◽  
Countable Number ◽  
Nuclear Operator ◽  
Approximation Numbers ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Nuclear Operators

AbstractIn this work, we establish an approach to constructing compact operators between arbitrary infinite-dimensional Banach spaces without a Schauder basis. For this purpose, we use a countable number of basic sequences for the sake of verifying the result of Morrell and Retherford. We also use a nuclear operator, represented as an infinite-dimensional matrix defined over the space $\ell _{1}$ℓ1 of all absolutely summable sequences. Examples of nuclear operators over the space $\ell _{1}$ℓ1 are given and used to construct operators over general Banach spaces with specific approximation numbers.

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.

scholarly journals The Approximation Numbers of Hardy-Type Operators on Trees

2001 ◽  
Vol 83 (2) ◽  
pp. 390-418 ◽  
Author(s):  
W. D. Evans ◽  
D. J. Harris ◽  
J. Lang
Keyword(s):  
Approximation Numbers ◽  
Hardy Type ◽  
Hardy Type Operators

scholarly journals Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

2005 ◽  
Vol 3 (1) ◽  
pp. 33-71 ◽  
Author(s):  
António M. Caetano ◽  
Dorothee D. Haroske
Keyword(s):  
Asymptotic Behaviour ◽  
Function Spaces ◽  
Approximation Numbers ◽  
Local Growth ◽  
Recent Developments ◽  
Limiting Case ◽  
Growth Envelopes

Continuity envelopes for the spaces of generalised smoothnessBpq(s,Ψ)(ℝn)andFpq(s,Ψ)(ℝn)are studied in the so-called supercriticals=1+n/p, paralleling recent developments for a corresponding limiting case for local growth envelopes of spaces of such a type. In addition, the power of the concept is used in proving conditions for some embeddings between function spaces to hold, as well as in the study of the asymptotic behaviour of approximation numbers of related embeddings.

scholarly journals About Approximation Numbers in Function Spaces

1998 ◽  
Vol 94 (3) ◽  
pp. 383-395 ◽  
Author(s):  
António M. Caetano
Keyword(s):  
Function Spaces ◽  
Approximation Numbers

Approximation Numbers for Polynomials

2019 ◽  
pp. 35-46
Author(s):  
H.-A. Braunss ◽  
H. Junek ◽  
E. Plewnia
Keyword(s):  
Approximation Numbers

scholarly journals A COMPARISON OF HOFER'S METRICS ON HAMILTONIAN DIFFEOMORPHISMS AND LAGRANGIAN SUBMANIFOLDS

2003 ◽  
Vol 05 (05) ◽  
pp. 803-811 ◽  
Author(s):  
YARON OSTROVER
Keyword(s):  
Symplectic Manifold ◽  
Isometric Embedding ◽  
Lagrangian Submanifolds ◽  
Canonical Embedding ◽  
Hamiltonian Diffeomorphisms

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold (M,ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π2(M) = 0, the canonical embedding of Ham (M) into ℒ, f ↦ graph (f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

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