Bernstein Polynomials and Modulus of Continuity

✍ Scribed by Zhongkai Li

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 89 KB
• Volume: 102
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1999.3374

✦ Synopsis

p n, j (x)= \ n j + x j (1&x) n& j of a given function f (x) on [0, 1], besides the convergence and approximation, preserve some properties of the original function. For example:

(i) if f (x) is non-decreasing, then for all n 1, the B n ( f; x) are nondecreasing;

(ii) if f (x) is convex, then for all n 1, the B n ( f; x) are convex and

x # [0, 1];

for other examples, cf. [7, Sect. 1.7; 5, Sect. 6.3]. Further studies on the convexity of the Bernstein polynomials can be found in [3, 4, 9]. Another property that the Bernstein polynomials preserve, proved by an elementary method in [2] (cf. [1] also), is that (iii) if f # Lip A +, then for all n 1, B n ( f; x) # Lip A +. A function f belongs to the Lipschitz class Lip A + where 0<+ 1 and A 0 if |( f ; t) At + for 0<t 1, where |( f ; t)=max |x2&x1| t | f (x 2 )& f (x 1 )| is the modulus of continuity of f (x). The interesting and important