Intro to Analysis in-class_Part_60

Intro to Analysis in-class_Part_60 - 7.5 Approximation by...

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Unformatted text preview: 7.5 Approximation by polynomials 125 Proof. By Weierstrass’ Thm, there is a sequence { g n } of real polynomials converging uniformly to | x | . Then let f n ( x ) := g n ( x )- g n (0) . 7.5.2 The Stone-Weierstrass Theorem We will isolate the properties of the polynomials which makes such an approximation possible. Definition 7.5.13. A family of functions A defined on a set X is an algebra iff (i) f + g ∈ A , (ii) fg ∈ A , and cf ∈ A whenever f,g ∈ A and c is a constant. (If A is an algebra of complex functions, then c ∈ C .) Definition 7.5.14. An algebra A is uniformly closed (or closed in the topology of uniform convergence) iff f n unif----→ f,f n ∈ A = ⇒ f ∈ A . The uniform closure of A is the set of all functions which are limits of uniformly convergent sequences of elements of A . Example 7.5.1. The polynomials on [ a,b ] form an algebra of functions. Weierstrass’ Thm states that C [ a,b ] is the uniform closure of the polynomials on [ a,b ]....
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_60 - 7.5 Approximation by...

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