Bohman-Korovkin theorem

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

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

imply convergence:

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

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 _ {n \rightarrow \infty } \| {g - L _ {n} g } \| = 0 $, $ g \in S $, imply $ \lim _ {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]).

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