In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function.
Marshall H. Stone considerably generalized the theorem[1] and simplified the proof.
The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on
The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space.
Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff spaces, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below.
The statement of the approximation theorem as originally discovered by Weierstrass is as follows: Weierstrass approximation theorem — Suppose f is a continuous real-valued function defined on the real interval [a, b].
A constructive proof of this theorem using Bernstein polynomials is outlined on that page.
For differentiable functions, Jackson's inequality bounds the error of approximations by polynomials of a given degree: if
is merely continuous, the convergence of the approximations can be arbitrarily slow in the following sense: for any sequence of positive real numbers
(Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.)
The set of all polynomial functions forms a subalgebra of C[a, b] (that is, a vector subspace of C[a, b] that is closed under multiplication of functions), 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 X and considers the algebra C(X, R) of real-valued continuous functions on X, with the topology of uniform convergence.
Now we may state: Stone–Weierstrass theorem (real numbers) — Suppose X is a compact Hausdorff space and A is a subalgebra of C(X, R) which contains a non-zero constant function.
This implies Weierstrass' original statement since the polynomials on [a, b] form a subalgebra of C[a, b] which contains the constants and separates points.
Let C0(X, R) be the space of real-valued continuous functions on X that vanish at infinity; that is, a continuous function f is in C0(X, R) if, for every ε > 0, there exists a compact set K ⊂ X such that |f| < ε on X \ K. Again, C0(X, R) is a Banach algebra with the supremum norm.
Then A is dense in C0(X, R) (given the topology of uniform convergence) if and only if it separates points and vanishes nowhere.
There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness.
As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.
Following Holladay (1957), consider the algebra C(X, H) of quaternion-valued continuous functions on the compact space X, again with the topology of uniform convergence.
Likewise Then we may state: Stone–Weierstrass theorem (quaternion numbers) — Suppose X is a compact Hausdorff space and A is a subalgebra of C(X, H) which contains a non-zero constant function.
Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which remains unsolved, is as follows: Conjecture — If a unital C*-algebra
In 1960, Jim Glimm proved a weaker version of the above conjecture.
Stone's original proof of the theorem used the idea of lattices in C(X, R).
A variant of the theorem applies to linear subspaces of C(X, R) closed under max:[7] Stone–Weierstrass theorem (max-closed) — Suppose X is a compact Hausdorff space and B is a family of functions in C(X, R) such that Then B is dense in C(X, R).
More precise information is available: Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop.
Bishop's theorem is as follows:[8] Bishop's theorem — Let A be a closed subalgebra of the complex Banach algebra C(X, C) of continuous complex-valued functions on a compact Hausdorff space X, using the supremum norm.
[9] Nachbin's theorem is as follows:[10] Nachbin's theorem — Let A be a subalgebra of the algebra C∞(M) of smooth functions on a finite dimensional smooth manifold M. Suppose that A separates the points of M and also separates the tangent vectors of M: for each point m ∈ M and tangent vector v at the tangent space at m, there is a f ∈ A such that df(x)(v) ≠ 0.
In 1885 it was also published in an English version of the paper whose title was On the possibility of giving an analytic representation to an arbitrary function of real variable.
[11][12][13][14][15] According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series".
[15][14] The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: