Korovkin systems of stochastic processes. (English) Zbl 0598.60058
Korovkin-systems for the space of \(L^ 1\)-continuous stochastic processes are examined. Sufficient conditions are given in order to decide whether a given system of random functions constitutes a Korovkin system, and it is shown how to obtain results about the approximation of stochastic processes.
For example, a constructive proof of a stochastic version of Weierstrass’ theorem and a trapezoidal rule for stochastic Riemann-Stieltjes integrals are derived.
For example, a constructive proof of a stochastic version of Weierstrass’ theorem and a trapezoidal rule for stochastic Riemann-Stieltjes integrals are derived.
Keywords:
Korovkin-systems; approximation of stochastic processes; stochastic version of Weierstrass’ theorem; trapezoidal rule for stochastic Riemann- Stieltjes integralsReferences:
| [1] | Berens, H., Lorentz, G.G.: Theorems of Korovkin type for positive linear operators on Banach lattices. In: Approximation theory (edited by G.G. Lorentz), pp. 1-30. New York: Academic Press 1973 · Zbl 0322.41017 |
| [2] | Bourbaki, N.: General topology, part 2. Reading: Addison-Wesley 1966 · Zbl 0145.19302 |
| [3] | Cheney, E.W.: Introduction to approximation theory. New York: McGraw Hill 1966 · Zbl 0161.25202 |
| [4] | Freud, G.: Über positive lineare Approximationsfolgen von stetigen reellen. Funktionen auf kompakten Mengen. ISNM 5, pp. 233-238. Basel: Birkhäuser 1964 |
| [5] | Gänssler, P., Stute, W.: Wahrscheinlichkeitstheorie. Berlin Heidelberg New York: Springer 1977 |
| [6] | Korovkin, P.P.: Linear operators and approximation theory. Delhi: Hindustan Publ. Corp. 1960 · Zbl 0094.10201 |
| [7] | Müller, M.W.: Sätze vom Bohmann-Korovkin-Typ für Banachsche Funktionenräume. ISNM 20, 292-299. Basel: Birkhäuser 1972 |
| [8] | Roth, W.: Families of convex subsets and Korovkin-type theorems in locally convex spaces. Preprint 763, Fachbereich Mathematik, TH Darmstadt 1983 |
| [9] | Schaefer, H.H.: Banach lattices and positive operators. Berlin Heidelberg New York: Springer 1974 · Zbl 0296.47023 |
| [10] | Scheffold, E.: Ein allgemeiner Korovkin-Satz für lokalkonvexe Vektorverbände. Math. Z.132, 209-214 (1973) · Zbl 0255.46007 · doi:10.1007/BF01213865 |
| [11] | Wolff, M.: On the universal Korovkin closure of subsets in vector lattices. J. Approx. Theory22, 243-253 (1978) · Zbl 0392.41017 · doi:10.1016/0021-9045(78)90056-4 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
