Fourier transform of tempered distributions - Imported Knowlpedia Content

Definition

Let $u$ be a tempered distribution on $\mathbb R^n$ . Its Fourier transform $\widehat u$ is the tempered distribution defined by

\langle\widehat u,\varphi\rangle =\langle u,\widehat\varphi\rangle \qquad \text{for every }\varphi\in\mathcal S(\mathbb R^n),

where $\varphi\mapsto\widehat\varphi$ is the Fourier transform on [[functional-analysis/schwartz-space|Schwartz space]] with kernel $e^{-2\pi i x\cdot\xi}$ . Because that transform is a continuous automorphism of $\mathcal S(\mathbb R^n)$ , this transpose operation is well-defined and is itself a linear automorphism of $\mathcal S'(\mathbb R^n)$ .

Compatibility with functions

If $u$ is induced by an integrable function $f$ , Fubini's theorem gives

\langle\widehat u,\varphi\rangle =\int_{\mathbb R^n}\widehat f(\xi)\varphi(\xi)\,d\xi.

Thus the dual definition agrees with the ordinary Fourier transform whenever both are available. It also assigns transforms to nonintegrable objects such as polynomials, plane waves, and derivatives of the Dirac distribution Hörmander, §7.1.

Differentiation and multiplication

With the displayed normalization,

\widehat{\partial^\alpha u}(\xi) =(2\pi i\xi)^\alpha\widehat u(\xi), \qquad \widehat{x^\alpha u}(\xi) =\left(-\frac{1}{2\pi i}\right)^{|\alpha|} \partial^\alpha\widehat u(\xi).

These identities hold distributionally. They turn constant-coefficient differential equations into algebraic multiplication equations and are a principal reason for working in $\mathcal S'$ .

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 Fourier transformation of distributions.
Robert S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, CRC Press, 1994. Publisher record. Relevant: chapters on tempered distributions and their Fourier transforms.