The
Weierstrass approximation theorem states that every
continuous function defined on an
interval [
a,
b] can be approximated as closely as desired by a
polynomial function.
Because polynomials are the simplest functions, and computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance.
[Marshall H. Stone]
? considerably generalized the theorem and simplified the proof; his result is known as the
Stone-Weierstrass theorem.
Weierstrass approximation theorem
Suppose
f is a continuous function defined on the interval [
a,
b] with
real values. For every ε>0, there exists a polynomial function
p with real coefficients such that for all
x in [
a,
b], we have |
f(
x) -
p(
x)| < ε.
The set C[a,b] of continuous real-valued functions on [a,b], together with the supremum norm ||f|| = supx in [a,b] |f(x)|, is a Banach algebra, (i.e. an associative algebra and a Banach space such that ||fg|| ≤ ||f|| ||g|| for all f, g). The set of all polynomial functions forms a subalgebra of C[a,b], and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C[a,b].
Stone-Weierstrass theorem, algebra version
The approximation theorem is generalized in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from other subalgebras of C(X) is investigated. The crucial property of the subalgebra is that it separates points: A subset A of C(X) is said to separate points, if for every two different points x and y in X and every two real numbers a and b there exists a function p in A with p(x) = a and p(y) = b. The formal statement is:
- If X is a compact Hausdorff space with at least two points and A is a subalgebra of the Banach algebra C(X) which separates points and contains a non-zero constant function, then A is dense in C(X).
This generalizes Weierstrass' statement since the polynomials form a subalgebra which separates points.
Applications
The Stone-Weierstrass theorem can be used to prove the following two statements:
- If f is a continuous real-valued function defined on the set [a,b] x [c,d] and ε>0, then there exists a polynomial function p in two variables such that |f(x,y) - p(x,y)| < ε for all x in [a,b] and y in [c,d].
- If X and Y are two compact Hausdorff spaces and f : XxY -> R is a continuous function, then for every ε>0 there exist n>0 and continuous functions f1, f2, ..., fn on X and continuous functions g1, g2, ..., gn on Y such that ||f - ∑figi|| < ε
Stone-Weierstrass theorem, lattice version
Let X be a compact Hausdorff space. A subset L of C(X) is called a lattice in C(X) if for any two elements f, g in L, the functions max(f,g) and min(f,g) also belong to L. The lattice version of the Stone-Weierstrass theorem states:
- If X is a compact Hausdorff space with at least two points and L is a lattice in C(X) which separates points, then A is dense in C(X).
- Need to cover the case of complex valued functions.