nLab tempered distribution

Idea

A tempered (or Schwartz) distribution is a distribution $u\in\mathcal{D}'(\mathbb{R}^n)$ that does not “grow too fast” – at most polynomial (or moderate/tempered) growth – at infinity (in all directions); in particular it is only defined on $\mathbb{R}^n$, not on any open subset. Formally, a tempered distribution is a continuous linear functional on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ of smooth functions with rapidly decreasing derivatives. The space of tempered distributions (with its natural topology) is denoted $\mathcal{S}'(\mathbb{R}^n)$. The main property is that its Fourier transform $\mathcal{F}u$ is well-defined, and is itself a tempered distribution; and that it naturally extends the standard Fourier transform $\mathcal{F}: L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$. This makes tempered distributions the natural setting for solving (linear) partial differential equations.

Definition

Definition

(tempered distributions)

For $n \in \mathbb{N}$, a tempered distribution on the Euclidean space $\mathbb{R}^n$ is a continuous linear functional on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ (this def.) of smooth functions with rapidly decreasing derivatives.

The topological vector space of tempered distributions is denoted $\mathcal{S}'(\mathbb{R}^n)$.

(e.g. Hörmander 90, def. 7.1.7)

Examples

Example

(compactly supported distributions are tempered distributions)

Every compactly supported distribution is a tempered distribution (def. ), yielding an inclusion

$$\mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n) \,.$$

Example

(square integrable functions induced tempered distributions)

Let $f \in L^p(\mathbb{R}^n)$ be a function in the $p$th Lebesgue space, e.g. for $p = 2$ this means that $f$ is a square integrable function. Then the operation of integration against the measure $f dvol$

$$g \mapsto \int_{x \in \mathbb{R}^n} g(x) f(x) dvol(x)$$

is a tempered distribution (def. ).

(e.g. Hörmander 90, below lemma 7.1.8)

Related concepts

• Fourier transform of distributions

References

• Lars Hörmander, section 7.1 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

• Richard Melrose, chapter 1 of Introduction to microlocal analysis, 2003 (pdf)

category: functional analysis