Stone-Weierstrass theorem

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Stone-Weierstrass theorem

[′stōn ′vī·ər‚sträs ‚thir·əm]
(mathematics)
If S is a collection of continuous real-valued functions on a compact space E, which contains the constant functions, and if for any pair of distinct points x and y in E there is a function ƒ in S such that ƒ(x) is not equal to ƒ(y), then for any continuous real-valued function g on E there is a sequence of functions, each of which can be expressed as a polynomial in the functions of S with real coefficients, that converges uniformly to g.
References in periodicals archive ?
Suppose we wish to approximate by polynomials on the whole real line, obtaining analogues of Weierstrass' Theorem.
Bernstein and Jackson had provided quantitative forms of Weierstrass' Theorem before the first World War, and it is natural to look for analogues in the weighted setting.
Weierstrass' theorem, weight, Sobolev spaces, weighted Sobolev spaces
If I is any compact interval, Weierstrass' Theorem says that C(I) is the largest set of functions which can be approximated by polynomials in the norm L'(I), if we identify, as usual, functions which are equal almost everywhere.
This is a consequence of Bernstein's pr[infinity]f of Weierstrass' Theorem (see e.
LUBINSKY, Weierstrass' theorem in the twentieth century: a selection, Quaestiones Mathematicae, 18 (1995), pp.
TOURIS, Weighted Weierstrass' theorem With first derivatives, preprint.
TOURIS, Weierstrass' theorem in weighted Sobolev spaces with k derivatives, Rocky Mt.
This is deduced from Bernstein's proof of Weierstrass' theorem, where the polynomials he builds approximate uniformly up to the k-th derivative any function in [C.