Fourier inversion theorem

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 → C {\displaystyle f:\mathbb {R} \to \mathbb {C} } satisfying certain conditions, and we use the convention for the Fourier transform that ( F f ) ( ξ ) := ∫ R e − 2 π i y ⋅ ξ f ( y ) d y , {\displaystyle ({\mathcal {F}}f)(\xi ):=\int _{\mathbb {R} }e^{-2\pi iy\cdot \xi }\,f(y)\,dy,} then f ( x ) = ∫ R e 2 π i x ⋅ ξ ( F f ) ( ξ ) d ξ . {\displaystyle 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 ) = ∬ R 2 e 2 π i ( x − y ) ⋅ ξ f ( y ) d y d ξ . {\displaystyle 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 {\displaystyle R} is the flip operator i.e. ⁠ ( R f ) ( x ) := f ( − x ) {\displaystyle (Rf)(x):=f(-x)} ⁠, then F − 1 = F R = R F . {\displaystyle {\mathcal {F}}^{-1}={\mathcal {F}}R=R{\mathcal {F}}.} The theorem holds if both f {\displaystyle f} and its Fourier transform are absolutely integrable (in the Lebesgue sense) and f {\displaystyle f} is continuous at the point ⁠ x {\displaystyle x} ⁠. However, even under more general conditions versions...