Some Korovkin-type theorems for stochastic processes. (English) Zbl 0985.60031

Pogány, Tibor K.

Theory Probab. Math. Stat. 61, 153-159 (2000) and Teor. Jmovirn. Mat. Stat. 61, 145-151 (2000).

The classical Banach-Steinhaus theorem states that a sequence of linear operators in linear normed spaces tends to some linear operator pointwise if and only if their norms are bounded and they converge strongly to this operator. P. P. Korovkin [“Lineare Operatoren und Approximationstheorie” (1959; Zbl 0094.10201)] proved that the condition on convergence can be weaken when the sequence of operators is positive. This paper deals with the classical Korovkin theorem which is extended to the class of finite second-order stochastic processes. Applying this result to the construction of a sequence of linear positive operators by I. I. Ibragimov and A. D. Gadzhiev [Sov. Math., Dokl. 11, 1092-1095 (1970); translation from Dokl. Akad. Nauk SSSR 193, 1222-1225 (1970; Zbl 0217.17302)] the author generates the Bernstein-Kholodovski, Baskakov and Hille-Szász-Mirakjan operators on the class of second-order processes. The quadratic mean truncation error upper bound is established for some of these operators.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60G12	General second-order stochastic processes
60G99	Stochastic processes
41A15	Spline approximation
41A36	Approximation by positive operators

Keywords:

Korovkin-type theorems; Banach-Steinhaus theorem; positive operators; second-order stochastic processes

Citations:

Zbl 0094.10201; Zbl 0217.17302