The Bernstein polynomial of degree n for / C[0, 1] is given by where Q') denotes n\/k\(n

Chapter 3, Problem 23

(choose chapter or problem)

The Bernstein polynomial of degree \(n\) for \(f \in C[0,1]\) is given by

\(B_{n}(x)=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}\),

where \(\left(\begin{array}{l} n \\ k \end{array}\right)\) denotes \(n ! / k !(n-k) !\). These polynomials can be used in a constructive proof of the Weierstrass Approximation Theorem 3.1 (see [Bart]) because \(\lim _{n \rightarrow \infty} B_{n}(x)=f(x)\), for each \(x \in[0,1]\).

a. Find \(B_{3}(x)\) for the functions \(\begin{array}{l}\text{i. } f(x)=x\\ \text{ii. } f(x)=1\end{array}\)

b. Show that for each \(k \leq n\),

\(\left(\begin{array}{l} n-1 \\ k-1 \end{array}\right)=\left(\begin{array}{l} k \\ n \end{array}\right)\left(\begin{array}{l} n \\ k \end{array}\right)\) .

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

\(1=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) x^{k}(1-x)^{n-k}, \text { for each } n\)

to show that, \(f(x)=x^{2}\),

\(B_{n}(x)=\left(\frac{n-1}{n}\right) x^{2}+\frac{1}{n} x\) .

d. Use part (c) to estimate the value of n necessary for \(\left|B_{n}(x)-x^{2}\right| \leq 10^{-6}\) to hold for all \(x\) in [0,1].