Fourier transform on Schwartz space - Imported Knowlpedia Content

Definition

For $f$ in the Schwartz space $\mathcal S(\mathbb R^n)$ , its Fourier transform is

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

The integral is a Lebesgue integral; rapid decay makes $f$ Lebesgue integrable. The map $\mathcal F:f\mapsto\widehat f$ is a continuous linear bijection $\mathcal S(\mathbb R^n)\to\mathcal S(\mathbb R^n)$ whose inverse is continuous. This topological automorphism, with the displayed normalization, is the Fourier transform on Schwartz space.

Why Schwartz space is preserved

Differentiation under the integral and integration by parts exchange derivatives with polynomial factors:

\partial_\xi^\alpha\widehat f(\xi) =\widehat{(-2\pi i x)^\alpha f}(\xi), \qquad (2\pi i\xi)^\beta\widehat f(\xi) =\widehat{\partial^\beta f}(\xi).

Every polynomially weighted derivative of $f$ remains integrable, so these identities bound every Schwartz seminorm of $\widehat f$ . They also show continuity of $\mathcal F$ in the Schwartz topology Hörmander, §7.1.

Structural role

Fourier transformation converts constant-coefficient differentiation into multiplication by a polynomial and translation into modulation. Its automorphism property therefore makes $\mathcal S(\mathbb R^n)$ a common invariant domain for both operations. Taking the transpose of this automorphism defines the Fourier transform of [[functional-analysis/tempered-distribution|tempered distributions]].

Conventions and scope

References

Lars Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, 2nd ed., Springer, 2003. DOI record. Relevant: §7.1 on the Fourier transform on Schwartz space.
Elias M. Stein and Rami Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003. DOI record. Relevant: Chapters 5–6 on the Fourier transform and Schwartz functions.