Stone–Weierstrass theorem

The approximation theorem of Stone - Weierstrass ( by Marshall Harvey Stone and Karl Weierstrass ) is a set of calculus, says the conditions under which one can approximate any continuous function arbitrarily well by simpler functions.

Set

Each subalgebra P of the algebra A of continuous real or complex functions on a compact Hausdorff space M,

• The points which are separated:
• At no point disappears:
• And is closed under complex conjugation,

Located with respect to the topology of uniform convergence dense in A.

This means: Every continuous function on M can be arbitrarily well approximated under the specified conditions by functions from P.

Conclusions

• This theorem is a generalization of the approximation theorem of Weierstrass that one can approximate any continuous function uniformly on a compact interval by polynomials. This special case can be easily derived from the above general proposition, if one takes as a sub- algebra P is the set of polynomials (see also Bernstein polynomials ).
• Another important consequence of (often also referred to as approximation theorem of Weierstrass ) is that every continuous function on the compact interval [0,2 π ] with the same value at 0 and 2π uniformly by trigonometric polynomials (that is, polynomials in sin ( x) and cos ( x) and linear combinations of sin ( nx) and cos ( nx ) n ∈ ℕ ) can be approximated (see also the article on Fourier series ).
• By means of the Alexandroff compactification, the set also transfers to the space of functions (see below) on a locally compact Hausdorff space.

History

1885 Weierstrass published a proof of his sentence. Regardless of several mathematicians found further evidence, such as Runge (1885), Picard (1891 ), Volterra (1897 ), Lebesgue (1898), Mittag-Leffler (1900), Fejér (1900), Lerch (1903 ), Landau (1908 ), de La Vallée Poussin ( 1912) and Bernstein ( 1912).