scholarly journals De Branges Type Lemma and Approximation in Weighted Spaces

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
I. Bucur ◽  
G. Paltineanu
Keyword(s):  
Compact Space ◽  
Compact Support ◽  
Scalar Function ◽  
Convex Cones ◽  
Weighted Spaces ◽  
Locally Compact ◽  
Approximation Theorems ◽  
Locally Compact Space ◽  
Bounded Functions ◽  
Vector Subspaces

AbstractThe purpose of this paper is to give some generalizations of de Branges Lemma for weighted spaces to obtain different approximation theorems in weighted spaces for algebras, vector subspaces or convex cones. We recall that the (original) de Branges Lemma (Proc Am Math Soc 10(5):822–824, 1959) was demonstrated for continuous scalar function on a compact space while, the weighted spaces are classes of continuous scalar functions on a locally compact space (e.g. the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity).

On Unsymmetric Dirichlet Forms

1973 ◽  
Vol 25 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Joanne Elliott
Keyword(s):  
Compact Space ◽  
Measure Space ◽  
Closed Subspace ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Finite Measure Space

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

Actions That Fiber and Vector Semigroups

1972 ◽  
Vol 24 (1) ◽  
pp. 29-37 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Compact Space ◽  
Lie Group ◽  
Fiber Bundle ◽  
Closed Subset ◽  
Locally Compact ◽  
Locally Compact Space

From [2], we can derive a criterion for determining when an action of a Lie group on a locally compact space leads to a fiber bundle. Here, we present an equivalent criterion which can be stated purely in the language of actions of groups on spaces. This is Theorem I. Using this result, we are able to give a version of a result of Home [1] for dimensions greater than one. This is done in Theorem IV and Corollary IVA. In Theorem II, we show that if a vector semigroup acts on a space X, then whenever the map t ↦ tx is 1 — 1 from onto x, it is in fact a homeomorphism. Also, is a closed subset of X. This is also a version of a result in [1].

ON ALGEBRAIC EQUATION WITH COEFFICIENTS FROM THE $ \beta $-UNIFORM ALGEBRA $ C_{\beta} (\Omega) $

2018 ◽  
Vol 52 (3 (247)) ◽  
pp. 161-165
Author(s):  
A.H. Kamalyan ◽  
M.I. Karakhanyan
Keyword(s):  
Algebraic Equation ◽  
Compact Space ◽  
Uniform Algebra ◽  
Locally Compact ◽  
Locally Compact Space

In this work the question of algebraic closeness of $ \beta $-uniform algebra $ A (\Omega) $ defined on locally compact space $ \Omega $ is investigated.

scholarly journals On the locally functionally countable subalgebra of C(X) on locally functionally countable subalgebra of C(X)

2015 ◽  
Vol 16 (2) ◽  
pp. 183 ◽  
Author(s):  
O. A. S. Karamzadeh ◽  
M. Namdari ◽  
S. Soltanpour
Keyword(s):  
Prime Ideal ◽  
Compact Space ◽  
Maximal Ideal ◽  
Regular Ring ◽  
Connected Subset ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Space

<p><br />Let $C_c(X)=\{f\in C(X) : |f(X)|\leq \aleph_0\}$, $C^F(X)=\{f\in C(X): |f(X)|&lt;\infty\}$, and $L_c(X)=\{f\in C(X) : \overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $|f(U)|\leq\aleph_0$, and $C_F(X)$ be the socle of $C(X)$ (i.e., the sum of minimal ideals of $C(X)$). It is shown that if $X$ is a locally compact space, then $L_c(X)=C(X)$ if and only if $X$ is locally scattered.<br />We observe that $L_c(X)$ enjoys most of the important properties which are shared by $C(X)$ and $C_c(X)$.<br /> Spaces $X$ such that $L_c(X)$ is regular (von Neumann) are characterized. Similarly to $C(X)$ and $C_c(X)$, it is shown that $L_c(X)$ is a regular ring if and only if it is $\aleph_0$-selfinjective.<br />We also determine spaces $X$ such that ${\rm Soc}{\big(}L_c(X){\big)}=C_F(X)$ (resp., ${\rm Soc}{\big(}L_c(X){\big)}={\rm Soc}{\big(}C_c(X){\big)}$). It is proved that if $C_F(X)$ is a maximal ideal in $L_c(X)$, then $C_c(X)=C^F(X)=L_c(X)\cong \prod\limits_{i=1}^n R_i$, where $R_i=\mathbb R$ for each $i$, and $X$ has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces $X$ for which $C_F(X)$ is a prime ideal in $L_c(X)$<br />are characterized and consequently for these spaces, we infer that $L_c(X)$ can not be isomorphic to any $C(Y)$. <br /><br /></p>

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