Difference between revisions of "Korovkin-type approximation theory"
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Originating from Korovkin's theorems (cf. [[Korovkin theorems|Korovkin theorems]]), this theory consists of a collection of results whose main objective is to investigate under what circumstances the convergence of a (particular) sequence (more generally, net) of linear operators acting on a [[Topological vector space|topological vector space]] is, in fact, a consequence of its convergence on special (possibly finite) subsets. | Originating from Korovkin's theorems (cf. [[Korovkin theorems|Korovkin theorems]]), this theory consists of a collection of results whose main objective is to investigate under what circumstances the convergence of a (particular) sequence (more generally, net) of linear operators acting on a [[Topological vector space|topological vector space]] is, in fact, a consequence of its convergence on special (possibly finite) subsets. | ||
| − | More precisely, let | + | More precisely, let $ E $ |
| + | and $ F $ | ||
| + | be topological vector spaces, and consider classes $ {\mathcal L} _ {0} $ | ||
| + | and $ {\mathcal L} $ | ||
| + | of continuous linear operators from $ E $ | ||
| + | to $ F $( | ||
| + | cf. also [[Linear operator|Linear operator]]; [[Topological vector space|Topological vector space]]). | ||
| − | For a linear operator | + | For a linear operator $ T \in {\mathcal L} _ {0} $, |
| + | a subset $ H \subset E $ | ||
| + | is called a Korovkin subset of $ E $( | ||
| + | for $ T $ | ||
| + | with respect to $ {\mathcal L} $) | ||
| + | if for every equicontinuous net $ ( L _ {i} ) _ {i \in I } ^ \leq $( | ||
| + | cf. also [[Equicontinuity|Equicontinuity]]) of operators in $ {\mathcal L} $ | ||
| + | satisfying | ||
| − | + | $$ \tag{a1 } | |
| + | { {\lim\limits } _ {i \in I } } ^ \leq L _ {i} ( h ) = T ( h ) \textrm{ for all } h \in H, | ||
| + | $$ | ||
one has | one has | ||
| − | + | $$ \tag{a2 } | |
| + | { {\lim\limits } _ {i \in I } } ^ \leq L _ {i} ( f ) = T ( f ) \textrm{ for all } f \in E. | ||
| + | $$ | ||
| − | One of the main problems in this theory is to state sufficient (and necessary) conditions under which a given subset is a Korovkin subset in | + | One of the main problems in this theory is to state sufficient (and necessary) conditions under which a given subset is a Korovkin subset in $ E $( |
| + | for $ T $ | ||
| + | with respect to $ {\mathcal L} $). | ||
| + | Such theorems are called Korovkin-type theorems. | ||
| − | Similar problems have been settled for nets | + | Similar problems have been settled for nets $ ( L _ {i} ) _ {i \in I } ^ \leq $ |
| + | of operators that are not necessarily equicontinuous or linear. | ||
| − | Another important aspect of the theory is the characterization of the Korovkin closure, or Korovkin shadow, of a given subset | + | Another important aspect of the theory is the characterization of the Korovkin closure, or Korovkin shadow, of a given subset $ H $( |
| + | for $ T $ | ||
| + | with respect to $ {\mathcal L} $), | ||
| + | which is defined as the subspace of all elements $ f \in E $ | ||
| + | such that if $ ( L _ {i} ) _ {i \in I } ^ \leq $ | ||
| + | is an equicontinuous net of operators in $ {\mathcal L} $ | ||
| + | satisfying (a1), then $ { {\lim\limits } _ {i \in I } } ^ \leq L _ {i} ( f ) = T ( f ) $. | ||
Korovkin-type approximation theory has been developed in the context of classical function spaces and in more abstract spaces, such as locally convex ordered spaces, Banach lattices, Banach algebras, Banach spaces, etc. | Korovkin-type approximation theory has been developed in the context of classical function spaces and in more abstract spaces, such as locally convex ordered spaces, Banach lattices, Banach algebras, Banach spaces, etc. | ||
Latest revision as of 22:15, 5 June 2020
Originating from Korovkin's theorems (cf. Korovkin theorems), this theory consists of a collection of results whose main objective is to investigate under what circumstances the convergence of a (particular) sequence (more generally, net) of linear operators acting on a topological vector space is, in fact, a consequence of its convergence on special (possibly finite) subsets.
More precisely, let $ E $ and $ F $ be topological vector spaces, and consider classes $ {\mathcal L} _ {0} $ and $ {\mathcal L} $ of continuous linear operators from $ E $ to $ F $( cf. also Linear operator; Topological vector space).
For a linear operator $ T \in {\mathcal L} _ {0} $, a subset $ H \subset E $ is called a Korovkin subset of $ E $( for $ T $ with respect to $ {\mathcal L} $) if for every equicontinuous net $ ( L _ {i} ) _ {i \in I } ^ \leq $( cf. also Equicontinuity) of operators in $ {\mathcal L} $ satisfying
$$ \tag{a1 } { {\lim\limits } _ {i \in I } } ^ \leq L _ {i} ( h ) = T ( h ) \textrm{ for all } h \in H, $$
one has
$$ \tag{a2 } { {\lim\limits } _ {i \in I } } ^ \leq L _ {i} ( f ) = T ( f ) \textrm{ for all } f \in E. $$
One of the main problems in this theory is to state sufficient (and necessary) conditions under which a given subset is a Korovkin subset in $ E $( for $ T $ with respect to $ {\mathcal L} $). Such theorems are called Korovkin-type theorems.
Similar problems have been settled for nets $ ( L _ {i} ) _ {i \in I } ^ \leq $ of operators that are not necessarily equicontinuous or linear.
Another important aspect of the theory is the characterization of the Korovkin closure, or Korovkin shadow, of a given subset $ H $( for $ T $ with respect to $ {\mathcal L} $), which is defined as the subspace of all elements $ f \in E $ such that if $ ( L _ {i} ) _ {i \in I } ^ \leq $ is an equicontinuous net of operators in $ {\mathcal L} $ satisfying (a1), then $ { {\lim\limits } _ {i \in I } } ^ \leq L _ {i} ( f ) = T ( f ) $.
Korovkin-type approximation theory has been developed in the context of classical function spaces and in more abstract spaces, such as locally convex ordered spaces, Banach lattices, Banach algebras, Banach spaces, etc.
Powerful and fruitful connections of this theory have been discovered not only with classical approximation theory, but also with fields such as: functional analysis; harmonic analysis; measure theory; probability theory; potential theory; and the theory of partial differential equations (cf. Differential equation, partial).
References
| [a1] | F. Altomare, M. Campiti, "Korovkin-type approximation theory and its application" , de Gruyter studies in math. , 17 , de Gruyter (1994) |
| [a2] | H. Bauer, "Approximation and abstract boundaries" Amer. Math. Monthly , 85 (1978) pp. 632–647 |
| [a3] | K. Donner, "Extension of positive operators and Korovkin theorem" , Lecture Notes in Mathematics , 904 , Springer (1982) |
| [a4] | K. Keimel, W. Roth, "Ordered cones and approximation" , Lecture Notes in Mathematics , 1517 , Springer (1992) |
Korovkin-type approximation theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Korovkin-type_approximation_theory&oldid=16676