216 7. Approximations and Inequalities
--y=at+b
- Suppose that f(t) is a given function and that p(t) is a polynomial
of degree not exceeding n. Let
K = max{ (f(t) - p(t)1 : a 5 t <_ b}.
Suppose that there are n + 2 distinct points tc,tr, t2,... , tn+l in the
interval such that
(i) If(ti) - p(ti)( = K (0 5 i 5 n + 1)
(ii) f(ti+l) - P(ti+l) = -[f(h) -P(h)] (0^5 i^5 n)
(i.e., the maximum distance between the graphs is achieved with al-
terna.te signs at least n + 2 times). The diagram illustrates a possible
situation when n = 3.
,‘--I
(a) Suppose that q(t) is a polynomial such that
max{lf(t) - q(t)] : a 5 t 5 b} < K.