Further applications of Fourier analysis are examined. Shannon’s sampling theorem is proven and discussed. The spectral properties of sampling applications are considered and a basic digital transmission system is shown. The transmission of signals with a linear multi-carrier system, such as WLAN or mobile data transmission, is treated as a current everyday application. The method is orthogonal frequency division multiplexing (OFDM), which uses the FFT and linear filters. Further sections examine the Heisenberg uncertainty principle and its consequences for the time-bandwidth product of signals. Closely related to this is the windowed Fourier transform (STFT) as a tool for time-frequency analysis. Inversion formulas for the STFT with continuous and discrete parameters are proven. The use of time windows in the DFT to reduce alias effects is discussed. In further sections, initial value problems for the homogeneous and inhomogeneous wave and heat equations in two and three dimensions are solved. The Fourier transform of distributions is used to solve these equations. The Huygens’ principle for waves is explained. For the heat equation, an inhomogeneous boundary value 3D problem is solved approximately as a further application of the FEM method and the solution is displayed graphically.