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Korovkin-type approximation theory

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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 and be topological vector spaces, and consider classes and of continuous linear operators from to (cf. also Linear operator; Topological vector space).

For a linear operator , a subset is called a Korovkin subset of (for with respect to ) if for every equicontinuous net (cf. also Equicontinuity) of operators in satisfying

(a1)

one has

(a2)

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 (for with respect to ). Such theorems are called Korovkin-type theorems.

Similar problems have been settled for nets 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 (for with respect to ), which is defined as the subspace of all elements such that if is an equicontinuous net of operators in satisfying (a1), then .

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)
How to Cite This Entry:
Korovkin-type approximation theory. F. Altomare (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Korovkin-type_approximation_theory&oldid=16676
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098