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Korovkin's first theorem states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101301.png" /> is an arbitrary sequence of positive linear operators on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101302.png" /> of real-valued continuous functions on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101303.png" /> (cf. [[Continuous functions, space of|Continuous functions, space of]]; [[Linear operator|Linear operator]]) and if
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101304.png" /></td> </tr></table>
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{{TEX|done}}
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101305.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101306.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101307.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101308.png" />), then
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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|Continuous functions, space of]]; [[Linear operator|Linear operator]]) and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k1101309.png" /></td> </tr></table>
+
$$
 +
{\lim\limits } _ {n \rightarrow \infty } L _ {n} ( h ) = h  \textrm{ uniformly  on  }  [ 0,1 ]
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013010.png" />.
+
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
  
The statement of Korovkin's second theorem is similar to that of the first theorem, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013011.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013012.png" /> (the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013013.png" />-periodic real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013014.png" />, endowed with the topology of [[Uniform convergence|uniform convergence]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013015.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013016.png" /> is taken from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013017.png" />.
+
$$
 +
{\lim\limits } _ {n \rightarrow \infty } L _ {n} ( f ) = f  \textrm{ uniformly  on }  [ 0,1 ]
 +
$$
  
These theorems were proved by P.P. Korovkin in 1953 ([[#References|[a3]]], [[#References|[a4]]]). In 1952, H. Bohman [[#References|[a2]]] had proved a result similar to Korovkin's first theorem but concerning sequences of positive linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013018.png" /> of the form
+
for all  $  f \in C [ 0,1 ] $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013019.png" /></td> </tr></table>
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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|uniform convergence]] on  $  \mathbf R $)
 +
and  $  h $
 +
is taken from the set  $  \{ e _ {0} ,  \cos  ,  \sin  \} $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013020.png" /> is a finite set of numbers in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013022.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013023.png" />). Therefore Korovkin's first theorem is also known as the [[Bohman–Korovkin theorem|Bohman–Korovkin theorem]]. However, T. Popoviciu [[#References|[a5]]] had already proved the essence of the theorem in 1950.
+
These theorems were proved by P.P. Korovkin in 1953 ([[#References|[a3]]], [[#References|[a4]]]). In 1952, H. Bohman [[#References|[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
  
Korovkin has tried to generalize his first theorem by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013024.png" /> with other finite subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013025.png" />. He has shown that if a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013026.png" />  "behaves like"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013028.png" /> (Korovkin's third theorem). Moreover, he showed that a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013029.png" /> "behaves like"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013030.png" /> if and only if it is a [[Chebyshev system|Chebyshev system]] of order two.
+
$$
 +
L ( f ) = \sum _ {i \in I } f ( a _ {i} ) \phi _ {i} f \in C [ 0,1 ] ,
 +
$$
  
The Korovkin theorems are simple yet powerful tools for deciding whether a given sequence of positive linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013031.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110130/k11013032.png" /> 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|Korovkin-type approximation theory]]).
+
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|Bohman–Korovkin theorem]]. However, T. Popoviciu [[#References|[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|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|Korovkin-type approximation theory]]).
  
 
See [[#References|[a1]]] for a modern and comprehensive exposition of these results and for (some) applications.
 
See [[#References|[a1]]] for a modern and comprehensive exposition of these results and for (some) applications.

Latest revision as of 22:15, 5 June 2020


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.

References

[a1] F. Altomare, M. Campiti, "Korovkin-type approximation theory and its application" , de Gruyter studies in math. , 17 , de Gruyter (1994)
[a2] H. Bohman, "On approximation of continuous and analytic functions" Ark. Math. , 2 (1952–1954) pp. 43–56
[a3] P.P. Korovkin, "On convergence of linear positive operators in the space of continuous functions" Dokl. Akad. Nauk. SSSR (N.S.) , 90 (1953) pp. 961–964 (In Russian)
[a4] P.P. Korovkin, "Linear operators and approximation theory" , Gordon&Breach (1960) (In Russian)
[a5] T. Popoviciu, "On the proof of the Weierstrass theorem using interpolation polynomials" Lucr. Ses. Gen. Stiintific. , 2 : 12 (1950) pp. 1664–1667
How to Cite This Entry:
Korovkin theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Korovkin_theorems&oldid=13016
This article was adapted from an original article by F. Altomare (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article