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
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