tempered distributions

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1: 1.16 Distributions

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§1.16(v) Tempered Distributions

… ► … ►For a detailed discussion of tempered distributions see Lighthill (1958). … ►

§1.16(vii) Fourier Transforms of Tempered Distributions

… ►The Fourier transform $ℱ\left(u\right)$ 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 $𝒯$ of rapidly decreasing functions on $ℝ$. … ►

2.6.11 $$⟨f,\varphi ⟩={\int }_0^{\mathrm{\infty }}f(t)\varphi (t)dt,$$ $\varphi \in 𝒯$.

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Symbols:

$dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral, $f(t)$: locally integrable function and $𝒯$: space of decreasing functions

Referenced by:

§2.6(ii)

Permalink:

http://dlmf.nist.gov/2.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

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3: 1.14 Integral Transforms

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1.14.1 $$ℱ\left(f\right)\left(x\right)=ℱf\left(x\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t){\mathrm{e}}^{\mathrm{i}xt}dt.$$

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Defines:

$ℱ\left(f\right)\left(s\right)$: Fourier transform

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral

Referenced by:

§1.14(v), §1.14(i), item (i), §1.17(ii), §1.8(iv), item (i)

Permalink:

http://dlmf.nist.gov/1.14.E1

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

This equation defines $ℱ\left(f\right)\left(x\right)$ or $ℱf\left(x\right)$ as the Fourier transform of functions of a single variable. An analogous notation is defined for the Fourier transform of tempereddistributions in (1.16.29) and the Fourier transform of special distributions in (1.16.38).

See also:

Annotations for §1.14(i), §1.14 and Ch.1

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4: Errata

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Several changes have been made to

1. make consistent use of the Fourier transform notations $ℱ\left(f\right)$, $ℱ\left(\varphi \right)$ and $ℱ\left(u\right)$ where $f$ is a function of one real variable, $\varphi$ 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. introduce the partial differential operator $𝐃$ in (1.16.30);

3. clarify the definition (1.16.32) of the partial differential operator $P(𝐃)$; and

4. clarify the use of $P(𝐃)$ and $P(𝐱)$ in (1.16.33), (1.16.34), (1.16.36) and (1.16.37).

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