Example of a thick polynomially convex compact subset of space ?2, with connected interior, on which not every continuous function analytic at interior points is uniformly approximable by polynomials

1974 ◽  
Vol 2 (2) ◽  
pp. 226-227 ◽  
Author(s):  
V. N. Senichkin
Keyword(s):  
Continuous Function ◽  
Compact Subset ◽  
Convex Compact ◽  
Convex Compact Subset ◽  
Polynomially Convex ◽  
Interior Points

scholarly journals Tietze Extension Theorem for n-dimensional Spaces

2014 ◽  
Vol 22 (1) ◽  
pp. 11-19 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Compact Subset ◽  
Closed Subset ◽  
Extension Theorem ◽  
Convex Compact ◽  
Arbitrary Subset ◽  
Empty Interior ◽  
Convex Compact Subset ◽  
Arbitrary Convex

Summary In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of En with a non-empty interior. This article is based on [20]; [23] and [22] can also serve as reference books.

scholarly journals Brouwer Fixed Point Theorem in the General Case

2011 ◽  
Vol 19 (3) ◽  
pp. 151-153 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Compact Subset ◽  
Point Theorem ◽  
Convex Compact ◽  
Empty Interior ◽  
Brouwer Fixed Point Theorem ◽  
Convex Compact Subset ◽  
Arbitrary Convex

Brouwer Fixed Point Theorem in the General Case In this article we prove the Brouwer fixed point theorem for an arbitrary convex compact subset of εn with a non empty interior. This article is based on [15].

scholarly journals Smoothness in spaces of compact operators

1988 ◽  
Vol 37 (2) ◽  
pp. 221-225 ◽  
Author(s):  
A. Sersouri
Keyword(s):  
Banach Spaces ◽  
Compact Subset ◽  
Compact Operators ◽  
Convex Compact ◽  
Convex Compact Subset

We prove that if X and Y are two (real) Banach spaces such that dim X ≥ 2 and dim Y ≥ 2, then the space K(X, Y) contains a convex compact subset C with dim C ≥ 2 (in the affine sense) which fails to be an intersection of balls. This improves two results of Ruess and Stegall.

Unisolvence on Multidimensional Spaces

1968 ◽  
Vol 11 (3) ◽  
pp. 469-474 ◽  
Author(s):  
Charles B. Dunham
Keyword(s):  
Continuous Function ◽  
Euclidean Space ◽  
Compact Subset ◽  
Compact Space ◽  
General Condition ◽  
Chebyshev Approximation ◽  
Continuous Functions ◽  
Dimensional Euclidean Space ◽  
Variable Degree ◽  
Convexity Conditions

In this note we consider the possibility of unisolvence of a family of real continuous functions on a compact subset X of m-dimensional Euclidean space. Such a study is of interest for two reasons. First, an elegant theory of Chebyshev approximation has been constructed for the case where the approximating family is unisolvent of degree n on an interval [α, β]. We study what sort of theory results from unisolvence of degree n on a more general space. Secondly, uniqueness of best Chebyshev approximation on a general compact space to any continuous function on X can be shown if the approximating family is unisolvent of degree n and satisfies certain convexity conditions. It is therefore of importance to Chebyshev approximation to consider the domains X on which unisolvence can occur. We will also study a more general condition on involving a variable degree.

scholarly journals An estimate for the total mean curvature in negatively curved spaces

2002 ◽  
Vol 66 (2) ◽  
pp. 267-273
Author(s):  
Albert Borbély
Keyword(s):  
Compact Subset ◽  
Mean Curvature ◽  
Smooth Boundary ◽  
Convex Compact ◽  
Empty Interior ◽  
Connected Manifold ◽  
Total Mean Curvature ◽  
Negatively Curved ◽  
Nonpositively Curved ◽  
Simply Connected

Let Mn be a nonpositively curved complete simply connected manifold and D ⊂ Mn be a convex compact subset with non-empty interior and smooth boundary. It is shown that the total mean curvature ∂D can be estimated in terms of volume and curvature bound.

Superresolution Rates in Prokhorov Metric

1996 ◽  
Vol 48 (2) ◽  
pp. 316-329 ◽  
Author(s):  
P. Doukhan ◽  
F. Gamboa
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Convex Hull ◽  
Compact Subset ◽  
Prokhorov Metric ◽  
System Extension

AbstractConsider the problem of recovering a probability measure supported by a compact subsetUof ℝmwhen the available measurements concern only some of its Ф-moments (Ф being an ℝkvalued continuous function onU). When thetrueФ-momentclies on the boundary of the convex hull of Ф(U), generalizing the results of [10], we construct asmallsetRα,δ(∊)such that any probability measureμsatisfyingisalmostconcentrated onRα,δ(∊). When Ф is a pointwiseT-system (extension ofT-systems), the study of the setRα,δ(∊)leads to the evaluation of the Prokhorov radius of the set.

AN ACCESS THEOREM FOR CONTINUOUS FUNCTIONS

2001 ◽  
Vol 64 (1) ◽  
pp. 81-92
Author(s):  
ALEXANDER BORICHEV ◽  
IGOR KLESCHEVICH
Keyword(s):  
Continuous Function ◽  
Compact Subset ◽  
Open Subset ◽  
Higher Dimensions ◽  
Continuous Functions ◽  
Continuous Map

Let f be a continuous function on an open subset Ω of ℝ2 such that for every x ∈ Ω there exists a continuous map γ : [−1, 1] → Ω with γ(0) = x and f ∘ γ increasing on [−1, 1]. Then for every γ ∈ Ω there exists a continuous map γ : [0, 1) → Ω such that γ(0) = y, f ∘ γ is increasing on [0; 1), and for every compact subset K of Ω, max{t : γ(t) ∈ K} < 1. This result gives an answer to a question posed by M. Ortel. Furthermore, an example shows that this result is not valid in higher dimensions.

A theorem on bounded synthesis

1971 ◽  
Vol 70 (1) ◽  
pp. 23-26
Author(s):  
E. Galanis
Keyword(s):  
Abelian Group ◽  
Continuous Function ◽  
Compact Subset ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Image Position

LetGbe a locally compact Abelian group.DEFINITION 1. A compact subset K ⊂ G is called Kroneclcer set if for every continuous function f on K of modulus identically one (|f(x)| = 1, ∀x ∈ K) and for every ε 0 there exists x ∈ Ĝ such that

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

Generalized ʎ-Closed Sets and (ʎ-ʏ)*-Continuous Function

2017 ◽  
Vol 4 (ICBS Conference) ◽  
pp. 1-17 ◽  
Author(s):  
Alias Khalaf ◽  
Sarhad Nami
Keyword(s):  
Continuous Function ◽  
Closed Sets
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