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Korovkin theorem in modular spaces

100%
Bardaro C., Mantellini I.
Commentationes Mathematicae
|
2007
|
tom 47
|
nr 2
EN
In this paper we obtain an extension of the classical Korovkin theorem in abstract modular spaces. Applications to some discrete and integral operators are discussed.
2
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Topological properties of spaces of linear operators on non-locally convex Orlicz spaces

100%
Oelke A.
Commentationes Mathematicae
|
2010
|
tom 50
|
nr 2
EN
We study the topological properties of the space \(\mathcal{L}(L^\varphi, X)\) of all continuous linear operators from an Orlicz space \(L^\varphi\) (an Orlicz function \(\varphi\) is not necessarily convex) to a Banach space \(X\). We provide the space \(\mathcal{L}(L^\varphi ,X)\) with the Banach space structure. Moreover, we examine the space \(\mathcal{L}_s (L^\varphi, X)\) of all singular operators from \(L^\varphi\) to \(X\).
3
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Approximate controllability of infinite dimensional systems of the n-th order

88%
Respondek J.
International Journal of Applied Mathematics and Computer Science
|
2008
|
tom 18
|
nr 2
199-212
EN
The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation. This innovative approach was possible owing to analyzing the n-th order linear system in the Frobenius form which generates a Jordan transition matrix of the Vandermonde form. We extensively used the fact that the knowledge of the inverse of a Jordan transition matrix enables us to directly verify the controllability by Chen's theorem. We used the explicit analytical form of the inverse Vandermonde matrix. This enabled us to obtain more general conditions for different types of controllability for infinite dimensional systems than the conditions existing in the literature so far. The methods introduced can be easily adapted to the analysis of other dynamic properties of the systems considered.
4
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Best approximation in spaces of bounded linear operators

77%
Lewicki G.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Chapter 0...............................................................................................................................................................................5    0.1. Introduction..................................................................................................................................................................5    0.2. Preliminary results.......................................................................................................................................................9 Chapter I..............................................................................................................................................................................16    I.1. Best approximation in finite-dimensional subspaces of ℒ(B,D)....................................................................................16    I.2. Kolmogorov's type criteria for spaces of compact operators; general case.................................................................26    I.3. Criteria for the space $K(C_K(T))$.............................................................................................................................30    I.4. The case of sequence spaces....................................................................................................................................38 Chapter II.............................................................................................................................................................................43    II.1. Extensions of linear operators from hyperplanes of $l^{(n)}_∞$.................................................................................43    II.2. Minimal projections onto hyperplanes of $l^{(n)}_1$...................................................................................................52    II.3. Strongly unique minimal projections onto hyperplanes of $l^{(n)}_∞$ and $l^{(n)}_1$...............................................59    II.4. Minimal projections onto subspaces of $l^{(n)}_∞$ of codimension two......................................................................71    II.5. Uniqueness of minimal projections onto subspace of $l^{(n)}_∞$ of codimension two................................................75    II.6. Strong unicity criterion in some space of operators....................................................................................................79 Chapter III.............................................................................................................................................................................83    III.1. Extensions of linear operators from finite-dimensional subspaces I...........................................................................83    III.2. Extensions of linear operators from finite-dimensional subspaces II..........................................................................90    III.3. Algorithms for seeking the constant $W_m$..............................................................................................................97 References..........................................................................................................................................................................99 Index..................................................................................................................................................................................102 Index of symbols................................................................................................................................................................102
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