9.4 *Two R esults of Weierstrass 573after a convenient number of terms, and thus were really using polynomials
to approximate functions. They relied on theorems such as Taylor series to
address the accuracy of these polynomial approximations. Weierstrass' theorem
guarantees that arbitrary continuous functions can indeed be approximated by
polynomials, but both the polynomials and the approximating behavior are
completely different from the familiar Taylor polynomials.
This is a deep theorem, and it has no "easy" proof. Weierstrass' original
proof used "singular integrals,'' a topic well beyond the scope of this course.
In subsequent decades intense work by many mathematicians produced quite a
few alternative proofs, most of which are also beyond the scope of this course.^12
The proof we give here is a modified version of a proof suggested by H. Lebesgue
in 1898 and included in [99]. I find it to be the most straightforward and most
easily understood, as well as intuitively appealing.
We begin our proof with the following:Definition 9.4.2 We say that a function f : [a, b] __, IR can be approx-
imated by polynomials if Ve > 0, there exists a polynomial p such that
'ix E [a, b], lf(x) - p(x)I < e; that is , relative to [a, b], Iii -Pll < e.
Lemma 9.4.3 A function f : [a, b] , IR can be approximated by poly-
nomials iff 3 sequence {pn} of polynomials such that Pn , f uniformly on
[a,b].
Lemma 9.4.4 The set P[a, b] of polynomials with domain [a, b] is a vector
space [in fact, a subspace of C[a, b]J.
Lemma 9.4.5 The set of all continuous f: [a, b] __,IR that can be approximated
by polynomials is a vector space [a subspace of C[a, b]J. Let us (temporarily) call
this space CAP[a,b].
Definition 9.4.6 A subset S of C[a, b] is said to be dense^13 in C[a, b] if V f E
C[a,b] and Ve> 0, 3 g ES 3 Iii - 911 < e.
Lemma 9.4. 7 A subset S of C[a, b] is dense^14 in C[a, b] iff V f E C[a, b], 3
sequence { sn} of elements of S that converges uniformly to f on [a, b].
Thus, our task is to prove that the polynomials are dense in C[a, b]. We
begin by obtaining an intermediate result, which requires another definition.
- For a summary of the various proofs see the survey paper [107] by Pinkus.
- Compare this with Exercise 3.2.29.
- Compare this with Theorem 3.2.21.