Tempered distribution - Imported Knowlpedia Content

Definition

A tempered distribution on $\mathbb R^n$ is a continuous linear functional on the Schwartz space $\mathcal S(\mathbb R^n)$ . The space of all tempered distributions is the topological dual

\mathcal S'(\mathbb R^n) =\mathcal L_{\mathrm{cont}}(\mathcal S(\mathbb R^n),\mathbb C).

Equivalently, $u\in\mathcal S'(\mathbb R^n)$ when there are $C>0$ and an integer $N$ such that

|u(\varphi)| \leq C\sum_{|\alpha|,|\beta|\leq N} \sup_{x\in\mathbb R^n}|x^\alpha\partial^\beta\varphi(x)|

for every $\varphi\in\mathcal S(\mathbb R^n)$ . Restriction to compactly supported test functions makes every tempered distribution a distribution, but not every distribution is tempered.

Examples and nonexamples

Every polynomially bounded locally integrable function defines a tempered distribution by integration against Schwartz functions. Finite measures, polynomials, the Dirac delta, and derivatives of the Dirac delta are tempered. By contrast, a locally integrable function with sufficiently rapid exponential growth need not define a functional on all of $\mathcal S(\mathbb R^n)$ .

Stable operations

Tempered distributions are closed under distributional differentiation and under multiplication by polynomials. Translation and multiplication by smooth functions whose derivatives grow at most polynomially also act continuously. These stability properties explain why $\mathcal S'$ is a natural setting for constant-coefficient differential equations.

Fourier duality and topology

The Fourier transform is an automorphism of $\mathcal S(\mathbb R^n)$ , so it extends to $\mathcal S'(\mathbb R^n)$ by duality. This extension includes objects such as plane waves and delta distributions that are not integrable functions. The notation $\mathcal S'$ specifies the continuous dual as a vector space; when convergence of tempered distributions is discussed, one must additionally specify the weak-star or strong dual topology.

References