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The starting point is the following theorem of P.P. Korovkin (1953) [[#References|[a6]]], [[#References|[a7]]]: for a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107001.png" /> of positive linear operators on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107002.png" /> into itself (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107003.png" /> is a compact interval and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107004.png" /> denotes the [[Banach space|Banach space]] of all continuous real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107005.png" /> with the supremum norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107006.png" />), the relations
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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/b/b110/b110700/b1107007.png" /></td> </tr></table>
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The starting point is the following theorem of P.P. Korovkin (1953) [[#References|[a6]]], [[#References|[a7]]]: for a given sequence  $  ( L _ {n} ) _ {n \leq 1 }  $
 +
of positive linear operators on the space  $  C ( A ) $
 +
into itself (where  $  A = [a,b] $
 +
is a compact interval and  $  C ( A ) $
 +
denotes the [[Banach space|Banach space]] of all continuous real-valued functions on  $  A $
 +
with the supremum norm  $  \| g \| = \sup  \{ {\| {g ( x ) } \| } : {x \in A } \} $),
 +
the relations
 +
 
 +
$$
 +
{\lim\limits } _ {n \rightarrow \infty } \left \| {e _ {k} - L _ {n} e _ {k} } \right \| = 0,  k = 0,1,2,
 +
$$
  
 
imply convergence:
 
imply convergence:
  
<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/b/b110/b110700/b1107008.png" /></td> </tr></table>
+
$$
 +
{\lim\limits } _ {n \rightarrow \infty } \left \| {f - L _ {n} f } \right \| = 0 \textrm{ for  any  }  f \in C ( A ) .
 +
$$
  
Here, the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b1107009.png" /> are defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070011.png" /> is the supremum norm on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070012.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070013.png" /> is positive (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070015.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070017.png" />, and the [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070018.png" /> is positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070019.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070020.png" />. One year earlier, in 1952, H. Bohman [[#References|[a2]]] had proved this theorem for positive operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070021.png" /> having a representation
+
Here, the functions $  e _ {k} $
 +
are defined by $  e _ {k} ( x ) = x  ^ {k} $
 +
and $  \| \cdot \| $
 +
is the supremum norm on the interval $  A $.  
 +
An element $  f \in C ( A ) $
 +
is positive (denoted by $  f \geq  0 $)  
 +
if $  f ( x ) \geq  0 $
 +
for all $  x \in A $,  
 +
and the [[Linear operator|linear operator]] $  L _ {n} $
 +
is positive if $  f \geq  0 $
 +
implies $  L _ {n} f \geq  0 $.  
 +
One year earlier, in 1952, H. Bohman [[#References|[a2]]] had proved this theorem for positive operators $  L _ {n} $
 +
having a representation
  
<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/b/b110/b110700/b11070022.png" /></td> </tr></table>
+
$$
 +
L _ {n} f = \sum _ {k = 0 } ^ { n }  f ( x _ {k,n }  ) \cdot \psi _ {k,n }
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070023.png" /> knots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070024.png" /> in the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070027.png" />.
+
with $  n + 1 $
 +
knots $  x _ {k,n }  $
 +
in the open interval $  ( 0,1 ) $,
 +
and $  \psi _ {k,n }  \geq  0 $
 +
on $  A = [0,1] $.
  
One says that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070028.png" /> are a Korovkin set for (positive linear operators on) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070029.png" />. Korovkin proved that a set of three functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070030.png" /> that replaces the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070031.png" /> must be a [[Chebyshev system|Chebyshev system]]; he also proved that there are no sets of only two functions such that convergence for these two functions implies convergence for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070032.png" /> (cf. also [[Korovkin theorems|Korovkin theorems]]; [[Korovkin-type approximation theory|Korovkin-type approximation theory]]).
+
One says that the functions $  e _ {0} ,e _ {1} ,e _ {2} $
 +
are a Korovkin set for (positive linear operators on) $  C[a,b] $.  
 +
Korovkin proved that a set of three functions in $  C[a,b] $
 +
that replaces the set $  \{ e _ {0} ,e _ {1} ,e _ {2} \} $
 +
must be a [[Chebyshev system|Chebyshev system]]; he also proved that there are no sets of only two functions such that convergence for these two functions implies convergence for all $  f \in C ( A ) $(
 +
cf. also [[Korovkin theorems|Korovkin theorems]]; [[Korovkin-type approximation theory|Korovkin-type approximation theory]]).
  
The results have been generalized to other compact Hausdorff spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070033.png" />. For example, on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070034.png" />, the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070035.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070038.png" /> is a Korovkin set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070039.png" />. As usual, one identifies the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070040.png" /> with the continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070041.png" />-periodic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070042.png" /> (cf, [[#References|[a1]]], [[#References|[a4]]]). On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070043.png" />-dimensional cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070044.png" />, the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070045.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070046.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070048.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070050.png" />) is a Korovkin set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070051.png" />, but not a minimal Korovkin set.
+
The results have been generalized to other compact Hausdorff spaces $  A $.  
 +
For example, on the circle $  \mathbf T $,  
 +
the set of functions $  \{ t _ {0} ,t _ {1} ,t _ {2} \} $
 +
with $  t _ {0} = e _ {0} $,  
 +
$  t _ {1} ( x ) = \cos  ( x ) $,  
 +
$  t _ {2} ( x ) = \sin  ( x ) $
 +
is a Korovkin set for $  C ( \mathbf T ) $.  
 +
As usual, one identifies the functions in $  C ( \mathbf T ) $
 +
with the continuous $  2 \pi $-
 +
periodic functions on $  \mathbf R $(
 +
cf, [[#References|[a1]]], [[#References|[a4]]]). On the $  d $-
 +
dimensional cube $  A = [0,1]  ^ {d} $,
 +
the set of $  2d + 1 $
 +
functions $  e _ {0} ,p _ {1} \dots p _ {d} ,q _ {1} \dots q _ {d} $(
 +
with $  p _ {k} ( x ) = x _ {k} $
 +
and $  q _ {k} ( x ) = x _ {k}  ^ {2} $
 +
for $  x = ( x _ {1} \dots x _ {d} ) \in A $
 +
and $  k = 1 \dots d $)  
 +
is a Korovkin set for $  C ( A ) $,  
 +
but not a minimal Korovkin set.
  
There is also the following generalization. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070052.png" /> be a Banach lattice, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070053.png" /> be the class of all positive linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070054.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070055.png" /> be a fixed subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070056.png" />. Then the Korovkin closure (or shadow, or Korovkin hull) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070058.png" /> is the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070059.png" /> with the property that for each sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070060.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070061.png" /> the relations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070063.png" />, imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070064.png" />. The problem is to find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070065.png" /> for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070066.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070067.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070068.png" /> is a Korovkin set (cf. [[#References|[a1]]], [[#References|[a5]]], [[#References|[a8]]]).
+
There is also the following generalization. Let $  X $
 +
be a Banach lattice, let $  {\mathcal L} $
 +
be the class of all positive linear operators on $  X $,  
 +
and let $  S $
 +
be a fixed subset of $  X $.  
 +
Then the Korovkin closure (or shadow, or Korovkin hull) $  \Sigma ( S ) $
 +
of $  S $
 +
is the set of all $  f \in X $
 +
with the property that for each sequence $  ( L _ {n} ) _ {n \geq 1 }  $
 +
in $  {\mathcal L} $
 +
the relations $  {\lim\limits } _ {n \rightarrow \infty }  \| {g - L _ {n} g } \| = 0 $,  
 +
$  g \in S $,  
 +
imply $  {\lim\limits } _ {n \rightarrow \infty }  \| {f - L _ {n} f } \| = 0 $.  
 +
The problem is to find $  \Sigma ( S ) $
 +
for a given $  S $;  
 +
if $  \Sigma ( S ) = X $,  
 +
then $  S $
 +
is a Korovkin set (cf. [[#References|[a1]]], [[#References|[a5]]], [[#References|[a8]]]).
  
In some cases one can prove a quantitative form of the Korovkin theorem, estimating the rate of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070069.png" /> in terms of the rate of convergence for the elements of the Korovkin set. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070070.png" /> one has obtained estimates in terms of the first- or second-order modulus of continuity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070071.png" />; for example:
+
In some cases one can prove a quantitative form of the Korovkin theorem, estimating the rate of convergence $  \| {f - L _ {n} f } \| $
 +
in terms of the rate of convergence for the elements of the Korovkin set. For $  A = [ -1,1] $
 +
one has obtained estimates in terms of the first- or second-order modulus of continuity $  \omega _ {i} $;  
 +
for example:
  
<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/b/b110/b110700/b11070072.png" /></td> </tr></table>
+
$$
 +
\left \| {f - L _ {n} f } \right \| \leq  \left \| f \right \| \cdot \alpha _ {n} + C \cdot \omega _ {1} ( f, \sqrt {\alpha _ {n} } ) ,  n = 1,2 \dots
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070073.png" /> and some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070074.png" />.
+
with $  \alpha _ {n} = \max  _ {k = 0,1,2 }  \| {e _ {k} - L _ {n} e _ {k} } \| $
 +
and some constant $  C > 0 $.
  
If some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070075.png" /> is a polynomial operator, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070076.png" /> is a polynomial of degree less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070077.png" />, then at least one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070078.png" /> can not be approximated better than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070079.png" />.
+
If some $  L _ {n} $
 +
is a polynomial operator, i.e., $  L _ {n} f $
 +
is a polynomial of degree less than or equal to $  n $,  
 +
then at least one of the functions $  e _ {k} $
 +
can not be approximated better than $  n ^ {-2 } $.
  
Similar results can be obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070080.png" />. Also, it is possible to estimate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070081.png" /> in terms of the moduli of smoothness corresponding to
+
Similar results can be obtained for $  A = \mathbf T $.  
 +
Also, it is possible to estimate $  | {f ( x ) - L _ {n} f ( x ) } | $
 +
in terms of the moduli of smoothness corresponding to
  
<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/b/b110/b110700/b11070082.png" /></td> </tr></table>
+
$$
 +
\alpha _ {n} ( x ) = \max  _ {k = 0,1,2 } \left | {e _ {k} ( x ) - L _ {n} e _ {k} ( x ) } \right | .
 +
$$
  
Furthermore, quantitative Korovkin theorems for positive linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110700/b11070083.png" />-spaces have been obtained (cf. [[#References|[a3]]], [[#References|[a4]]]).
+
Furthermore, quantitative Korovkin theorems for positive linear operators on $  L _ {p} $-
 +
spaces have been obtained (cf. [[#References|[a3]]], [[#References|[a4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Altomare,  M. Campiti,  "Korovkin-type approximation theory and its applications" , de Gruyter  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bohman,  "On approximation of continuous and of analytic functions"  ''Arkiv. Mat. (2)'' , '''1'''  (1952)  pp. 43–56</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.A. DeVore,  "The approximation of continuous functions by positive linear operators" , ''Lecture Notes in Mathematics'' , '''293''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.A. DeVore,  G.G. Lorentz,  "Constructive approximation" , Springer  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Donner,  "Extension of positive operators and Korovkin theorems" , ''Lecture Notes in Mathematics'' , '''904''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.P. Korovkin,  "On convergence of linear positive operators in the space of continuous functions"  ''Dokl. Akad. Nauk. SSSR'' , '''90'''  (1953)  pp. 961–964  (In Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.P. Korovkin,  "Linear operators and approximation theory" , Hindustan Publ. Corp.  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G.G. Lorentz,  M. von Golitschek,  Y. Makovoz,  "Constructive approximation: advanced problems" , Springer  (1996)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Altomare,  M. Campiti,  "Korovkin-type approximation theory and its applications" , de Gruyter  (1994)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Bohman,  "On approximation of continuous and of analytic functions"  ''Arkiv. Mat. (2)'' , '''1'''  (1952)  pp. 43–56</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.A. DeVore,  "The approximation of continuous functions by positive linear operators" , ''Lecture Notes in Mathematics'' , '''293''' , Springer  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.A. DeVore,  G.G. Lorentz,  "Constructive approximation" , Springer  (1993)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Donner,  "Extension of positive operators and Korovkin theorems" , ''Lecture Notes in Mathematics'' , '''904''' , Springer  (1982)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.P. Korovkin,  "On convergence of linear positive operators in the space of continuous functions"  ''Dokl. Akad. Nauk. SSSR'' , '''90'''  (1953)  pp. 961–964  (In Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P.P. Korovkin,  "Linear operators and approximation theory" , Hindustan Publ. Corp.  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G.G. Lorentz,  M. von Golitschek,  Y. Makovoz,  "Constructive approximation: advanced problems" , Springer  (1996)</TD></TR></table>

Revision as of 10:59, 29 May 2020


The starting point is the following theorem of P.P. Korovkin (1953) [a6], [a7]: for a given sequence $ ( L _ {n} ) _ {n \leq 1 } $ of positive linear operators on the space $ C ( A ) $ into itself (where $ A = [a,b] $ is a compact interval and $ C ( A ) $ denotes the Banach space of all continuous real-valued functions on $ A $ with the supremum norm $ \| g \| = \sup \{ {\| {g ( x ) } \| } : {x \in A } \} $), the relations

$$ {\lim\limits } _ {n \rightarrow \infty } \left \| {e _ {k} - L _ {n} e _ {k} } \right \| = 0, k = 0,1,2, $$

imply convergence:

$$ {\lim\limits } _ {n \rightarrow \infty } \left \| {f - L _ {n} f } \right \| = 0 \textrm{ for any } f \in C ( A ) . $$

Here, the functions $ e _ {k} $ are defined by $ e _ {k} ( x ) = x ^ {k} $ and $ \| \cdot \| $ is the supremum norm on the interval $ A $. An element $ f \in C ( A ) $ is positive (denoted by $ f \geq 0 $) if $ f ( x ) \geq 0 $ for all $ x \in A $, and the linear operator $ L _ {n} $ is positive if $ f \geq 0 $ implies $ L _ {n} f \geq 0 $. One year earlier, in 1952, H. Bohman [a2] had proved this theorem for positive operators $ L _ {n} $ having a representation

$$ L _ {n} f = \sum _ {k = 0 } ^ { n } f ( x _ {k,n } ) \cdot \psi _ {k,n } $$

with $ n + 1 $ knots $ x _ {k,n } $ in the open interval $ ( 0,1 ) $, and $ \psi _ {k,n } \geq 0 $ on $ A = [0,1] $.

One says that the functions $ e _ {0} ,e _ {1} ,e _ {2} $ are a Korovkin set for (positive linear operators on) $ C[a,b] $. Korovkin proved that a set of three functions in $ C[a,b] $ that replaces the set $ \{ e _ {0} ,e _ {1} ,e _ {2} \} $ must be a Chebyshev system; he also proved that there are no sets of only two functions such that convergence for these two functions implies convergence for all $ f \in C ( A ) $( cf. also Korovkin theorems; Korovkin-type approximation theory).

The results have been generalized to other compact Hausdorff spaces $ A $. For example, on the circle $ \mathbf T $, the set of functions $ \{ t _ {0} ,t _ {1} ,t _ {2} \} $ with $ t _ {0} = e _ {0} $, $ t _ {1} ( x ) = \cos ( x ) $, $ t _ {2} ( x ) = \sin ( x ) $ is a Korovkin set for $ C ( \mathbf T ) $. As usual, one identifies the functions in $ C ( \mathbf T ) $ with the continuous $ 2 \pi $- periodic functions on $ \mathbf R $( cf, [a1], [a4]). On the $ d $- dimensional cube $ A = [0,1] ^ {d} $, the set of $ 2d + 1 $ functions $ e _ {0} ,p _ {1} \dots p _ {d} ,q _ {1} \dots q _ {d} $( with $ p _ {k} ( x ) = x _ {k} $ and $ q _ {k} ( x ) = x _ {k} ^ {2} $ for $ x = ( x _ {1} \dots x _ {d} ) \in A $ and $ k = 1 \dots d $) is a Korovkin set for $ C ( A ) $, but not a minimal Korovkin set.

There is also the following generalization. Let $ X $ be a Banach lattice, let $ {\mathcal L} $ be the class of all positive linear operators on $ X $, and let $ S $ be a fixed subset of $ X $. Then the Korovkin closure (or shadow, or Korovkin hull) $ \Sigma ( S ) $ of $ S $ is the set of all $ f \in X $ with the property that for each sequence $ ( L _ {n} ) _ {n \geq 1 } $ in $ {\mathcal L} $ the relations $ {\lim\limits } _ {n \rightarrow \infty } \| {g - L _ {n} g } \| = 0 $, $ g \in S $, imply $ {\lim\limits } _ {n \rightarrow \infty } \| {f - L _ {n} f } \| = 0 $. The problem is to find $ \Sigma ( S ) $ for a given $ S $; if $ \Sigma ( S ) = X $, then $ S $ is a Korovkin set (cf. [a1], [a5], [a8]).

In some cases one can prove a quantitative form of the Korovkin theorem, estimating the rate of convergence $ \| {f - L _ {n} f } \| $ in terms of the rate of convergence for the elements of the Korovkin set. For $ A = [ -1,1] $ one has obtained estimates in terms of the first- or second-order modulus of continuity $ \omega _ {i} $; for example:

$$ \left \| {f - L _ {n} f } \right \| \leq \left \| f \right \| \cdot \alpha _ {n} + C \cdot \omega _ {1} ( f, \sqrt {\alpha _ {n} } ) , n = 1,2 \dots $$

with $ \alpha _ {n} = \max _ {k = 0,1,2 } \| {e _ {k} - L _ {n} e _ {k} } \| $ and some constant $ C > 0 $.

If some $ L _ {n} $ is a polynomial operator, i.e., $ L _ {n} f $ is a polynomial of degree less than or equal to $ n $, then at least one of the functions $ e _ {k} $ can not be approximated better than $ n ^ {-2 } $.

Similar results can be obtained for $ A = \mathbf T $. Also, it is possible to estimate $ | {f ( x ) - L _ {n} f ( x ) } | $ in terms of the moduli of smoothness corresponding to

$$ \alpha _ {n} ( x ) = \max _ {k = 0,1,2 } \left | {e _ {k} ( x ) - L _ {n} e _ {k} ( x ) } \right | . $$

Furthermore, quantitative Korovkin theorems for positive linear operators on $ L _ {p} $- spaces have been obtained (cf. [a3], [a4]).

References

[a1] F. Altomare, M. Campiti, "Korovkin-type approximation theory and its applications" , de Gruyter (1994)
[a2] H. Bohman, "On approximation of continuous and of analytic functions" Arkiv. Mat. (2) , 1 (1952) pp. 43–56
[a3] R.A. DeVore, "The approximation of continuous functions by positive linear operators" , Lecture Notes in Mathematics , 293 , Springer (1972)
[a4] R.A. DeVore, G.G. Lorentz, "Constructive approximation" , Springer (1993)
[a5] K. Donner, "Extension of positive operators and Korovkin theorems" , Lecture Notes in Mathematics , 904 , Springer (1982)
[a6] P.P. Korovkin, "On convergence of linear positive operators in the space of continuous functions" Dokl. Akad. Nauk. SSSR , 90 (1953) pp. 961–964 (In Russian)
[a7] P.P. Korovkin, "Linear operators and approximation theory" , Hindustan Publ. Corp. (1960) (In Russian)
[a8] G.G. Lorentz, M. von Golitschek, Y. Makovoz, "Constructive approximation: advanced problems" , Springer (1996)
How to Cite This Entry:
Bohman-Korovkin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohman-Korovkin_theorem&oldid=22151
This article was adapted from an original article by H.-B. Knoop (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article