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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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    Yes. The$r$e 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.2012-10-09

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