# tempered distributions

(0.001 seconds)

## 4 matching pages

##### 1: 1.16 Distributions
###### §1.16(v) TemperedDistributions
For a detailed discussion of tempered distributions see Lighthill (1958). …
###### §1.16(vii) Fourier Transforms of TemperedDistributions
The Fourier transform $\mathscr{F}\left(u\right)$ of a tempered distribution is again a tempered distribution, and …
##### 2: 2.6 Distributional Methods
To each function in this equation, we shall assign a tempered distribution (i. …, a continuous linear functional) on the space $\mathcal{T}$ of rapidly decreasing functions on $\mathbb{R}$. …
2.6.11 $\left\langle f,\phi\right\rangle=\int_{0}^{\infty}f(t)\phi(t)\mathrm{d}t,$ $\phi\in\mathcal{T}$.
##### 3: Errata
• Subsection 1.16(vii)

Several changes have been made to

1. (i)

make consistent use of the Fourier transform notations $\mathscr{F}\left(f\right)$, $\mathscr{F}\left(\phi\right)$ and $\mathscr{F}\left(u\right)$ where $f$ is a function of one real variable, $\phi$ is a test function of $n$ variables associated with tempered distributions, and $u$ is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));

2. (ii)

introduce the partial differential operator $\mathbf{D}$ in (1.16.30);

3. (iii)

clarify the definition (1.16.32) of the partial differential operator $P(\mathbf{D})$; and

4. (iv)

clarify the use of $P(\mathbf{D})$ and $P(\mathbf{x})$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

• ##### 4: 1.14 Integral Transforms
1.14.1 $\mathscr{F}\left(f\right)\left(x\right)=\mathscr{F}\mskip-3.0mu f\mskip 3.0mu % \left(x\right)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(t)e^{ixt}\mathrm{% d}t.$