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.

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