Kenneth Kuttler

5.10. THE STONE WEIERSTRASS APPROXIMATION THEOREM 141

≤ ∥ f − pn∥p

∏k=1|bk−ak| (5.5)

With this, it is easy to prove a rudimentary Fubini theorem valid for continuous functions.

Theorem 5.9.6 f : ∏pk=1 [ak,bk]→ R be continuous. Then for (i1, · · · , ip) any per-

mutation of (1, · · · , p) ,∫ bi1

aip

· · ·∫ bip

aip

f (x)dxip · · ·dxi1 =∫ b1

· · ·∫ bp

f (x)dxp · · ·dx1

If f ≥ 0, then the iterated integrals are nonnegative if each ak ≤ bk.

Proof: Let ∥pn− f∥∏

pk=1[ak,bk]

→ 0 where pn is a polynomial. Then from 5.5,

∫ bi1

ai1

· · ·∫ bip

aip

f (x)dxip · · ·dxi1 = limn→∞

∫ bi1

aip

· · ·∫ bip

aip

pn (x)dxip · · ·dxi1

= limn→∞

∫ b1

· · ·∫ bp

pn (x)dxp · · ·dx1 =∫ b1

· · ·∫ bp

f (x)dxp · · ·dx1 ■

You could replace f with f XG where XG (x) = 1 if x ∈ G and 0 otherwise provided eachsection of G consisting of holding all variables constant but 1, consists of finitely manyintervals. Thus you can integrate over all the usual sets encountered in beginning calculus.

5.10 The Stone Weierstrass Approximation TheoremThere is a profound generalization of the Weierstrass approximation theorem due to Stone.It has to be one of the most elegant things available. It holds on locally compact Hausdorffspaces but here I will show the version which is valid on compact sets. Later the moregeneral version is discussed.

Definition 5.10.1 A is an algebra of functions if A is a vector space and if when-ever f ,g ∈A then f g ∈A .

To begin with assume that the field of scalars is R. This will be generalized later.Theorem 5.6.2 implies the following corollary. See Corollary 5.6.3.

Corollary 5.10.2 The polynomials are dense in C ([a,b]).

Here is another approach to proving this theorem. It is the original approach used byWeierstrass. Let m ∈ N and consider cm such that

∫ 1−1 cm

(1− x2

)m dx = 1. Then

1 = 2∫ 1

0cm(1− x2)m

dx≥ 2cm

∫ 1

0(1− x)m dx = 2cm

1m+1

so cm ≤ m+1. Then∫ 1

cm(1− x2)m

dx+∫ −δ

−1cm(1− x2)m

dx≤ 2(m+1)(

1−δ2)m

5.10. THE STONE WEIERSTRASS APPROXIMATION THEOREM 141P< ||f — Pall [] lox — al (5.5)k=1With this, it is easy to prove a rudimentary Fubini theorem valid for continuous functions.Theorem 5.9.6 /: TTR; [ax,b«] 4 R be continuous. Then for (i},+++ ,ip) any per-mutation of (1,---,P);iy Dip by bp[~ _ fe x) dx; +++ Axi, = [~ f (@) dxp---dx1If f = 0, then the iterated integrals are nonnegative if each ax < br.Proof: Let || Pn — f\lpP_, fa;,b4] > 9 Where Pn is a polynomial. Then from 5.5,it i bi, Dip[- (x) dx, - --dx;, = lim ee Pn (x) dx;, -+-dXxi,a, nro qj dipby bp by bp<lim [ |. Pn (ee) daip--dxy = | EF (a) dxp--dxy Inoo ay ap a YapYou could replace f with f 2G where 2G (x) = 1 if x € G and 0 otherwise provided eachsection of G consisting of holding all variables constant but 1, consists of finitely manyintervals. Thus you can integrate over all the usual sets encountered in beginning calculus.5.10 The Stone Weierstrass Approximation TheoremThere is a profound generalization of the Weierstrass approximation theorem due to Stone.It has to be one of the most elegant things available. It holds on locally compact Hausdorffspaces but here I will show the version which is valid on compact sets. Later the moregeneral version is discussed.Definition 5.10.1 7 is an algebra of functions if & is a vector space and if when-ever f,g © & then fg E &.To begin with assume that the field of scalars is R. This will be generalized later.Theorem 5.6.2 implies the following corollary. See Corollary 5.6.3.Corollary 5.10.2 The polynomials are dense in C ([a, b}).Here is another approach to proving this theorem. It is the original approach used byWeierstrass. Let m € N and consider c,, such that fi Cm (1 —x)"dx = 1. Then1SO Cm <m+1. Then[ew( — eyact [en (1-x°) "dx <2(m+1) (1-8?)"