Convergence of q-Bernstein polynomials
by
Sofiya Ostrovska
Atilim University, Department of Mathematics, 06836 Incek, Ankara, Turkey

In 1912 S.N. Bernstein found the proof of the Weierstrass Approximation Theorem based on the Law of Large Numbers for a sequence of Bernoulli trials. He constructed, for any continuous function f on [0, 1], a sequence of polynomials

Bn(f;x):= n å k=0  f æ è  k n ö ø æ è n k ö ø xk(1-x)n-k,   n=1, 2, ...

(1)

and proved that the sequence converges to f as n tends to infinity uniformly with respect to x in [0, 1]. These polynomials (1), called Bernstein polynomials, possess a lot of remarkable properties. They have been studied intensively, and their connections with different branches of analysis have been investigated.

In 1997 G.M. Phillips introduced generalized Bernstein polynomials based on the q-integers or q-Bernstein polynomials B_n(f, q;x).

Definition. Let f: [0, 1] --> C,  q > 0. The q-Bernstein polynomial of f is

Bn(f, q;x): = n å k=0  f æ è  [k]q [n]q ö ø é ë n k ù û q  xk n-1-k Õ s=0  (1-qsx),     n=1, 2, ...

(2)

(An empty product is taken to be equal 1.)

Note that for q=1, the polynomials B_n(f, 1;x) are classical Bernstein polynomials (1). If q =/= 1, one gets a new class of polynomials. It was proved that convergence properties of q-Bernstein polynomials for q =/= 1 differ essentially from those in the classical case.

In this talk we present new results concerning approximating properties of q-Bernstein polynomials. Both cases q < 1 and q > 1 are considered. We show that in the case 0 < q < 1 the sequence {B_n(f, q;x)} is not approximating for a function f unless f is linear. On the other hand, we prove that in the case q > 1 approximating properties of q-Bernstein polynomials can be better than in the classical case.

Date received: April 14, 2003