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tempered distribution

Contents

Idea

A tempered (or Schwartz) distribution is a distribution u𝒟( n)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 n\mathbb{R}^n, not on any open subset. Formally, a tempered distribution is a continuous linear functional on the Schwartz space 𝒮( n)\mathcal{S}(\mathbb{R}^n) of smooth functions with rapidly decreasing derivatives. The space of tempered distributions (with its natural topology) is denoted 𝒮( n)\mathcal{S}'(\mathbb{R}^n). The main property is that its Fourier transform u\mathcal{F}u is well-defined, and is itself a tempered distribution; and that it naturally extends the standard Fourier transform :L 2( n)L 2( n)\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 nn \in \mathbb{N}, a tempered distribution on the Euclidean space n\mathbb{R}^n is a continuous linear functional on the Schwartz space 𝒮( n)\mathcal{S}(\mathbb{R}^n) (this def.) of smooth functions with rapidly decreasing derivatives.

The topological vector space of tempered distributions is denoted 𝒮( n)\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

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

(square integrable functions induced tempered distributions)

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

g x ng(x)f(x)dvol(x) 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)

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)

Last revised on November 2, 2017 at 14:25:44. See the history of this page for a list of all contributions to it.