Quadrature Formulas Constructed by Using Certain Linear Positive Operators
Let [a,b] be a compact interval of the real line IR. It is known that the classical theorem of Bohman-Korovkin states that in order that a sequence of positive linear operators (Lm), mapping into itself the space C[a,b] of continuous real-valued functions on [a,b], equipped with the uniform norm, to have the property that, for any f∈C[a,b], if m → ∞ we have lim Lmf = f, uniformly on [a,b], it is necessary and sufficient that such a convergence occur for a triplet of “test functions” from C[a,b], forming a so called Korovkin system. For C[a,b] the three monomials e0,e1,e2 where ej(x) := xj (j = 0,1,2), represent such a system.
Unable to display preview. Download preview PDF.
- 3.MOLDOVAN, G., Discrete convolutions and linear positive operators. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 15 (1972), 31–44.Google Scholar
- 4.POPOVICIU, T., Sur le reste dans certaines formules linéaires d’approximation de l’analyse. Mathematica (Cluj), 1(24) (1959), 95–142.Google Scholar
- 5.STANCU, D.D., Approximation of functions by a new class of linear polynomial operators. Rev. Roum. Math. Pures Appl. 13 (1968), 1173–1194.Google Scholar
- 6.STANCU, D.D., On a generalization of the Bernstein polynomials. Studia Univ. Babes-Bolyai, Ser. Math.-Phys., 14 (1969), 31–45 (Rumanian).Google Scholar
- 7.STANCU, D.D., Use of linear interpolation for constructing a class of Bernstein polynomials. Studii Cercet. Matem. (Bucharest), 28 (1976), 369–379 (Rumanian).Google Scholar
- 8.STANCU, D.D., Approximation of functions by means of a new generalized Bernstein operator. (To appear).Google Scholar