Unknown

7.2. Approximation on an Interval 217

Argue that the graph of q must cross that of p in at least n + 1

places. (Use a sketch.) Deduce that degq(t) 2 n + 1.

(b) Suppose, instead, for the polynomial in (a) that

max{]f(t) - q(t)1 : a 5 t 5 b} = K.

Argue that either q(t) = p(t) or deg q(t) 2 n + 1.

(c) Conjecture a characterization of the polynomial p(t) of degree

not exceeding n for which

m=UfW - PWI : a 5 t 5 b1

is as small as possible. Do you think that the polynomial p(t) is

uniquely determined?

4. (a) Solve th e f o 11 owing problem for k: = 1,2,3. Find that polynomial
   pk(t) of degree not exceeding Jz - 1 which minimizes

max{]tk -pk(t)l : -1 5 t 5 1)

over all polynomials of degree not exceeding Ic - 1.

(b) Let Ck = tk -pk(t). Sketch the graphs of Cl, Cz, Cs. Roughly

speaking, what should be graph of Ck look like?

(c) Does this exercise have anything to do with Exercise 1.3.15 and

Exploration E.6?

5. Bernstein polynomials. Let f(t) (0 5 t < 1) be a function defined on
   the closed unit interval [0, 11. The B ernstein polynomial of order n
   corresponding to f(t) is defined by

B(f, n;t) = 2 f(k/n) ( z ) tk(l - t)n-k.

k=O

(a) Verify that

B(f, l;t) = mu1 - t) + f(lY

B(f,2;t) = f(O)(l - t)2 + 2f(l/2)t(l -t) + f(l)t”

B(f, 3; t) = f(O)(l - t)3 + 3f(1/3)t(l - t>z

+ 3f(2/3)t2(1 - t) + f(l)t3.

(b) Prove that, if f(t) = 1 and g(t) = t for^0 5 t^5 1, then

B(f,n;t) = 1 B(g,n;t) = t.

(c) What is B(t2,n; t)? These are not all the same. Check the cases

n = 2,3,4 to discover the pattern.