Lecture I . Approximatio n t o x

k

It i s a long establishe d rul e o f publi c speakin g tha t on e begin s a talk wit h a joke.

We intend t o follow thi s rule to th e extreme b y in fact making our whole first lectur e a kind

of joke. Th e joke i s that we will dedicat e ourselve s to approximating tha t "simplest " of all

functions,

xk

(b y lower degree polynomials o r rational functions , to be sure).

Joke o r no joke, however, one of our conclusions will be that, even on this rudimen-

tary level, rational function s d o a better job of approximating than do th e polynomials. Thi s

message will reappear throughout thes e talk s and may even be thought o f a s our central

theme. Thu s it seems that we may have less of a joke tha n a light version of ou r heavy

preaching. B e that as it may, it shoul d turn out that thes e results, as well as the very ele-

mentary method s of proof, will be quite amusing.

The situation regardin g polynomials was analyzed b y Ted Rivlin and myself som e time

ago, and the story i s roughly this : Th e function x

k

ca n be "effectively " approximate d b y

nth degre e polynomials as long as n is "much larger" than \fk.

More recently I considered th e corresponding problem for rational functions an d found

that no such condition o n n was necessary! Thu s the rationa l functio n stor y i s this: Th e

function x

k

ca n be "effectively" approximate d b y nth degre e rational functions a s long as

n is large (independent o f k).

In short th e collection of function s x k i s uniformly approximate b y nth degre e ra-

tional functions , but definitely no t b y nth degre e polynomials.W e have, then, another ex-

ample illustrating a qualitative superiority o f rationals over polynomials. I t is not just a

matter of spee d of convergenc e (e~^ n versu s I In) but o f th e very existence o f (uniform )

convergence.W e turn now t o the precise forms of th e above statement.

First o f all we introduce som e necessary notation whic h we will use throughout thes e

lectures.

Nl.l. ll/(Jc)| | = sup,_ lfl,|/(a:)|.

N1.2. U

n

i s the set of all polynomials o f degre e a t most n.

N1.3. n

m n

i s the set of al l rational functions wit h numerator o f degree m and de-

nominator o f degre e n.

N1.4. E

n

(f) = distanc e (/, n„ ) = mf

p{x)Gn

Jf(x) - P(x)\l

Nl'5- R m,n(f) = Stanc e (f

f

Yl

m n

) = inf

r(x)en

||/(x ) - r(x)\\; also we abbreviate

N1.6.W e write a 0 to mean |a | |j8| .

We can now state

1