APPROXIMATION WIT H RATIONA L FUNCTION S 5

However, going back to (1.14), we find we are able to make the estimate

/ sup|F(0) - P(6)\ ^- f* 11 + e ie + • • • + e^1 )W \2 dd

which, by ParsevaPs theorem, simply says

(1.18) /*sup|F(0)-P(0)| .

A comparison of (1.17) an d (1.18) completes th e lemma.

Next we need

LEMMA

1.2. Suppose a, are nonnegative, nonincreasing, and logarithmically concave

(i.e. satisfy af a

] + l

• a-_x) then Zjl , a y e • Maxy/ay.

Before proceeding to th e proof of this lemma, however, let us observe how it, together

with Lemm a 1.1, enables us to obtain our lower bound.

So suppose tha t p(x) i s any even polynomial of degree n. Thus , with x = cos(0/2) as

in (1.12), P(0) = p(x) i s a trigonometric polynomia l of degree w/2. Lemm a 1.1applie s with

F(6) = (cos(0/2))* and we obtain

(1.19) II* * -p(x)\\ % Max A4/+n/2 -

Hence, after verifying th e monotonicity an d logarithmic concavity of the binomia l co-

efficients, A,, w e may apply Lemm a 1.2 with a* = A,+n,2 t o conclude tha t

0- 2 0 ) MaxM

/ + n / 2

\ Z A

f+Hl2

= l -Pkn

(again by the definition of the Pk

n

), and so (1.19) together with (1.20) proves Theorem 1.1.

All that is missing, then, is the proof of Lemma 1.2 and this is our next joke. We

won't prove it! Afte r all , the proof is in print in the Newman, Rivlin paper and moreover

the integral form is really just a s good, and-well—here's the alternate:

LEMMA

1.2. Let f(x) be positive and logarithmically concave, then

f~f(x)dxe Sup xf(x).

PROOF.

I f xf(x) i s unbounded, there is nothing to prove and so we may normalize

matters so that xf(x) 1. No w w e write f(x) =

e~~g^

an d deduce that:

(1.21) g(x) i s convex an d satisfies g(x) log x

and we wish to thereby conclud e tha t

(1.22) f~e~*

Mdxe.

But, g(x) is convex and log x is concave, and so by th e "principl e of the separating hy-

perplane" we are assured of a straight lin e between thei r graphs. Tha t is to say, there exist

a, b such that