Positive linear operators generated by analytic functions

Abstract

Let phi be a power series with positive Taylor coefficients {a(k)}(k=0)(infinity) and non-zero radius of convergence r <= infinity. Let xi x, 0 <= x <= r be a random variable whose values alpha(k), k = 0, 1,..., are independent of x and taken with probabilities a(k)x(k)/phi(x), k = 0, 1,.... The positive linear operator (A(phi)f)(x) := E[f(xi x)] is studied. It is proved that if E(xi(x)) = x, E(xi(2)(x)) = qx(2) + bx + c, q, b, c is an element of R, q > 0, then A(phi) reduces to the Szasz-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupas, operator in the case q > 1.