scholarly journals Best Simultaneous approximation of quasi-continuous functions by monotone functions

1991 ◽  
Vol 50 (3) ◽  
pp. 391-408 ◽  
Author(s):  
Salem M. A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Continuous Functions ◽  
Unit Interval ◽  
Closed Convex Cone ◽  
Monotone Functions ◽  
Measurable Functions ◽  
Best Simultaneous Approximation ◽  
Nondecreasing Functions

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

scholarly journals On the monotone simultaneous approximation on [0, 1]

1988 ◽  
Vol 38 (3) ◽  
pp. 401-411 ◽  
Author(s):  
Salem M.A. Sahab
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Simultaneous Approximation ◽  
Closed Interval ◽  
Continuous Functions ◽  
Closed Convex Cone ◽  
Measurable Functions ◽  
Decreasing Functions

Let Ω denote the closed interval [0, 1] and let bA denote the set of all bounded, approximately continuous functions on Ω. Let Q denote the Banach space (sup norm) of quasi-continuous functions on Ω. Let M denote the closed convex cone in Q comprised of non-decreasing functions. Let hp, 1 < p < ∞, denote the best Lp-simultaneaous approximation to the bounded measurable functions f and g by elements of M. It is shown that if f and g are elements of Q, then hp converges unifornily to a best L1-simultaneous approximation of f and g. We also show that if f and g are in bA, then hp is continuous.

scholarly journals Best Simultaneous Approximation in Orlicz Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-7 ◽  
Author(s):  
M. Khandaqji ◽  
Sh. Al-Sharif
Keyword(s):  
Banach Space ◽  
Simultaneous Approximation ◽  
Closed Subspace ◽  
Orlicz Spaces ◽  
Unit Interval ◽  
Luxemburg Norm ◽  
Integrable Functions ◽  
Best Simultaneous Approximation ◽  
Distance Formula

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

Cone-Monotone Functions: Differentiability and Continuity

2005 ◽  
Vol 57 (5) ◽  
pp. 961-982 ◽  
Author(s):  
Jonathan M. Borwein ◽  
Xianfu Wang
Keyword(s):  
Banach Spaces ◽  
Convex Cone ◽  
Continuous Functions ◽  
Monotone Functions ◽  
Empty Interior ◽  
Space Of Continuous Functions

AbstractWe provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone K with non-empty interior. We also show that the set of nowhere K-monotone functions has a σ-porous complement in the space of continuous functions endowed with the uniform metric.

scholarly journals Best simultaneous approximation on metric spaces via monotonous norms

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3777-3787
Author(s):  
Mona Khandaqji ◽  
Aliaa Burqan
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Finite Number ◽  
Metric Space ◽  
Measure Space ◽  
Simultaneous Approximation ◽  
Metric Spaces ◽  
Closed Subspace ◽  
Integrable Functions ◽  
Best Simultaneous Approximation

For a Banach space X, L?(T,X) denotes the metric space of all X-valued ?-integrable functions f : T ? X, where the measure space (T,?,?) is a complete positive ?-finite and ? is an increasing subadditive continuous function on [0,?) with ?(0) = 0. In this paper we discuss the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y?X to a finite number of elements in X.

An example regarding Kalton's paper ‘isomorphisms between spaces of vector-valued continuous functions’

2021 ◽  
pp. 1-5
Author(s):  
Félix Cabello Sánchez
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Continuous Functions ◽  
Unit Interval ◽  
Finite Covering ◽  
Covering Dimension ◽  
Striking Result ◽  
Vector Valued ◽  
Locally Convex

Abstract The paper alluded to in the title contains the following striking result: Let $I$ be the unit interval and $\Delta$ the Cantor set. If $X$ is a quasi Banach space containing no copy of $c_{0}$ which is isomorphic to a closed subspace of a space with a basis and $C(I,\,X)$ is linearly homeomorphic to $C(\Delta ,\, X)$ , then $X$ is locally convex, i.e., a Banach space. We will show that Kalton result is sharp by exhibiting non-locally convex quasi Banach spaces $X$ with a basis for which $C(I,\,X)$ and $C(\Delta ,\, X)$ are isomorphic. Our examples are rather specific and actually, in all cases, $X$ is isomorphic to $C(K,\,X)$ if $K$ is a metric compactum of finite covering dimension.

Some families of sublinear correspondences

2019 ◽  
Vol 25 (1) ◽  
pp. 91-95
Author(s):  
Parvaneh Najmadi ◽  
Masoumeh Aghajani
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Closed Convex Cone ◽  
Nonempty Convex

Abstract Let K be a closed convex cone in a real Banach space, {H\colon K\to\operatorname{cc}(K)} a continuous sublinear correspondence with nonempty, convex and compact values in K, and let {f\colon\mathbb{R}\to\mathbb{R}} be defined by {f(t)=\sum_{n=0}^{\infty}a_{n}t^{n}} , where {t\in\mathbb{R}} , {a_{n}\geq 0} , {n\in\mathbb{N}} . We show that the correspondence {F^{t}(x)\mathrel{\mathop{:}}=\sum_{n=0}^{\infty}a_{n}t^{n}H^{n}(x),(x\in K)} is continuous and sublinear for every {t\geq 0} and {F^{t}\circ F^{s}(x)\subseteq\sum_{n=0}^{\infty}c_{n}H^{n}(x)} , {x\in K} , where {c_{n}=\sum_{k=0}^{n}a_{k}a_{n-k}t^{k}s^{n-k}} , {t,s\geq 0} .

On a Multivalued Differential Problem

2003 ◽  
Vol 13 (07) ◽  
pp. 1877-1882 ◽  
Author(s):  
Andrzej Smajdor
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Convex Compact ◽  
Closed Convex Cone ◽  
Continuous Linear ◽  
Differential Problem ◽  
Linear Set ◽  
Nonempty Convex ◽  
The Family

Let K be a closed convex cone with the nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. Assume that two continuous linear set-valued functions G, Ψ : K → cc(K) are given. The following problem is considered: [Formula: see text] for t ≥ 0 and x ∈ K, where DtΦ(t, x) denotes the Hukuhara derivative of Φ(t, x). with respect to t.

On the Structure of Finite Level and ω-Decomposable Borel Functions

2013 ◽  
Vol 78 (4) ◽  
pp. 1257-1287 ◽  
Author(s):  
Luca Motto Ros
Keyword(s):  
Banach Space ◽  
Elementary Proof ◽  
Borel Function ◽  
Continuous Functions ◽  
Full Description ◽  
Countable Union ◽  
Space Theory ◽  
Measurable Functions ◽  
Borel Functions

AbstractWe give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σα0-measurable functions (for every fixed 1 ≤ α < ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.

On Best Simultaneous Approximation in Normed Linear Spaces

1974 ◽  
Vol 17 (4) ◽  
pp. 523-527 ◽  
Author(s):  
D. S. Goel ◽  
A. S. B. Holland ◽  
C. Nasim ◽  
B. N. Sahney
Keyword(s):  
Simultaneous Approximation ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Best Simultaneous Approximation ◽  
Image Position

Let S be a non-empty family of real valued continuous functions on [a, b]. Diaz and McLaughlin [1], [2], and Dunham [3] have considered the problem of simultaneously approximating two continuous functions f1 and f2 by elements of S. If || • || denotes the supremum norm, then the problem is to find an element * ∈ S if it exists, for which

Commutativity of set-valued cosine families

2014 ◽  
Vol 12 (12) ◽  
Author(s):  
Andrzej Smajdor ◽  
Wilhelmina Smajdor
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Real Banach Space ◽  
Nonempty Interior ◽  
Convex Compact ◽  
Closed Convex Cone ◽  
Cosine Family ◽  
Cosine Families ◽  
Nonempty Convex ◽  
The Family

AbstractLet K be a closed convex cone with nonempty interior in a real Banach space and let cc(K) denote the family of all nonempty convex compact subsets of K. If {F t: t ≥ 0} is a regular cosine family of continuous additive set-valued functions F t: K → cc(K) such that x ∈ F t(x) for t ≥ 0 and x ∈ K, then $F_t \circ F_s (x) = F_s \circ F_t (x)fors,t \geqslant 0andx \in K$.

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