The Bernstein polynomial of degree n for f C [0, 1] is given by Where (n/k)

The Bernstein polynomial of degree n for f ˆˆ C [0, 1] is given by

Where (n/k) denotes n!/k!(n ˆ’ k)!. These polynomials can be used in a constructive proof of the Weierstrass Approximation Theorem 3.1 (see [Bart]) because limn†’ˆž Bn(x) = f (x), for each x ˆˆ [0, 1].
a. Find B3(x) for the functions
i. f (x) = x
ii. f (x) = 1
b. Show that for each k ‰¤ n,

Use part (b) and the fact, from (ii) in part (a), that

to show that, for f (x) = x2,
Bn(x) = (n - 1/n) x2 + 1/nx.
d. Use part (c) to estimate the value of n necessary for |Bn(x) ˆ’x2 |‰¤ 10ˆ’6 to hold for all x in [0, 1].

Transcribed Image Text:

k-0 (二)-()(i) k-1 t for each n, K--D