Generalization of Bernstein's polynomials to the infinite interval

On the Extensions of Bernstein Polynomials to the Infinite Interval

“…Now our kernel (5) satisfies the hypothesis of Lemma 4. In fact the inequality (8) shows that (7) holds with Ci = 1 and since /" (ux)"…”

“…Introduction. If the function f(x) is defined in the infinite interval 0^x< oo, M. Kac,2 J. Favard [4], and also O. Szasz [8] considered the transform / " (ux)'…”

“…(1) Pu(x) = «-»» E /("/«) ^-7-> « > 0, Now if/(x) is continuous in (0, oo) it is known (see [8] or [4]) that lim,,,*, Pfu(x) =f(x) uniformly in (0, oo); for the interval [0, 1 ] the corresponding familiar result holds for the Bernstein polynomials (see [4] or [l] for a recent approach to the general approximation prob-…”

A representation theorem and approximation operators arising from inequalities involving differential operators

“…In §3 we shall introduce some approximation operators which arise naturally from our discussion and which generalize the Szász operators [8], while in §4 we shall define by means of the GVjS's, some sequence-to-function summability methods which generalize the [J,f(x)] transforms [3].…”

Some Results for Borel Transforms

“…These operators are generalizations of operators which have been studied by Szasz [8] where f(0) =z+(z*-l)1'2 cos <f> and t(u) =t+(t2-l)1'2 cosh u. (The branch of (z2 -l)112 is chosen so that z+(z2 -I)112 lies outside the unit circle.…”