DefinitionIn mathematics, the Fourier transform of a periodic function is a generalization of the Fourier series to non-periodic functions and is used to represent a function as a sum of fundamental frequencies. This property implies harmonics in the frequency domain, meaning that the Fourier transform of a periodic function is nonzero only at integer multiples of the fundamental frequency. In such varied areas as signal processing, communication systems or mathematical physics (to name just a few) the analysis of such functions is crucial to the understanding of periodic signals, to the synthesis of waveforms, or to the solution of differential equations with periodic boundary conditions. For example, the Fourier transform of the periodically defined function f(x) with period T is defined as F(ω)=∑n=−∞∞cnδ(ω−T2πn), where cn are the Fourier coefficients of the function.
