# Korovkin theorems

Korovkin's first theorem states that if $( L _ {n} ) _ {n \geq 1 }$ is an arbitrary sequence of positive linear operators on the space $C [ 0,1 ]$ of real-valued continuous functions on the interval $[ 0,1 ]$( cf. Continuous functions, space of; Linear operator) and if

$${\lim\limits } _ {n \rightarrow \infty } L _ {n} ( h ) = h \textrm{ uniformly on } [ 0,1 ]$$

for all $h \in \{ e _ {0} ,e _ {1} ,e _ {2} \}$, where $e _ {k} ( t ) = t ^ {k}$( $0 \leq t \leq 1$, $k = 0,1,2$), then

$${\lim\limits } _ {n \rightarrow \infty } L _ {n} ( f ) = f \textrm{ uniformly on } [ 0,1 ]$$

for all $f \in C [ 0,1 ]$.

The statement of Korovkin's second theorem is similar to that of the first theorem, but $C [ 0,1 ]$ is replaced by $C _ {2 \pi } ( \mathbf R )$( the space of $2 \pi$- periodic real-valued functions on $\mathbf R$, endowed with the topology of uniform convergence on $\mathbf R$) and $h$ is taken from the set $\{ e _ {0} , \cos , \sin \}$.

These theorems were proved by P.P. Korovkin in 1953 ([a3], [a4]). In 1952, H. Bohman [a2] had proved a result similar to Korovkin's first theorem but concerning sequences of positive linear operators on $C [ 0,1 ]$ of the form

$$L ( f ) = \sum _ {i \in I } f ( a _ {i} ) \phi _ {i} , f \in C [ 0,1 ] ,$$

where $( a _ {i} ) _ {i \in I }$ is a finite set of numbers in $[ 0,1 ]$ and $\phi _ {i} \in C [ 0,1 ]$( $i \in I$). Therefore Korovkin's first theorem is also known as the Bohman–Korovkin theorem. However, T. Popoviciu [a5] had already proved the essence of the theorem in 1950.

Korovkin has tried to generalize his first theorem by replacing $\{ e _ {0} ,e _ {1} ,e _ {2} \}$ with other finite subsets of $C [ 0,1 ]$. He has shown that if a subset $\{ f _ {1} \dots f _ {n} \} \subset C [ 0,1 ]$" behaves like" $\{ e _ {0} ,e _ {1} ,e _ {2} \}$, then $n > 2$( Korovkin's third theorem). Moreover, he showed that a subset $\{ f _ {0} ,f _ {1} ,f _ {2} \} \subset C [ 0,1 ]$" behaves like" $\{ e _ {0} ,e _ {1} ,e _ {2} \}$ if and only if it is a Chebyshev system of order two.

The Korovkin theorems are simple yet powerful tools for deciding whether a given sequence of positive linear operators on $C [ 0,1 ]$ or $C _ {2 \pi } ( \mathbf R )$ is an approximation process. Furthermore, they have been the source of a considerable amount of research in several other fields of mathematics (cf. Korovkin-type approximation theory).

See [a1] for a modern and comprehensive exposition of these results and for (some) applications.

How to Cite This Entry:
Korovkin theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Korovkin_theorems&oldid=47520
This article was adapted from an original article by F. Altomare (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article