Editing Fourier inversion theorem (section)

==Relation to Fourier series==

{{hatnote|When considering the Fourier series of a function it is conventional to rescale it so that it acts on <math>[0, 2 \pi]</math> (or is <math>2 \pi</math>-periodic). In this section we instead use the somewhat unusual convention taking <math>f</math> to act on <math>[0, 1]</math>, since that matches the convention of the Fourier transform used here.}}

The Fourier inversion theorem is analogous to the [[convergence of Fourier series]]. In the Fourier transform case we have
:<math>f\colon\mathbb{R}^n\to\mathbb{C},\quad\hat f\colon\mathbb{R}^n\to\mathbb{C},</math>
:<math>\hat f(\xi):=\int_{\mathbb{R}^n} e^{-2\pi iy\cdot\xi} \, f(y)\,dy,</math>
:<math>f(x)=\int_{\mathbb{R}^n} e^{2\pi ix\cdot\xi} \, \hat f(\xi)\,d\xi.</math>
In the Fourier series case we instead have
:<math>f\colon[0,1]^n\to\mathbb{C},\quad\hat f\colon\mathbb{Z}^n\to\mathbb{C},</math>
:<math>\hat f(k):=\int_{[0,1]^n} e^{-2\pi iy\cdot k} \, f(y)\,dy,</math>
:<math>f(x)=\sum_{k\in\mathbb{Z}^n} e^{2\pi ix\cdot k} \, \hat f(k).</math>

In particular, in one dimension <math>k \in \mathbb Z</math> and the sum runs from <math>- \infty</math> to <math>\infty</math>.