Stone-Weierstrass theorem
Encyclopedia
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 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.
Marshall H. Stone considerably generalized the theorem and simplified the proof . 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 [a,b], an arbitrary compact
Hausdorff space
X is considered, and instead of the algebra
of polynomial functions, approximation with elements from more general subalgebras of C(X) is investigated. 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 space
s, 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.
A constructive proof of this theorem (for ƒ real-valued) using Bernstein polynomial
s is outlined on that page.
coefficients; there are only countably many polynomials with rational coefficients. Since C[a,b] is Hausdorff
and separable it follows that C[a,b] has cardinality equal to 2ℵ0 — the same cardinality as the cardinality of the reals
.
, (i.e. an associative algebra
and a Banach space
such that for all f, g). The set of all polynomial functions forms a subalgebra of C[a,b] (i.e. 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. He wants to find subalgebras of C(X,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 X is said to separate points if, for every two different points x and y in X there exists a function p in A with p(x) not equal to p(y).
The statement of Stone–Weierstrass is:
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.
with the supremum norm. A subalgebra A of is said to vanish nowhere if not all of the elements of A simultaneously vanish at a point; that is, for every x in X, there is some f in A such that . The theorem generalizes as follows:
This version clearly implies the previous version in the case when X is compact, since in that case . There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness.
The theorem has many other applications to analysis, including:
The complex unital *-algebra generated by S consists of all those functions that can be obtained 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. As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.
s inside C(X,R); that is, subsets B of C(X,R) such that for every f, g in B, the functions f+g and max{f,g} are also in B. The boolean ring version of the Stone–Weierstrass theorem states :
A similar version of the theorem applies to lattices
in C(X,R). A subset L of C(X,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:
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:
. 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
. See also .
) is freely available from the digital online archive of the Berlin Brandenburgische Akademie der Wissenschaften:
Important historical works of Stone include:
.; 21 (5), 237–254.
and Vladimir Tikhomirov: 2005, Princeton University Press
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, the Weierstrass approximation theorem states that every continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
defined on an interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[a,b] can be uniformly approximated as closely as desired by a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
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
Polynomial interpolation
In numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial: given some points, find a polynomial which goes exactly through these points.- Applications :...
. The original version of this result was established by Karl Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....
in 1885.
Marshall H. Stone considerably generalized the theorem and simplified the proof . 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 [a,b], an arbitrary compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
X is considered, and instead of the algebra
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
of polynomial functions, approximation with elements from more general subalgebras of C(X) is investigated. The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space
Continuous functions on a compact Hausdorff space
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by C, is a vector space with respect to the pointwise addition of functions...
.
Further, there is a generalization of the Stone–Weierstrass theorem to noncompact Tychonoff space
Tychonoff space
In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces.These conditions are examples of separation axioms....
s, namely, any continuous function on a Tychonoff space is approximated uniformly on compact sets
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
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
Mergelyan's theorem
Mergelyan's theorem is a famous result from complex analysis, a branch of mathematics, named after the Armenian mathematician Sergei Nikitovich Mergelyan and was put forward by him in 1951. It states the following:...
, 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:- Suppose ƒ is a continuous complexComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
-valued function defined on the real interval [a,b]. For every , there exists a polynomial function p over C such that for all x in [a,b], we have , or equivalently, the supremum norm . If ƒ is real-valued, the polynomial function can be taken over R.
A constructive proof of this theorem (for ƒ real-valued) using Bernstein polynomial
Bernstein polynomial
In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials....
s 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 rationalRational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
coefficients; there are only countably many polynomials with rational coefficients. Since C[a,b] is Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
and separable it follows that C[a,b] has cardinality equal to 2ℵ0 — the same cardinality as the cardinality of the reals
Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by |\mathbb R| or \mathfrak c ....
.
Stone–Weierstrass theorem, real version
The set C[a,b] of continuous real-valued functions on [a,b], together with the supremum norm , is a Banach algebraBanach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
, (i.e. an associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
and a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
such that for all f, g). The set of all polynomial functions forms a subalgebra of C[a,b] (i.e. a vector subspace
Linear subspace
The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
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
Topology glossary
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology...
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. He wants to find subalgebras of C(X,R) which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it separates points
Separating set
In mathematics a set of functions S from a set D to a set C is called a separating set for D or said to separate the points of D if for any two distinct elements x and y of D, there exists a function f in S so that f ≠ f.- Examples :* The singleton set consisting of the identity function on R...
: a set A of functions defined on X is said to separate points if, for every two different points x and y in X there exists a function p in A with p(x) not equal to p(y).
The statement of Stone–Weierstrass is:
- Suppose X is a compact Hausdorff space and A is a subalgebra of C(X,R) which contains a non-zero constant function. Then A is dense in C(X,R) if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it separates points.
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.
Locally compact version
A version of the Stone–Weierstrass theorem is also true when X is only locally compact. Let be the space of real-valued continuous functions on X which vanish at infinity; that is, a continuous function f is in if, for every , there exists a compact set such that on . Again, is a Banach algebraBanach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
with the supremum norm. A subalgebra A of is said to vanish nowhere if not all of the elements of A simultaneously vanish at a point; that is, for every x in X, there is some f in A such that . The theorem generalizes as follows:
- Suppose X is a locally compact Hausdorff space and A is a subalgebra of . Then A is dense in (given the topology of uniform convergence) if and only if it separates points and vanishes nowhere.
This version clearly implies the previous version in the case when X is compact, since in that case . There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness.
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 and , then there exists a polynomial function p in two variables such that for all x in and y in .
- If X and Y are two compact Hausdorff spaces and is a continuous function, then for every there exist and continuous functions f1, f2, …, fn on X and continuous functions g1, g2, …, gn on Y such that .
The theorem has many other applications to analysis, including:
- Fourier seriesFourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
: The set of linear combinations of functions is dense in C([0,1]/{0,1}), where we identify the endpoints of the interval [0,1] to obtain a circle. An important consequence of this is that the en are an orthonormal basisOrthonormal basisIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
of the space L2([0,1])Lp spaceIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
of square-integrable functionSquare-integrable functionIn mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...
s on [0,1].
Stone–Weierstrass theorem, complex version
Slightly more general is the following theorem, where we consider the algebra C(X,C) of complex-valued continuous functions on the compact space X, again with the topology of uniform convergence. This is a C*-algebra with the *-operation given by pointwise complex conjugation.- Let X be a compact Hausdorff space and let S be a subset of C(X,C) which separates pointsSeparating setIn mathematics a set of functions S from a set D to a set C is called a separating set for D or said to separate the points of D if for any two distinct elements x and y of D, there exists a function f in S so that f ≠ f.- Examples :* The singleton set consisting of the identity function on R...
. Then the complex unital *-algebra generated by S is dense in C(X,C).
The complex unital *-algebra generated by S consists of all those functions that can be obtained 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. As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.
Lattice and boolean ring versions
Let X be a compact Hausdorff space. Stone's original proof of the theorem used the idea of boolean ringBoolean ring
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists only of idempotent elements....
s inside C(X,R); that is, subsets B of C(X,R) such that for every f, g in B, the functions f+g and max{f,g} are also in B. The boolean ring version of the Stone–Weierstrass theorem states :
- Suppose X is a compact Hausdorff space and B is a family of functions in C(X,R) such that
- B separates points.
- B contains the constant function 1.
- If f ∈ B then af ∈ B for all constants a ∈ R.
- B is a boolean ring; that is, if f, g ∈ B, then f+g ∈ B and max{f,g} ∈ B.
- Then B is dense in C(X,R).
A similar version of the theorem applies to lattices
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
in C(X,R). A subset L of C(X,R) is called a lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
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 X is a compact Hausdorff space with at least two points and L is a lattice in C(X,R) with the property that for any two distinct elements x and y of X 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(X,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
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
|f| which in turn can be approximated by polynomials in f.
More precise information is available:
- Suppose X is a compact Hausdorff space with at least two points and L is a lattice in C(X,R). The function φ in C(X,R) belongs to the closureClosure (topology)In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
of L iffIFFIFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
for each pair of distinct points x and y in X and for each ε > 0 there exists some f in L for which |f(x) - φ(x)| < ε and |f(y) - φ(y)| < ε.
Bishop's theorem
Another generalization of the Stone–Weierstrass theorem is due to Errett BishopErrett Bishop
Errett Albert Bishop was an American mathematician known for his work on analysis. He is the father of constructive analysis, because of his 1967 Foundations of Constructive Analysis, where he proved most of the important theorems in real analysis by constructive methods.-Life:Errett Bishop's...
. Bishop's theorem is as follows :
- Let A be a closed subalgebra of the Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
C(X,C) of continuous complex-valued functions on a compact Hausdorff space X. Suppose that f ∈ C(X, C) has the following property:- f|S ∈ AS for every maximal set S ⊂ X such that AS contains no non-constant real functions.
- Then f ∈ A.
gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem
Hahn–Banach theorem
In mathematics, the Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear functionals defined on every normed...
. See also .
See also
- Runge's phenomenonRunge's phenomenonIn the mathematical field of numerical analysis, Runge's phenomenon is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree...
shows that finding a polynomial P such that ƒ(x) = P(x) for some finely spaced x = xn is a bad way to attempt to find a polynomial approximating ƒ uniformly. However, as is shown in Walter RudinWalter RudinWalter Rudin was an American mathematician, for most of his career a Professor of Mathematics at the University of Wisconsin–Madison....
's Principles of Mathematical Analysis, one can easily find a polynomial P uniformly approximating ƒ by convolving ƒ with a polynomial kernel. - Mergelyan's theoremMergelyan's theoremMergelyan's theorem is a famous result from complex analysis, a branch of mathematics, named after the Armenian mathematician Sergei Nikitovich Mergelyan and was put forward by him in 1951. It states the following:...
, concerning polynomial approximations of complex functions.
Historical works
The historical publication of Weierstrass (in German languageGerman language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
) 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:
.; 21 (5), 237–254.
Books
'Optimization: Insights and Applications', Jan BrinkhuisJan Brinkhuis
Prof. dr. Jan Brinkhuis is Associate Professor of Finance and Mathematical Methods and Techniques at the Econometric Institute of Erasmus University Rotterdam, Rotterdam. He obtained his PhD in 1981 with the disseration "Embedding Problems and Galois Modules"...
and Vladimir Tikhomirov: 2005, Princeton University Press