= Fourier inversion theorem
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{title2=$\mathcal F^{-1}$}
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= Fourier inversion
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The <Fourier inversion theorem> makes decomposition into <frequencies> reversible: a complete <Fourier transform>, including its <wave phases>, determines the original <function> up to equality <almost everywhere>. With the angular-frequency convention
$$
\widehat f(\xi)=\int_{\mathbb R^n}e^{-ix\cdot\xi}f(x)\,dx,
$$
if $f$ and $\widehat f$ belong to the <L1 space>, then
$$
f(x)=\frac1{(2\pi)^n}\int_{\mathbb R^n}e^{ix\cdot\xi}\widehat f(\xi)d\xi
$$
<almost everywhere>. The right side is a continuous representative of $f$. If $f$ itself is continuous, equality holds everywhere, as explained by the <continuous version of Fourier inversion>. The integrability hypotheses ensure that both displayed <integrals> are ordinary absolutely convergent <integrals>. For arbitrary elements of the <L2 space>, the <Plancherel theorem> instead supplies inversion in the <L2 space> norm.
Each factor $e^{ix\cdot\xi}$ is a <Fourier mode>; the inverse <integral> adds them with the complex weights $\widehat f(\xi)/(2\pi)^n$. The normalization is part of the recipe. In one dimension, writing ordinary <frequency> as $\nu=\xi/(2\pi)$ gives the equivalent pair
$$
F(\nu)=\int_{\mathbb R}f(t)e^{-2\pi i\nu t}\,dt,\qquad f(t)=\int_{\mathbb R}F(\nu)e^{2\pi i\nu t}\,d\nu,
$$
where $F(\nu)=\widehat f(2\pi\nu)$. Reversing both exponential signs gives another equivalent convention, but the forward and inverse formulas must use opposite signs consistently.
For example, the <Fourier transform of a Gaussian> gives
$$
f(x)=e^{-ax^2},\qquad \widehat f(\xi)=\sqrt{\frac\pi a}\,e^{-\xi^2/(4a)},\qquad a>0.
$$
<Fourier inversion> says that a weighted continuum of <oscillations> reconstructs this smooth <Gaussian function>. Narrowing the original <Gaussian function> by increasing $a$ broadens its <frequency spectrum>. This is a concrete demonstration that a sharply localized shape needs a wider range of <frequencies>.
Inversion is what makes a <frequency> calculation useful: transform a <function>, solve the simpler problem there, then return to the original variables. A brief proof outline is to multiply $\widehat f(\xi)$ by $e^{-\epsilon|\xi|^2}$, $\epsilon>0$. The inverse <integral> becomes <convolution> with a normalized <Gaussian function>, an <approximate identity>. As $\epsilon$ tends to zero, <dominated convergence> gives the unregularized inverse <integral>, while the <approximate identity> recovers $f$ <almost everywhere>. Joseph Fourier's \i[Théorie analytique de la chaleur] (1822), Chapter IX, article 417, contains a cosine double-integral representation antecedent to this modern formula; the hypotheses above are the modern statement.https://fr.wikisource.org/wiki/Th%C3%A9orie_analytique_de_la_chaleur/Chapitre_9[Fourier 1822, Chapter IX, article 417: cosine integral representation]{ref}
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