Bohman-Korovkin theorem

The starting point is the following theorem of P.P. Korovkin (1953) [a6], [a7]: for a given sequence of positive linear operators on the space into itself (where is a compact interval and denotes the Banach space of all continuous real-valued functions on with the supremum norm ), the relations

imply convergence:

Here, the functions are defined by and is the supremum norm on the interval . An element is positive (denoted by ) if for all , and the linear operator is positive if implies . One year earlier, in 1952, H. Bohman [a2] had proved this theorem for positive operators having a representation

with knots in the open interval , and on .

One says that the functions are a Korovkin set for (positive linear operators on) . Korovkin proved that a set of three functions in that replaces the set 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 (cf. also Korovkin theorems; Korovkin-type approximation theory).

The results have been generalized to other compact Hausdorff spaces . For example, on the circle , the set of functions with , , is a Korovkin set for . As usual, one identifies the functions in with the continuous -periodic functions on (cf, [a1], [a4]). On the -dimensional cube , the set of functions (with and for and ) is a Korovkin set for , but not a minimal Korovkin set.

There is also the following generalization. Let be a Banach lattice, let be the class of all positive linear operators on , and let be a fixed subset of . Then the Korovkin closure (or shadow, or Korovkin hull) of is the set of all with the property that for each sequence in the relations , , imply . The problem is to find for a given ; if , then 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 in terms of the rate of convergence for the elements of the Korovkin set. For one has obtained estimates in terms of the first- or second-order modulus of continuity ; for example:

with and some constant .

If some is a polynomial operator, i.e., is a polynomial of degree less than or equal to , then at least one of the functions can not be approximated better than .

Similar results can be obtained for . Also, it is possible to estimate in terms of the moduli of smoothness corresponding to

Furthermore, quantitative Korovkin theorems for positive linear operators on -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)