q-Bernstein polynomials and their iterates

✍ Scribed by Sofiya Ostrovska

• Book ID: 104142852
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 242 KB
• Volume: 123
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1016/s0021-9045(03)00104-7

✦ Synopsis

Let B n ð f ; q; xÞ; n ¼ 1; 2; y be q-Bernstein polynomials of a function f : ½0; 1-C: The polynomials B n ð f ; 1; xÞ are classical Bernstein polynomials. For qa1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q41 with respect to both n and q: Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in fz: jzjoq þ eg the rate of convergence of fB n ð f ; q; xÞg to f ðxÞ in the norm of C½0; 1 has the order q Àn (versus 1=n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials fB jn n ð f ; q; xÞg; where both n-N and j n -N; are studied. It is shown that for qAð0; 1Þ the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j n -N: