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Uniform Limit of a Sequence of Functions

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Prove that any continuous function g :[0,pi}->Reals is a uniform limit of a sequence of functions g_n(x)= b_0 +b_1 cos(x) + ...+b_n cos(nx).

I think Stone's theorem is needed here. Here is the version of Stone's theorem for the Reals.

Let f:[a,b]-> Real be continuous. Let A be an algebra of functions defined over [a,b] such that A separates points on [a,b] and vanishes at no points on [a,b].

Then there exists a sequence f_n in A such that f_n converges uniformly to f on [a,b] as n tends to infinity.

So for this question I need to show that
A is an algebra. This would mean that for f_n,g_n in A, and a scalar c in the Reals
f_n+g_n is in A, (f_n)(g_n) is in A and of course c(f_n) is in A.

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Solution Preview

You can take A to be the algebra of functions generated by {cos nx}_{n=0}^infinity.

An algebra is actually a vector space on the basis of generating functions. So, all linear combination
b_0(cos 0x) + b_1 cos x + b_2 cos 2x +... ...

Solution Summary

This solution contextualizes continuous functions.