Weierstrass approximation theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 28/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem


00:02:20 1 Weierstrass approximation theorem
00:02:56 1.1 Applications
00:03:59 2 Stone–Weierstrass theorem, real version
00:06:22 2.1 Locally compact version
00:07:30 2.2 Applications
00:07:43 3 Stone–Weierstrass theorem, complex version
00:08:54 4 Stone–Weierstrass theorem, quaternion version
00:09:25 5 Stone–Weierstrass theorem, C*-algebra version
00:11:19 6 Lattice versions
00:13:07 7 Bishop's theorem
00:15:51 8 Nachbin's theorem
00:16:58 9 See also
00:18:46 10 Notes
00:20:02 11 References
00:21:06 11.1 Historical works
00:21:29 12 External links



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SUMMARY
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In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial 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. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.
Marshall H. Stone considerably generalized the theorem (Stone 1937) and simplified the proof (Stone 1948).  His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions both regressive and progressive: 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.  The Stone–Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space.
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.
A different generalization of Weierstrass' original theorem is Mergelyan's theorem, which generalizes it to functions defined on certain subsets of the complex plane.