Understanding the Convergence of Fourier Series for Periodic Functions

[Swimmingly!]

Hey. I'm looking for a proof of:
Theorem: If f \in C^1(\mathbb{T}), then the Fourier series converges to f uniformly (and hence also pointwise.)

I have looked around for it, googled, etc, but I only found proofs which used theorem they did not prove. (Or I misunderstood what they said.)
I'd really like to truly understand why they converge, be it uniformly or pointwise. If anyone could either link me to a proof or do it, it'd be great. Thanks.

 

[micromass]

Check Folland's "Fourier analysis and applications", Theorem 2.5

 

[brmath]

Sorry, I don't know what C^1(T) is. Are these complex functions? And what domain is T?

 

[Swimmingly!]

Check Folland's "Fourier analysis and applications", Theorem 2.5

Completely answered my question. Thanks a lot!

Sorry, I don't know what C^1(T) is. Are these complex functions? And what domain is T?

C^k is the set of functions such that: There exist continuous derivatives of 0th, 1st, 2nd... and kth order.
C^1(T) probably means that f is periodic of period 2π or something of the sorts. The number of senseful meanings is not that big.

 

[brmath]

Thanks Swimmingly. ThenC^1 would be real functions that have just one continuous derivative and you guess T means they are periodic. I would guess if they are not periodic one could construct examples where the Fourier series wouldn't converge at all. There is always the famous {x^2sin1/x} which has exactly one continuous derivative at 0.

 

[Office_Shredder]

Bold faced T usually refers to a torus - in this case I assume the one dimensional torus (which is the circle, equivalently we are discussing periodic functions on the real line)

 

[brmath]

Office_Shredder- Thanks for the clarification. He most likely meant periodic functions on the real line. I personally am more than willing to consider functions on the unit circle -- quite often it helps.

Also, for all, please substitute x^3sin1/x for x^2sin1/x as per my previous post. The first does have one continuous derivative at x = 0. The second has only a discontinuous derivative at x = 0.