The Approximation of All Continuous Functions on [0, 1] by q-Bernstein Polynomials in the Case q → 1+

Abstract

Since for q > 1, the q-Bernstein polynomials Bn,q(f ; .) are not positive linear operators on C[0, 1], their convergence properties are not similar to those in the case 0 < q ≤ 1. It has been known that, in general, Bn,q n (f ; .) does not approximate f ∈ C[0, 1] if qn → 1 + , n → ∞, unlike in the case qn → 1 − . In this paper, it is shown that if 0 ≤ qn − 1 = o(n −1 3 −n ), n → ∞, then for any f ∈ C[0, 1], we have: Bn,q n (f ; x) → f (x) as n → ∞, uniformly on [0,1].

Approximation byq-Bernstein Polynomials in the Caseq→1+

“…On the other hand, − −1 ( ; ) = 0 = − ( ; ) for ∈ { − −1 , − }, which completes the proof of (20). From (14), (19), and (20), we get…”

“…In the case > 1, , are not positive linear operators on [0, 1], and the lack of positivity makes the investigation of convergence in the case > 1 essentially more difficult. There are many unexpected results concerning convergence of -Bernstein polynomials in the case > 1 (see [2,[12][13][14][15][16][17]). For example, the rate of approximation by -Bernstein polynomials ( > 1) in [0, 1] for functions analytic in { : | | < + } is − versus 1/ for the classical Bernstein polynomials, while, for some infinitely differentiable functions on [0, 1], their sequences of -Bernstein polynomials ( > 1) may be divergent (see [12]).…”

“…Indeed, Ostrovska showed in [13] that if − 1 ↓ 0 slower than (ln )/ , then the sequence ( , ( )) may not be approximating for some ∈ [0, 1] (e.g., ( ) = √ ). However, in [14] Ostrovska showed that if → 1 + fast enough, the sequence ( , ( )) is approximating for any ∈ [0, 1]: a sufficient condition is = 1 + ( −1 3 − ).…”