Quadrature Formulas Constructed by Using Certain Linear Positive Operators

Abstract

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 (L_m), 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 L_mf = 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 e_j(x) := x^j (j = 0,1,2), represent such a system.