Fourier inversion on Schwartz space - Imported Knowlpedia Content

Statement

Let $f\in\mathcal S(\mathbb R^n)$ , and define its Fourier transform by

\widehat f(\xi)=\int_{\mathbb R^n}e^{-2\pi i x\cdot\xi}f(x)\,dx .

The Fourier inversion theorem on Schwartz space states that

f(x)=\int_{\mathbb R^n}e^{2\pi i x\cdot\xi}\widehat f(\xi)\,d\xi

for every $x\in\mathbb R^n$ . Equivalently, $\widehat{\widehat f}(x)=f(-x)$ . Thus the inverse Fourier transform is $\mathcal F^{-1}g(x)=\widehat g(-x)$ , and inversion holds both pointwise and in the Schwartz topology.

Proof mechanism

One regularizes the inverse integral by a Gaussian factor, interchanges the resulting absolutely convergent integrals using Fubini's theorem, and evaluates the Gaussian Fourier transform. As the regularization parameter tends to zero, the Gaussian approximate identity converges to $f$ . Rapid decay controls the limiting passage and permits differentiation, yielding convergence in every Schwartz seminorm Stein–Shakarchi, Chapters 5–6.

Consequences

Inversion proves that the Fourier transform is injective and surjective on $\mathcal S(\mathbb R^n)$ . Together with the continuity estimates for weighted derivatives, it makes the transform a topological automorphism. The reflection formula also determines the fourth-power identity $\mathcal F^4=\mathrm{id}$ under this normalization.

Conventions and scope

References

Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003. DOI record. Relevant: Chapters 5–6, Fourier inversion for Schwartz functions.
Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Wiley, 1999. Publisher record. Relevant: Chapter 8 on Fourier analysis.