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I know that if two continuous $2\pi$ -periodic functions $f,g$ have the same Fourier coefficients then $f=g$.

Is the assumption about $2\pi$ periodicity of functions essential?

Thanks

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I am not sure whether this is what you want. But basically Fourier series is only defined for periodic functions, or equivalently, those functions with compact support. For other functions you have to use fourier transform. –  Hui Yu Oct 9 '12 at 16:27
    
Let's assume that $f,g$ are defined only on $[0,2\pi]$ and continuous. Then we could define Fourier coefficients for $f$ and $g$ in the same way as for periodic functions. –  Richard Oct 9 '12 at 16:39
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Yes. There is actually no difference between this case and periodic case. Because if your functions are supported on this interval, then you can extend them periodically to the real line. On the other hand, if your functions are periodic then all information is contained in one period, which is the interval. –  Hui Yu Oct 9 '12 at 16:41

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