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.
(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)
(compactly supported distributions are tempered distributions)
Every compactly supported distribution is a tempered distribution (def. ), yielding an inclusion
(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 $f dvol$
is a tempered distribution (def. ).
(e.g. Hörmander 90, below lemma 7.1.8)
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.