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Question: exercise 95 use the stoneweierstrass theorem to prove that polynomials...

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Exercise 9.5. Use the Stone-Weierstrass Theorem to prove that polynomials are universal appproximators. Stone-Weierstrass Theorem (Rudin [1976]). Let Z be a set of real continuous functions on a compact set U. If (i) Z is an algebra, that is, the set Z is closed under addition, multiplication, and scalar multiplication; ii) Z separates points on U, that is, for every z, y E U, zメy, there exists f E Z such that f(x) f(y); and (ii) Z vanishes at no point of U, that is, for each z E U there exists f e Z such that f(x) 0; then for any real continuous function g(x) on U and arbitrary > 0, there exists f E Z such that suprev lf(x) - g(x) < e.

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