Stone–Weierstrass theorem
- Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval ["a","b"] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are the simplest functions, and 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.
Marshall H. Stone considerably generalized the theorem ::Let "A" be a closed subalgebra of the Banach space "C"("X",C) of continuous complex-valued functions on a compact Hausdorff space "X". Suppose that "f" ∈ "C"("X", C) has the following property::* "f"|"S" ∈ "A""S" for every maximal set "S" ⊂ "X" such that "A""S" contains no non-constant real functions.:Then "f" ∈ "A".
harvtxt|Glicksberg|1962 gives a short proof of Bishop's theorem using the Krein–Milman theorem in an essential way, as well as the Hahn–Banach theorem. See also harvtxt|Rudin|1973|loc=§5.7.
See also
* Runge's phenomenon shows that finding a polynomial P such that f(x)=P(x) for some finely spaced x=x_n is a bad way to attempt to find a polynomial approximating f uniformly. However, as is shown in Rudin's Principles of Mathematical Analysis, one can easily find a polynomial P uniformly approximating f by convolving f with a polynomial kernel.
References
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Historical works
The historical publication of Weierstrass (in German language) is freely available from the digital online archive of the " [http://bibliothek.bbaw.de/ Berlin Brandenburgische Akademie der Wissenschaften] ":
* K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. "Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin", 1885 (II). : [http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1885-2&seite:int=109 Erste Mitteilung] (part 1) pp. 633–639, [http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1885-2&seite:int=272 Zweite Mitteilung] (part 2) pp. 789–805.
Important historical works of Stone include:
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*; 21 (5), 237–254.
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