Stone-Weierstrass theorem

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly 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, especially in polynomial interpolation. The original version of this result has been established by Karl Weierstraß in 1885.

Marshall H. Stone considerably generalized the theorem (Stone, 1937) and simplified the proof (Stone, 1948); his result is known as the Stone-Weierstrass theorem. The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated.

Further, there is a generalization of the Stone-Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space space is approximated uniformly on compact sets by algebras of the type appearing in the Stone-Weierstrass theorem and described below.

Weierstrass approximation theorem

The statement of Weierstrass's theorem is as follows:

Suppose f is a continuous real-valued function defined on the interval [a,b]. 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.

A constructive proof of this theorem using Bernstein polynomials is outlined on that page.

Applications

As a consequence of the Weierstrass approximation theorem, one can show that the space C[a,b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients.

Stone-Weierstrass theorem, real version

The set C[a,b] of continuous real-valued functions on [a,b], together with the supremum norm ||f|| = sup_{x 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 starts with an arbitrary compact Hausdorff space K and considers the algebra C(K,R) of real-valued continuous functions on K, with the topology of uniform convergence. He wants to find subalgebras of C(K,R) which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: A set A of functions defined on K is said to separate points if, for every two different points x and y in K there exists a function p in A with p(x) not equal to p(y).

The statement of Stone-Weierstrass is:

Suppose K is a compact Hausdorff space and A is a subalgebra of C(K,R) which contains a non-zero constant function. Then A is dense in C(K,R) if and only if it separates points.

This implies Weierstrass' original statement since the polynomials on [a,b] form a subalgebra of C[a,b] which separates points.

Applications

The Stone-Weierstrass theorem can be used to prove the following two statements which go beyond Weierstrass's result.

• 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, ..., f_n on X and continuous functions g₁, g₂, ..., g_n on Y such that ||f - ∑f_ig_i|| < ε

Stone-Weierstrass theorem, complex version

Slightly more general is the following theorem, where we consider the algebra C(K,C) of complex-valued continuous functions on the compact space K, again with the topology of uniform convergence. This is a C*-algebra with the *-operation given by pointwise complex conjugation.

Let K be a compact Hausdorff space and let S be a subset of C(K,C) which separates points. Then the complex unital *-algebra generated by S is dense in C(K,C).

The complex unital *-algebra generated by S consists of all those functions that can be gotten from the elements of S by throwing in the constant function 1 and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times.

This theorem implies the real version, because if a sequence of complex-valued functions uniformly approximate a given function f, then the real parts of those functions uniformly approximate the real part of f.

Stone-Weierstrass theorem, lattice version

Let K be a compact Hausdorff space. A subset L of C(K,R) is called a lattice 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:

Suppose K is a compact Hausdorff space with at least two points and L is a lattice in C(K,R) with the property that for any two distinct elements x and y of K and any two real numbers a and b there exists an element f in L with f(x) = a and f(y) = b. Then L is dense in C(K,R).

The above versions of Stone-Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value |f| which in turn can be approximated by polynomials in f.

More precise information is available:

Suppose K is a compact Hausdorf space with at least two points and L is a lattice in C(K,R). The function φ in C(K,R) belongs to the closure of L iff for each pair of distinct points x and y in K and for each ε > 0 there exists some f in L for which |f(x) - φ(x)| < ε and |f(y) - φ(y)| < ε.

References

The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften :

• K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885 (II). Erste Mitteilung (part 1) pp. 633–639, Zweite Mitteilung (part 2) pp. 789–805.

Important historical works of Stone include:

• M. H. Stone (1937). Applications of the Theory of Boolean Rings to General Topology. Transactions of the American Mathematical Society 41 (3), 375–481.
• M. H. Stone (1948). The Generalized Weierstrass Approximation Theorem. Mathematics Magazine 21 (4), 167–184 and 21 (5), 237–254.