Question: Is it true for every function that the characteristic function is equal to the conjugate of its Fourier transform, or there are any exceptions? How

Is it true for every function that the characteristic function is equal to the conjugate of its Fourier transform, or there are any exceptions? How to prove that?

I mean, every time I try to get the density function of a random variable from the given characteristic function, I just replace every "i" in characteristic function with "-i", and then resort to table of Fourier transform pairs to find the answer. Is it feasible all the time? And what about the vice versa (a density function to a characteristic function)?

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