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.[1] His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass 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
X
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.
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:
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
f
n\inN
pn
n
\lVertf-pn\rVert\leq
| \pi | |||
|
\lVertf(k)\rVert
However, if
f
(an)n\inN
f
\lVertf-p\rVert>an
p
n
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 of uniform convergence. 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 Stone–Weierstrass 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 Stone–Weierstrass that weaken the assumption of local compactness.[4]
The Stone–Weierstrass 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
C(X,\Complex)
X
The complex unital *-algebra generated by
S
S
This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function,
fn\tof
\operatorname{Re}fn\to\operatorname{Re}f
S\subsetC(X,\Reals)\subsetC(X,\Complex),
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
| q-iqi-jqj-kqk | |
| 4 |
| -qi-iq+jqk-kqj | |
| 4 |
| -qj-iqk-jq+kqi | |
| 4 |
| -qk+iqj-jqk-kq | |
| 4 |
Then we may state:
The space of complex-valued continuous functions on a compact Hausdorff space
X
C(X,\Complex)
ak{A}
ak{A}
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 Stone–Weierstrass theorem states:
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 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:
Suppose is a compact Hausdorff space with at least two points and is a lattice in . The function belongs to the closure of if and only if for each pair of distinct points x and y in and for each there exists some for which and .
Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows:
gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem: the process of . See also .
Nachbin's theorem gives an analog for Stone–Weierstrass 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.[5] [6] [7] According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series".[8] [9]
The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften: