A POINTWISE APPROXIMATION THEOREM FOR LINEAR COMBINATIONS OF BERNSTEIN POLYNOMIALS

Recently, Z. Ditzian gave an interesting direct estimate for Bernstein polynomials. In this paper we give direct and inverse results of this type for linear combinations of Bernstein polynomials.

However, Ditzian did not consider the inverse result in [3].We did give such an inverse result in [6], where we obtained the following equivalence.
In this paper we consider linear combinations of Bernstein polynomials, that is where n i and C i satisfy [5] (1.4) We recall that is equivalent to the K-functional We write ω r ϕ λ (f, t) ∼ K ϕ λ (f, t r ), i.e., there exists a constant C such that (1.7) ).Now we state our main result.

Inverse Theorem
In this section we prove the inverse part of (1.8).
To prove Theorem 2 we need some new notation and some lemmas.We use the following notation.
We also need the following lemmas which will be proved in the next section.

Now we prove (3.1).
Proof of (3.1).Since B n (f, x) preserves linear functions, we consider only . So, we may assume f ∈ C 0 .From (3.2) we have and we may choose g ∈ C r λ such that By the assumption of Theorem 2, one has Using Lemma 3.1 and (3.7) we have (3.9) From (3.6), (3.8) and (3.9) we obtain and this implies, via the Berens-Lorentz lemma [1], that if α < r then On the other hand, notice that δ From (3.9)-(3.12),0 and by choosing an appropriate g, we have The proof of (3.1) is complete.
For n > 4r we have Obviously I 1 + I 3 ≤ 2((3r + 1)!) r /n r and, by simple computation, we have By this and δ 2r n For x ∈ E n we use the expression (cf.[5]) n , we have by using Hölder inequality twice .
Proof of (3.4).We recall that [5] (4.3) For 0 < k < n − r, by [5, p. 155], Hence we have, as in [5], ∆ r For k = 0, note that u ∈ (0, r n ) implies δ n (u) ∼ 1 √ n .Thus, we have Similarly for k = n − r we have ) By a simple computation, it is easy to get (cf.[2]) and Using this and Hölder's inequality we obtain which is the stated result.