Stone-Weierstrass theorem

His result is known as the Stone-Weierstrass theorem.

This is a generalization of the Stone-Weierstrass theorem.

The boolean ring version of the Stone-Weierstrass theorem states :

First pass from polynomial to continuous functional calculus by using the Stone-Weierstrass theorem.

Uniqueness follows from application of the Stone-Weierstrass theorem.

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

A version of the Stone-Weierstrass theorem is also true when X is only locally compact.

In fact, if G is a matrix group, then the result follows easily from the Stone-Weierstrass theorem .

The Stone-Weierstrass theorem holds for C(X).

The Stone-Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space.

The Stone-Weierstrass theorem substantially generalized Weierstrass's theorem on the uniform approximation of continuous functions by polynomials.

The Stone-Weierstrass theorem mentioned above, for example, relies on Banach algebras which are both Banach spaces and algebras.

If a reservoir has fading memory and input separability, with help of a powerful readout, it can be proven the liquid state machine is a universal function approximator using Stone-Weierstrass theorem.

Bernstein polynomials thus afford one way to prove the Stone-Weierstrass theorem that every real-valued continuous function on a real interval 'a','b' can be uniformly approximated by polynomial functions over 'R'.

This first result resembles the Stone-Weierstrass theorem in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an algebraic characterization.

A basic result is that the trigonometric polynomials are dense in the space of continuous functions on the unit circle, with the uniform norm ; this is a special case of the Stone-Weierstrass theorem.

Indeed, by the Stone-Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.

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.

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.