# Bohman-Korovkin theorem

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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]).

How to Cite This Entry:
Bohman-Korovkin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bohman-Korovkin_theorem&oldid=46095
This article was adapted from an original article by H.-B. Knoop (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article