In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, the Weierstrass approximation theorem states that every
continuous function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
defined on a closed
interval can be
uniformly approximated as closely as desired by a
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
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
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points in the dataset.
Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no ...
. The original version of this result was established by
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
in
1885 using the
Weierstrass transform.
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 , an arbitrary
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
is considered, and instead of the
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
of polynomial functions, a variety of other families of continuous functions on
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
In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space X with values in the real or complex numbers. This space, denoted by \mathcal(X), is a v ...
.
Further, there is a generalization of the Stone–Weierstrass theorem to noncompact
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. A Tychonoff space is any completely regular space that is also a ...
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 Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
Statement
:Let K be a compact subset of the complex plane \mathbb C such that \mathbb C \setm ...
, which generalizes it to functions defined on certain subsets of the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
.
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 polynomial
In the mathematics, mathematical field of numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of #Bernstein basis polynomials, Bernstein basis polynomials. The idea is named after mathematician Sergei Nata ...
s is outlined on that page.
Degree of approximation
For differentiable functions,
Jackson's inequality
In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function or of it ...
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
.
Applications
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
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
coefficients; there are only
countably many polynomials with rational coefficients. Since is
metrizable
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...
and separable it follows that has
cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
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.)
Stone–Weierstrass theorem, real version
The set of continuous real-valued functions on , together with the supremum norm is a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
, (that is, an
associative algebra
In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...
and a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
such that for all ). The set of all polynomial functions forms a subalgebra of (that is, a
vector subspace
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
of that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is
dense
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
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
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E as the function domain i ...
. 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.
Locally compact version
A version of the Stone–Weierstrass theorem is also true when is only
locally compact
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
. Let be the space of real-valued continuous functions on that
vanish at infinity In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the othe ...
; that is, a continuous function is in if, for every , there exists a compact set such that on . Again, is a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
with the
supremum norm
In mathematical analysis, the uniform norm (or ) assigns, to real- or complex-valued bounded functions defined on a set , the non-negative number
:\, f\, _\infty = \, f\, _ = \sup\left\.
This norm is also called the , the , the , or, when t ...
. 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 theorem 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 is a continuous real-valued function defined on the set and , then there exists a polynomial function in two variables such that for all in and in .
* If and are two compact Hausdorff spaces and is a continuous function, then for every there exist and continuous functions on and continuous functions on such that .
Stone–Weierstrass theorem, complex version
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
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...
with the *-operation given by pointwise
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
.
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.
*
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
: The set of linear combinations of functions is dense in , where we identify the endpoints of the interval to obtain a circle. An important consequence of this is that the are an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
of the space
of
square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
s on .
Stone–Weierstrass theorem, quaternion 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
*its scalar part is the real number
.
Likewise
*the scalar part of is which is the real number
.
*the scalar part of is which is the real number
.
*the scalar part of is which is the real number
.
Then we may state:
Stone–Weierstrass theorem, C*-algebra version
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 Stone–Weierstrass theorem, which remains unsolved, is as follows:
In 1960,
Jim Glimm proved a weaker version of the above conjecture.
Lattice versions
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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
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 .
Bishop's theorem
Another generalization of the Stone–Weierstrass theorem is due to
Errett Bishop
Errett Albert Bishop (July 14, 1928 – April 14, 1983) was an American mathematician known for his work on analysis. He is best known for developing constructive analysis in his 1967 ''Foundations of Constructive Analysis'', where he proved mos ...
. Bishop's theorem is as follows:
gives a short proof of Bishop's theorem using the
Krein–Milman theorem
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs).
This theorem generalizes to infinite-dimensional spaces and to arbitra ...
in an essential way, as well as the
Hahn–Banach theorem
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
: the process of . See also .
Nachbin's theorem
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:
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''.
According to the mathematician Yamilet Quintana, Weierstrass "suspected that any
analytic functions
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
could be represented by
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
".
[ (arXiv 0611034v3). Citing: D. S. Lubinsky, ''Weierstrass' Theorem in the twentieth century: a selection'', in ''Quaestiones Mathematicae''18 (1995), 91–130.][ (arXiv 0611038v2).]
See also
*
Müntz–Szász theorem
*
Bernstein polynomial
In the mathematics, mathematical field of numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of #Bernstein basis polynomials, Bernstein basis polynomials. The idea is named after mathematician Sergei Nata ...
*
Runge's phenomenon
In 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 over a set of equispaced interpolation ...
shows that finding a polynomial such that for some finely spaced is a bad way to attempt to find a polynomial approximating uniformly. A better approach, explained e.g. in , p. 160, eq. (51) ff., is to construct polynomials uniformly approximating by taking the convolution of with a family of suitably chosen polynomial kernels.
*
Mergelyan's theorem Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
Statement
:Let K be a compact subset of the complex plane \mathbb C such that \mathbb C \setm ...
, concerning polynomial approximations of complex functions.
Notes
References
* .
* .
*
Jan Brinkhuis & Vladimir Tikhomirov (2005) ''Optimization: Insights and Applications'',
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial ...
.
*
* .
*
* .
* .
Historical works
The historical publication of Weierstrass (in
German language
German (, ) is a West Germanic language in the Indo-European language family, mainly spoken in Western Europe, Western and Central Europe. It is the majority and Official language, official (or co-official) language in Germany, Austria, Switze ...
) 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.
External links
*
{{DEFAULTSORT:Stone-Weierstrass Theorem
Theory of continuous functions
Theorems in mathematical analysis
Theorems in approximation theory
1885 in science
1937 in science
19th century in mathematics
20th century in mathematics