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Stone-Weierstrass theorem

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.

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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 of the theorem 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 on [a,b] form a subalgebra of C[a,b] which separates points.

Note that the above theorem is also true if we replace the assumption that A separate points with the slightly weaker assumption that for every two different points x and y in X there exists a function p in A with p(x) not equal to p(y).

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.



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