Approximation by q -Bernstein Polynomials in the Case q → 1 +

الملخص EN

Let B n , q ( f ; x ) , q ∈ ( 0 , ∞ ) be the q -Bernstein polynomials of a function f ∈ C [ 0,1 ] .

It has been known that, in general, the sequence B n , q n ( f ) with q n → 1 + is not an approximating sequence for f ∈ C [ 0,1 ] , in contrast to the standard case q n → 1 - .

In this paper, we give the sufficient and necessary condition under which the sequence B n , q n ( f ) approximates f for any f ∈ C [ 0,1 ] in the case q n > 1 .

Based on this condition, we get that if 1 < q n < 1 + ln 2 / n for sufficiently large n , then B n , q n ( f ) approximates f for any f ∈ C [ 0,1 ] .

On the other hand, if B n , q n ( f ) can approximate f for any f ∈ C [ 0,1 ] in the case q n > 1 , then the sequence ( q n ) satisfies lim ¯ n → ∞ n ( q n - 1 ) ≤ ln 2 .