[Proof] Fourier Coefficients = zero => function zero

[nonequilibrium]

Hello!

How do I prove

If an integrable function f, 2pi-periodic, has all its Fourier coefficients equal to zero, then f is almost everywhere zero itself.

?

Thank you!

(it can be proven by using the convergence of the Fourier series in L_p-norm, but I want to use the above result to prove the convergence in L_2-norm, so I want to avoid that)

 

[marcusl]

Can you use Parseval's theorem? If the Fourier coefficients are zero, then the time-integrated power in the function is also zero, so the function itself must be zero.

 

[mathman]

Can you use Parseval's theorem? If the Fourier coefficients are zero, then the time-integrated power in the function is also zero, so the function itself must be zero.

Parseval's theorem is the L2 convergence theorem, which is what he is trying to prove.

 

[nonequilibrium]

Hm thank you both.

So is there another suggestion?

 

[mathman]

You can Google Parseval's theorem.

 

[Vanadium 50]

It's always hard to suggest what to do for a proof, because we don't know what the starting point is, but I would be inclined to use uniqueness of the Fourier transformation plus the fact that a Fourier transformation is (more or less) its own inverse.