Variations on a theorem of Korovkin. (English) Zbl 1132.41331

Lomelí, Héctor E.; García, César L.

Am. Math. Mon. 113, No. 8, 744-750 (2006).

The Korovkin theorem states that a sequence of positive operators defined on \(C[0,1]\) converges to the identity if it converges uniformly for the three functions \(1\), \(x\), and \(x^2\). This theorem can be used to provide elegant and simple proofs for the Weierstrass theorem, the Fejér theorem for the convergence of the Cesàro means of Fourier series, the convergence of Fejér-Hermite interpolation and other classical convergence results.
The Korovkin theorem has been extended to vastly wider settings and there are two varieties of proofs. This paper provides a short, elementary proof that the authors have taught in third-year undergraduate courses. The proof applies to positive operators defined on the space of continuous functions defined on a compact metric space. The authors demonstrate the applicability of their approach by demonstrating that their result immediately recovers five fundamental theorems, among them the ones mentioned above.

Reviewer: Daniel Wulbert (MR2375207)

MSC:

41A36	Approximation by positive operators
41-02	Research exposition (monographs, survey articles) pertaining to approximations and expansions