APPROXIMATION WIT H RATIONA L FUNCTION S

3

Indeed letting Z —* 0 show s that C must be 1/7(0) and so (1.7) may be written as

(1.8)

5(jc) =

l-/(Z)//(0)

Hence, using (1.8) and (1.6), we may write our requirement, (1.5), as the inequality

Z/(l - 7(Z)//(0) ) - Z e . Thi s in turn simplifies t o (Z + e)/(Z ) e/(0 ) whic h is to say

that

(1.9) o n [0 , 1], (Z + e)I(Z) take s its maximum at 0.

(This finally is our calculus problem in sharp focus. I t poses a simple "differentiat e

and be done" exercise. Actuall y a bit of finesse is again necessary and we find it more to

the point t o differentiate twice. )

We will show in fact tha t (Z 4- e)I(Z) i s convex on [0 , 1]. Thi s forces th e maximum

to occu r a t a n endpoint and since 7(1) = 0 by (1.6), the statement (1.9) must follow. T o

verify th e convexity w e have, by (1.6),

((Z + e)/(Z)) " = 2/'(Z ) + ( Z + e)/"(Z )

-2(1 -Z llk

(1 -Z

1'*)"- 1

= -2( 1 -Z 1'*))n " 4 - (Z + e)n(\ -Z llk)n~l • iz1 /*" 1

k

((n + 2k)Z^

k

+ neZ

1^-1

- 2k),

so that all we need to show is that

(1.10) (n + 2k)Z 1/k +neZ l/k-1 -2k0 o n [0,1].

But if we write

2k-2

xjk

W

2k + n

and recall the definition o f e from (1.6), we find that we can rewrite (1.10) as (2k - 2)/W +

2Wk~l 2k,or

(1.11) (W-lXOf*-

1 +Wk~2

+--- + l ) - * ) 0 .

These factors are both nonnegative i f W 1 and both negative if W 1 and in either

case (1.11), and hence ou r theorem, is proved.

It is now time to address ourselves to the (slightly) more serious proof o f Theorem 1.1.

This must be viewed as a more sophisticated ite m as it has at its core a Fourier Series rather

than a mere power series as we just sa w used in Theorem 1.2. Wha t is more t o the point is

that we now require a lower bound which we did not need for Theorem 1.2. Indee d a direct

calculation o f a certain elementary Fourie r Series easily gives us our upper bound on E

n(xk),

and what we then require is a theorem which tells us that, under such and such circum-

stances, the Fourier Series is just about th e best one can do. This , then, is the point o f the

Newman-Rivlin Theorem 1.1an d here, then, are the details.

PROOF O F THEOREM

1.1W . e explicitly trea t th e case of even k and n, the other cases