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
1
http://dx.doi.org/10.1090/psapm/036/864363
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