In the diploma seminar, I studied the Fourier and discrete Fourier transforms. I first presented the Fourier transform and some of its properties, and then the motivation for the discretization and the introduction of the discrete Fourier transform and some of its properties. I also presented a algorithm for faster calculation of the discrete Fourier transform, called fast Fourier transform. The main goal of the diploma seminar is to prove the uncertainty principle for Fourier and discrete Fourier transforms. The formulations in the countinous and the discrete cases differ, but they give essentially equaivalent results. The essence of the Heisenberg principle can be stated as follows: we can never simultaneously know exactly where we are in time space and where we are in frequency space.
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