The Fourier Inversion Formula
Theorem (Fourier Inversion Formula)
Suppose \(f\) and \(\hat{f}\) are both continuous and absolutely improperly Riemann integrable on \({\mathbb R}^n\) (i.e., the improper integrals of \(|f|\) and \(|\hat{f}|\) exist.) Then for each \(x \in {\mathbb R}^n\),
\[ f(x) = \int_{-\infty}^{\infty} e^{2 \pi i x \cdot \xi} \hat{f}(\xi) d \xi. \]
Corollary
The Fourier inversion formula holds for all Schwartz functions \(f\).
The proof of the Fourier inversion formula can be understood in terms of three major steps:
Step 1 (Approximating the Inversion Integral)
We will show that for each \(x \in {\mathbb R}^n\),
\[{}\int e^{2 \pi i x \cdot \xi} \hat{f}(\xi) d\xi{}\]
\[{}= \lim_{\eta \rightarrow 0^+} \int e^{2 \pi i x \cdot \xi} e^{-\eta \pi ||\xi||^2} \hat{f}(\xi) d \xi.{}\]
Here \(||\xi||\) denotes the Euclidean norm of \(\xi\).
Proof