Proceedings of Symposia in Applied Mathematics

Volume 36, 1986

APPROXIMATION OF FUNCTIONS

Ronald A. DeVore

Approximation Theory began at the end of the last century with the study

of the approximation of functions by polynomials and rational functions. It is

a broad subject which interacts with various aspects of real, complex and

functional analysis. Some of its recent popularity comes from its importance in

the development of numerical algorithms and the solution of problems of

optimization.

One hundred years ago, Weierstrass proved his famous theorem on the

approximation of continuous functions by algebraic polynomials. Undoubtedly,

everyone of you has seen this theorem. But, in order to guarantee that we are

all starting at the same point, let's begin with a formulation of this theorem

which uses the notation of Approximation Theory.

We want to approximate functions f which are continuous on an interval

I:=[a,b]. We let C(I) denote the set of all such functions and let

||f||:- sup |f(x)|,

xsl

be its norm. We are interested in approximating f by algebraic polynomials

P(x)= a^ + a^ + ... + a x of degree at most n. If I I denotes the set of

o l n ° n

a l l such polynomials, we le t E (f) be the error of approximation to f from II :

(o.i) Enf :s= inf Mf-pM-

Pe n

n

With this, we have

THEOREM 0.1. (Weierstrass [W]) If fsC(I), then E (f)-K) as n-°°.

In other words, each continuous function can be approximated arbitrarily well

in the uniform norm by polynomials. There are many wonderful proofs of

© 1986 American Mathematical Society

0160-7634/86 $1.00 + $.25 per page

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http://dx.doi.org/10.1090/psapm/036/864363