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In this paper we will study the asymptotic behaviour of certain widths of the embeddings $ mathcal{A}_omega(mathbb{T}^d) to L_p(mathbb{T}^d)$, $2le p le infty$, and $ mathcal{A}_omega(mathbb{T}^d) to mathcal{A}(mathbb{T}^d)$, where $mathcal{A}_{omega}(mathbb{T}^d)$ is the weighted Wiener class and $mathcal{A}(mathbb{T}^d)$ is the Wiener algebra on the $d$-dimensional torus $mathbb{T}^d$. Our main interest will consist in the calculation of the associated asymptotic constant. As one of the consequences we also obtain the asymptotic constant related to the embedding $id: C^m_{rm mix}(mathbb{T}^d) to L_2(mathbb{T}^d)$ for Weyl and Bernstein numbers.
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Let $0<p,qleq infty$ and denote by $mathcal{S}_p^N$ and $mathcal{S}_q^N$ the corresponding Schatten classes of real $Ntimes N$ matrices. We study the Gelfand numbers of natural identities $mathcal{S}_p^Nhookrightarrow mathcal{S}_q^N$ between Schatten classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$. This extends classical results for finite-dimensional $ell_p$ sequence spaces by E. Gluskin to the non-commutative setting and complements bounds previously obtained by B. Carl and A. Defant, A. Hinrichs and C. Michels, and J. Chavez-Dominguez and D. Kutzarova.
We develop real Paley-Wiener theorems for classes ${mathcal S}_omega$ of ultradifferentiable functions and related $L^{p}$-spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of depth 1 embedded nets with a single vertex quotient graph. Additionally, we classify embeddings of n-fold copies of pcu with all connected components in a parallel orientation and n vertices in a repeat unit, and determine their maximal symmetry periodic isotopes. We also introduce the methodology of linear graph knots on the flat 3-torus [0, 1)^3. These graph knots, with linear edges, are spatial embeddings of the labelled quotient graphs of an embedded net which are associated with its periodicity bases.
Let $PW_S^1$ be the space of integrable functions on $mathbb{R}$ whose Fourier transform vanishes outside $S$, where $S = [-sigma,-rho]cup[rho,sigma]$, $0<rho<sigma$. In the case $rho>sigma/2$, we present a complete description of the extreme points of the unit ball of $PW_S^1$. This description is no longer true if $rho<sigma/2$. For $rho>sigma/2$ we also show that every $f in PW^1_S, , |f|_1 =1,$ can be represented as $f = (f_1 + f_2)/2$ where $f_1$ and $f_2$ are extreme.
In this paper, we characterize hypercyclic sequences of weighted translation operators on an Orlicz space in the context of locally compact hypergroups.
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