(k = 0,1,2,... , n) is given for x # zk for any k byP(X) = wwwwhereN(x) = g[(-l)kyk sin uk/(X - xk)]
k=O
and
D(x) = &(-I)’ sin uk/(x - xk)].
k=O228 7. Approximations and Inequalities- Determine all pairs (p, q) of real numbers for which the inequality
IA/i?- PX - Q ) I (1/2)(Jz - 1)is true for each x for which 0 < x 5 1.- Prove that the polynomial p with degree not exceeding n that assumes
the value yk at the values
xk = COS t‘k where uk = [(2k + l)?r/(2n + 2)]Hints
Chapter 7
1.9. (c) Prove by induction.1.20. Observe that
2’=1+( ;)+( ;)+...+( 3.2.6. (a) Look at the equation for the eigenfunction when t = 0,l. What
does this imply if k # l?3.1. (a) One term is (x2 - y2)2.3.3. For the n = 2k case, pair the yr off and apply the AGM inequality
to each pair.4.1. Apply the CSB inequality to {(l-x), (z-y), (y-z),%} and {l,l, l,l}.4.2. Express the difference of the two sides as a sum of squares.4.3. Let ri be the zeros. Apply the CSB inequality to {ri} and {r,:‘}.