In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.
Marshall H. Stone considerably generalized the theorem and simplified the proof. His result is known as the StoneâÂÂWeierstrass theorem. The StoneâÂÂWeierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact Hausdorff space is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on are shown to suffice, as is detailed below. The StoneâÂÂWeierstrass 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 StoneâÂÂWeierstrass 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 StoneâÂÂWeierstrass theorem and described below.
A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane.
The statement of the approximation theorem as originally discovered by Weierstrass is as follows:
The page for Bernstein polynomials outlines a constructive proof of the above theorem.
For differentiable functions, Jackson's inequality bounds the error of approximations by polynomials of a given degree: if has a continuous k-th derivative, then for every there exists a polynomial of degree at most such that .
However, if is merely continuous, the convergence of the approximations can be arbitrarily slow in the following sense: for any sequence of positive real numbers decreasing to 0 there exists a function such that for every polynomial of degree at most .
As a consequence of the Weierstrass approximation theorem, one can show that the space 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. Since is metrizable and separable it follows that has cardinality at most . (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 continuous real-valued functions on , together with the supremum norm is a Banach algebra, (that is, an associative algebra and a Banach space such that for all ). The set of all polynomial functions forms a subalgebra of (that is, a vector subspace of that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is dense in .
Stone starts with an arbitrary compact Hausdorff space and considers the algebra of real-valued continuous functions on , with the topology induced by the supremum norm. He wants to find subalgebras of which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set of functions defined on is said to separate points if, for every two different points and in there exists a function in with . Now we may state:
This implies Weierstrass' original statement since the polynomials on form a subalgebra of which contains the constants and separates points.
A version of the StoneâÂÂWeierstrass theorem is also true when is only locally compact. Let be the space of real-valued continuous functions on that vanish at infinity; that is, a continuous function is in if, for every , there exists a compact set such that on . Again, is a Banach algebra with the supremum norm. A subalgebra of is said to vanish nowhere if not all of the elements of simultaneously vanish at a point; that is, for every in , there is some in such that . The theorem generalizes as follows:
This version clearly implies the previous version in the case when is compact, since in that case . There are also more general versions of the StoneâÂÂWeierstrass theorem that weaken the assumption of local compactness.
The StoneâÂÂWeierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result.
Slightly more general is the following theorem, where we consider the algebra of complex-valued continuous functions on the compact space , again with the topology of uniform convergence. This is a C*-algebra with the *-operation given by pointwise complex conjugation.
The complex unital *-algebra generated by consists of all those functions that can be obtained from the elements of by throwing in the constant function 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 net of complex-valued functions uniformly approximates a given function, , then the real parts of those functions uniformly approximate the real part of that function, , and because for real subsets, taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated.
As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.
The following is an application of this complex version.
Following , consider the algebra of quaternion-valued continuous functions on the compact space , again with the topology of uniform convergence.
If a quaternion is written in the form
Likewise
Then we may state:
The space of complex-valued continuous functions on a compact Hausdorff space i.e. is the canonical example of a unital commutative C*-algebra . The space X may be viewed as the space of pure states on , with the weak-* topology. Following the above cue, a non-commutative extension of the StoneâÂÂWeierstrass theorem, which remains unsolved, is as follows:
In 1960, Jim Glimm proved a weaker version of the above conjecture.
Let be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in . A subset of is called a lattice if for any two elements , the functions also belong to . The lattice version of the StoneâÂÂWeierstrass theorem states:
The above versions of StoneâÂÂWeierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the absolute value which in turn can be approximated by polynomials in . A variant of the theorem applies to linear subspaces of closed under max:
More precise information is available:
Another generalization of the StoneâÂÂWeierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows:
gives a short proof of Bishop's theorem using the KreinâÂÂMilman theorem in an essential way, as well as the HahnâÂÂBanach theorem: the process of . See also .
Nachbin's theorem gives an analog for StoneâÂÂWeierstrass theorem for algebras of complex valued smooth functions on a smooth manifold. Nachbin's theorem is as follows:
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. According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series".
The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: