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Stone–Weierstrass theorem. 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.
Der Approximationssatz von Stone-Weierstraß (nach Marshall Harvey Stone und Karl Weierstraß) ist ein Satz aus der Analysis, der sagt, unter welchen Voraussetzungen man jede stetige Funktion durch einfachere Funktionen beliebig gut approximieren kann. Inhaltsverzeichnis. 1 Satz. 2 Folgerungen. 3 Historie. 4 Verallgemeinerungen. 5 Literatur.
The Stone-Weierstrass theorem is an approximation theorem for continuous functions on closed intervals. It says that every continuous function on the interval \([a,b]\) can be approximated as accurately desired by a polynomial function. Polynomials are far easier to work with than continuous functions and allow mathematicians and computers to ...
Theorem 1 (Approximationssatz fur reellwertige Funtionen). Sei X ein kompakter metrischer Raum und R C(X; R) ein Funktionenring, so dass. alle konstanten Funktionen f(x) = c mit c 2 R enthalt, und. Punkte trennt. Dann ist R. C(X; R) dicht, d.h. zu jeder stetigen Funktion f : X ! R existiert eine Folge ffngn 1 mit fn 2 R und fn f.
One useful theorem in analysis is the Stone-Weierstrass Theorem, which states that any continuous complex function over a compact interval can be approximated to an arbitrary degree of accuracy with a sequence of polynomials.
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The StoneWeierstrass Theorem 2 More examples would show that the behaviour of Pn at the end points only becomes wilder as n increases. You can see what the problem is, at least roughly—Lagrange’s polynomial doesn’t behave locally—i.e. singular behaviour of the function at a point can affect behaviour of the approximation far away from ...
Vor 6 Tagen · This theorem is a generalization of the Weierstrass approximation theorem. If X is any compact space, let A be a subalgebra of the algebra C(X) over the reals R with binary operations + and ×. Then, if A contains the constant functions and separates the points of X (i.e., for any two distinct points x and y of X, there is some ...
