Theorem 1 Let . Then the sequence convergence to f uniformly on for each if and only if .
The proof of the above theorem follows along the lines of [10], Theorem 2], thus we omit the details.
Let be fixed. We define and for
(4)
Using the fact that [14], we have
Using (3) and (4), it is easy to prove that
For , , we define the modulus of continuity as follows: . We shall show the following theorem.
Theorem 2 Let then for each the sequence converges to uniformly on . Furthermore, .
The proof of the above theorem follows along the lines of [10], Theorem 3], thus we omit the details.
Remark 3 We may observe that, for , we have , where means that and , and means that there exists a positive constant C independent of n such that . Hence, the estimate of Theorem 2 is sharp in the following sense: the sequence in Theorem 2 cannot be replaced by any other sequence decreasing to zero more rapidly as .
Lemma 3 [15]
Let L be a positive linear operator on , which reproduces linear functions. If , then if and only if f is linear.
Remark 4 Since for consequence of Lemma 3 we have the following:
Theorem 3 Let be fixed and let . Then for all if and only if f is linear.
Remark 5 Let be fixed and let . Then the sequence does not approximate unless f is linear. This is completely in contrast to the classical Bernstein polynomials, by which approximates for any .
Theorem 4 For any , converges to f uniformly on as .
Next, we establish a Voronovskaja type asymptotic formula for the operators :
Theorem 5 Let f be bounded and integrable on the interval , second derivative of f exists at a fixed point and such that as , then
The proof of the above lemma follows along the lines of [16], Theorem 3]; thus, we omit the details.