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November 22nd, 2009, 12:40 PM   #1
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help me or give ideal

I am using the fejer-cesaro to prove the weierstrass theorem. I need help of finding this, "simply convert the given function f : [a, b] ? C into a symmetric function g over an interval of size 2(b ? a), identify the end points of the new interval so that it is isomorphic to T". I tried but I couldn't get the correct interval.

"Weierstrassí Theorem says that given any continuous function over a ?nite inverval and an arbitrarily small envelope around it, we can ?nd a polynomial that ?ts inside that envelope in that interval. To see why this is implied by Fejerís Theorem, simply convert the given function f : [a, b] ? C into a symmetric function g over an interval of size 2(b ? a), identify the end points of the new interval so that it is isomorphic to T, and use Fejerís Theorem to conclude that ?n (g, .) is a trigonometric polynomial close to g (and hence f ). To see why Weierstrassí Theorem implies Fejerís Theorem, recall that cos rt can be expressed as a degree r polynomial in cos t. Use this to express the promised trigonometric polynomial P (t) as a linear combination of cos rt and sin rt with ?n ? r ? n."
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