Stone–Weierstrass theorem

Template:Short description In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval Template:Math 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<ref>Template:Citation</ref> and simplified the proof.<ref>Template:Citation; 21 (5), 237–254.</ref> 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 Template:Math, an arbitrary compact Hausdorff space Template:Mvar is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on <math>X</math> are shown to suffice, as is detailed below. 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.

A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane.

Weierstrass approximation theorem

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.

Degree of approximation

For differentiable functions, Jackson's inequality bounds the error of approximations by polynomials of a given degree: if <math>f</math> has a continuous k-th derivative, then for every <math>n\in\mathbb N</math> there exists a polynomial <math>p_n</math> of degree at most <math>n</math> such that <math>\lVert f-p_n\rVert \leq \frac\pi 2\frac 1{(n+1)^k} \lVert f^{(k)}\rVert</math>.<ref>Template:Cite book</ref>

However, if <math>f</math> is merely continuous, the convergence of the approximations can be arbitrarily slow in the following sense: for any sequence of positive real numbers <math>(a_n)_{n\in\mathbb N}</math> decreasing to 0 there exists a function <math>f</math> such that <math>\lVert f-p\rVert > a_n</math> for every polynomial <math>p</math> of degree at most <math>n</math>.<ref>Template:Cite journal</ref>

Applications

As a consequence of the Weierstrass approximation theorem, one can show that the space Template:Math 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 Template:Math is metrizable and separable it follows that Template:Math has cardinality at most Template:Math. (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.)

Stone–Weierstrass theorem, real version

The set Template:Math of continuous real-valued functions on Template:Math, together with the supremum norm Template:Math is a Banach algebra, (that is, an associative algebra and a Banach space such that Template:Math for all Template:Math). The set of all polynomial functions forms a subalgebra of Template:Math (that is, a vector subspace of Template:Math that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is dense in Template:Math.

Stone starts with an arbitrary compact Hausdorff space Template:Mvar and considers the algebra Template:Math of real-valued continuous functions on Template:Mvar, with the topology of uniform convergence. He wants to find subalgebras of Template:Math which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points: a set Template:Mvar of functions defined on Template:Mvar is said to separate points if, for every two different points Template:Mvar and Template:Mvar in Template:Mvar there exists a function Template:Mvar in Template:Mvar with Template:Math. Now we may state:

This implies Weierstrass' original statement since the polynomials on Template:Math form a subalgebra of Template:Math which contains the constants and separates points.

Locally compact version

A version of the Stone–Weierstrass theorem is also true when Template:Mvar is only locally compact. Let Template:Math be the space of real-valued continuous functions on Template:Mvar that vanish at infinity; that is, a continuous function Template:Math is in Template:Math if, for every Template:Math, there exists a compact set Template:Math such that Template:Math on Template:Math. Again, Template:Math is a Banach algebra with the supremum norm. A subalgebra Template:Mvar of Template:Math is said to vanish nowhere if not all of the elements of Template:Mvar simultaneously vanish at a point; that is, for every Template:Mvar in Template:Mvar, there is some Template:Math in Template:Mvar such that Template:Math. The theorem generalizes as follows:

This version clearly implies the previous version in the case when Template:Mvar is compact, since in that case Template:Math. There are also more general versions of the Stone–Weierstrass theorem that weaken the assumption of local compactness.<ref name=Willard>Template:Cite book</ref>

Applications

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

• If Template:Math is a continuous real-valued function defined on the set Template:Math and Template:Math, then there exists a polynomial function Template:Mvar in two variables such that Template:Math for all Template:Mvar in Template:Math and Template:Mvar in Template:Math.Template:Citation needed
• If Template:Mvar and Template:Mvar are two compact Hausdorff spaces and Template:Math is a continuous function, then for every Template:Math there exist Template:Math and continuous functions Template:Math on Template:Mvar and continuous functions Template:Math on Template:Mvar such that Template:Math. Template:Citation needed

Stone–Weierstrass theorem, complex version

Slightly more general is the following theorem, where we consider the algebra <math>C(X, \Complex)</math> of complex-valued continuous functions on the compact space <math>X</math>, 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 <math>S</math> consists of all those functions that can be obtained from the elements of <math>S</math> by throwing in the constant function Template:Math 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, <math>f_n\to f</math>, then the real parts of those functions uniformly approximate the real part of that function, <math>\operatorname{Re}f_n\to\operatorname{Re}f</math>, and because for real subsets, <math>S\subset C(X,\Reals)\subset C(X,\Complex),</math> 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.

• Fourier series: The set of linear combinations of functions Template:Math is dense in Template:Math, where we identify the endpoints of the interval Template:Math to obtain a circle. An important consequence of this is that the Template:Math are an orthonormal basis of the space [[Lp space|Template:Math]] of square-integrable functions on Template:Math. Template:Citation needed

Stone–Weierstrass theorem, quaternion version

Following Template:Harvtxt, consider the algebra Template:Math of quaternion-valued continuous functions on the compact space Template:Mvar, again with the topology of uniform convergence.

If a quaternion Template:Math is written in the form <math display=inline>q = a + ib + jc + kd</math>

• its scalar part Template:Math is the real number <math>\frac{q - iqi - jqj - kqk}{4}</math>.

Likewise

• the scalar part of Template:Math is Template:Math which is the real number <math>\frac{-qi - iq + jqk - kqj}{4}</math>.
• the scalar part of Template:Math is Template:Math which is the real number <math>\frac{-qj - iqk - jq + kqi}{4}</math>.
• the scalar part of Template:Math is Template:Math which is the real number <math>\frac{-qk + iqj - jqk - kq}{4}</math>.

Then we may state: Template:Math theorem

Stone–Weierstrass theorem, C*-algebra version

The space of complex-valued continuous functions on a compact Hausdorff space <math>X</math> i.e. <math>C(X, \Complex)</math> is the canonical example of a unital commutative C*-algebra <math>\mathfrak{A}</math>. The space X may be viewed as the space of pure states on <math>\mathfrak{A}</math>, with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which remains unsolved, is as follows:

In 1960, Jim Glimm proved a weaker version of the above conjecture.

Lattice versions

Let Template:Mvar be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in Template:Math. A subset Template:Mvar of Template:Math is called a lattice if for any two elements Template:Math, the functions Template:Mathalso belong to Template:Mvar. 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 Template:Math which in turn can be approximated by polynomials in Template:Math. A variant of the theorem applies to linear subspaces of Template:Math closed under max:<ref>Template:Citation</ref>

More precise information is available:

Suppose Template:Mvar is a compact Hausdorff space with at least two points and Template:Mvar is a lattice in Template:Math. The function Template:Math belongs to the closure of Template:Mvar if and only if for each pair of distinct points x and y in Template:Mvar and for each Template:Math there exists some Template:Math for which Template:Math and Template:Math.

Bishop's theorem

Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows:<ref>Template:Citation</ref>

Template:Harvtxt 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 Template:Harvtxt. See also Template:Harvtxt.

Nachbin's theorem

Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold.<ref>Template:Citation</ref> Nachbin's theorem is as follows:<ref>Template:Citation</ref>

Editorial history

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.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite book</ref><ref name="arxiv_0611038v2" /><ref name="arxiv_0611034v3" /> According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series".<ref name="arxiv_0611034v3">Template:Cite journal (arXiv 0611034v3). Citing: D. S. Lubinsky, Weierstrass' Theorem in the twentieth century: a selection, in Quaestiones Mathematicae18 (1995), 91–130.</ref><ref name="arxiv_0611038v2">Template:Cite journal (arXiv 0611038v2).</ref>

See also

• Müntz–Szász theorem
• Bernstein polynomial
• Runge's phenomenon shows that finding a polynomial Template:Mvar such that Template:Math for some finely spaced Template:Math is a bad way to attempt to find a polynomial approximating Template:Math uniformly. A better approach, explained e.g. in Template:Harvtxt, p. 160, eq. (51) ff., is to construct polynomials Template:Mvar uniformly approximating Template:Math by taking the convolution of Template:Math with a family of suitably chosen polynomial kernels.
• Mergelyan's theorem, concerning polynomial approximations of complex functions.

References

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• Jan Brinkhuis & Vladimir Tikhomirov (2005) Optimization: Insights and Applications, Princeton University Press Template:Isbn Template:Mr.
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Historical works

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). Template:Pb Erste Mitteilung (part 1) pp. 633–639, Zweite Mitteilung (part 2) pp. 789–805.

External links

• Template:Springer