Korovkin theorems

Korovkin's first theorem states that if is an arbitrary sequence of positive linear operators on the space of real-valued continuous functions on the interval (cf. Continuous functions, space of; Linear operator) and if

for all , where (, ), then

for all .

The statement of Korovkin's second theorem is similar to that of the first theorem, but is replaced by (the space of -periodic real-valued functions on , endowed with the topology of uniform convergence on ) and is taken from the set .

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 of the form

where is a finite set of numbers in and (). 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 with other finite subsets of . He has shown that if a subset "behaves like" , then (Korovkin's third theorem). Moreover, he showed that a subset "behaves like" 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 or 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