Concept

Fourier inversion theorem

Summary
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f:\R \to \Complex satisfying certain conditions, and we use the convention for the Fourier transform that :(\mathcal{F}f)(\xi):=\int_{\mathbb{R}} e^{-2\pi iy\cdot\xi} , f(y),dy, then :f(x)=\int_{\mathbb{R}} e^{2\pi ix\cdot\xi} , (\mathcal{F}f)(\xi),d\xi. In other words, the theorem says that :f(x)=\iint_{\mathbb{R}^2} e^{2\pi i(x-y)\cdot\xi} , f(y),dy,d\xi. This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R is the flip operator i.e. (Rf)(x) := f(-x), then :\mathcal{F}^{-1}=\mathca
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