The theorem of approximation of Weierstrass affirms that all function continues definite on a compact can be also approximate close one wants by a polynomial function. Because the functions polynomials are the simplest functions, and that the computers can directly evaluate the polynomials, this theorem has at the same time a practical and theoretical interest. Marshall H. Stone considerably generalized the theorem and simplified the demonstration; it is this improvement which is called the theorem of Stone-Weierstrass .

Theorem of approximation of Weierstrass

Let us suppose that F is a continuous function definite on the interval at real values . For all ε>0, there exists a function polynomial p with real coefficients such as for all X in, we have | F ( X ) - p ( X )| < ε.

This property can be also expressed in the following form: if F is a continuous function on, there exists a continuation $(P\_n)$ of polynomials converging uniformly towards F on. Below for example, the continuation of polynomials converges towards the absolute value on the interval.

While being reduced by change of variables to the interval, Bernstein gave a constructive demonstration of it by proving that one could take:

$P\_n\; (X)\; =\; \backslash \; sum\_\; \{k=0\}\; ^n\; F\; (\{K\; \backslash \; over\; N\})\; \{N\; \backslash \; choose\; K\}\; x^k\; (1-x)\; ^\; \{n-k\}$

The unit C () of the continuous functions on to actual values, provided with the infinite standard $||F||=\; \backslash \; sup\_\; \{X\; \backslash \; in\}\; |F\; (X)|$, is a Algèbre of Banach, (i.e a associative Algèbre and a Espace of Banach such as for all F and G , || fg || ≤ || F || || G ||). The whole of the polynomial functions forms a subalgebra of C (), and the theorem of approximation of Weierstrass says that this subalgebra is dense in C ().

Trigonometrical version

For any function F continues T - periodic, there exists a continuation $(T\_n)$ of trigonometrical polynomials which converges uniformly towards F .

The theorem of Fejér, resulting from the theory of the Fourier series, gives a concrete example of such continuation.

More general forms: theorem of Stone-Weierstrass

The theorem of Stone-Weierstrass, algebraic version

The theorem of approximation spreads in two directions: in the place of the compact interval, a separate Space or spaces of compact Hausdorff arbitrary X can be considered, and in the place of the algebra of the functions polynomials, an approximation with elements of other subalgebras of C ( X ) can be considered. The crucial property, that the subalgebra must check, is that it separates the points : it is said that a subset has C ( X ) so separates the points for any couple from points different X and there from X and any couple of real numbers has and B there exists a function p of has such as p ( X ) = has and p ( there ) = B . Formally the theorem is written:

if X is a space of compact Hausdorff having at least two elements, and if has is a subalgebra of the algebra of Banach C ( X ) which separates the points and contains a nonnull constant function, then has is dense in C ( X ).

That generalizes the theorem of Weierstrass since the polynomials on form a subalgebra of C which separates the points.

Let us notice that the preceding theorem remains also true if we replace the assertion that has separates the points with the slightly weaker assertion than for any couple from points distinct X and there from X , it exists a function p in has such as p ( X ) that is to say distinct from p ( there ).

Applications

The theorem of Stone-Weierstrass can be used to show the two following proposals:

• if F is continuous function with actual values definite on paving stone X and if ε is real strictly positive, then there exists a polynomial function p with two variables such as for all X in and there in, | F ( X , there ) - p ( X , there )| < ε.

• if X and Y is two compact spaces of Hausdorff and if $f:\; X\; \backslash \; times\; Y\; \backslash \; rightarrow\; \backslash \; mathbb\; R$ is a continuous function, then for all ε>0 it exists N >0 and of the continuous functions f₁ , f₂ ,…, f_n on X and of the continuous functions g₁ , g₂ ,…, g_n on Y such as || F - ∑ f_ig_i || < ε

The theorem of Stone-Weierstrass, version lattice

That is to say X a space of compact Hausdorff. A subset L of C ( X ) is called a lattice C ( X ) so for two unspecified elements F , G of L , the functions max ( F , G ) and min ( F , G ) also belongs to L . The version lattice of the theorem of Stone-Weierstrass affirms that:

if X is a space of compact Hausdorff with at least two points and if L is a lattice of C ( X ) which separates the points, then L is dense in C ( X ).

See too

• Theorem of Chudnovsky

• Approximation of Bernstein