Approximation operators of binomial type

Our objective is to present a unified theory of the approximation operators of binomial type by exploiting the main technique of the so- called "umbral calculus" or "finite operator calculus" (see [18], [20]-[22]). Let us consider the basic sequence (b(n))(n greater than or equal to 0) associated to a certain delta operator Q. By supposing that b(n)(x) greater than or equal to 0, x is an element of [0, infinity), our purpose is to put in evidence some approximation properties of the linear positive operators (L-n(Q))(n greater than or equal to 1) which are defined on C[0, 1] by L(n)(Q)f = Sigma(k=0)(n)beta(n,k)(Q)f(k/n), beta(n,k)(Q)(x) := 1/b(n)(n) (kn) b(k)(nx)b(n-k)(n-nx). This paper is first of all concerned with the construction of such operators. The construction method is performed by means of umbral calculus. The Bernstein operators are in fact (L-n(D))(n greater than or equal to 1), D- being the derivative. It is shown that such operators leave invariant the cone of the convex functions of higher order and there are establishhed some identities between the elements of the "Q-basis" {beta(n,0)(Q), ..., beta(n,n)(Q)}. Such representations turn out to be useful since they may be used to investigate qualitative properties of the operators. AMS Subject Classification : 41A36.