1 Algebraic and Analytic Methods Topics of Discussion 1.15 Summability Methods 1.17 Integral and Series Representations of the Dirac Delta

§1.16 Distributions

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Referenced by:: §1.16(ix), §2.6(i), Erratum (V1.2.0) §1.16
Permalink:: http://dlmf.nist.gov/1.16
Modifications (effective with 1.0.15):: An entire new subsection, §1.16(viii), was added to supply the Fourier transform of special distributions.
See also:: Annotations for Ch.1

§1.16(i) Test Functions
§1.16(ii) Derivatives of a Distribution
§1.16(iii) Dirac Delta Distribution
§1.16(iv) Heaviside Function
§1.16(v) Tempered Distributions
§1.16(vi) Distributions of Several Variables
§1.16(vii) Fourier Transforms of Tempered Distributions
§1.16(viii) Fourier Transforms of Special Distributions
§1.16(ix) References for Section 1.16

§1.16(i) Test Functions

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Defines:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function
Keywords:: convergence, definition, distribution, distributions, functions, generalized function, linear functional, linear functionals, of a function, of compact support, regular, singular, support, support of, test function space, test functions
Notes:: See Wong (1989, pp. 241–248).
Referenced by:: §1.16(v)
Permalink:: http://dlmf.nist.gov/1.16.i
Modification (effective with 1.2.0):: Just below (1.16.1) “distribution” was replaced with “distribution, or generalized function”. The description of regular below (1.16.2) was changed.
Addition (effective with 1.2.0):: Equation (1.16.3_5) and a sentence immediately above it were added.
See also:: Annotations for §1.16 and Ch.1

Let $\phi$ be a function defined on an open interval , which can be infinite. The closure of the set of points where $\phi\not=0$ is called the support of $\phi$ . If the support of $\phi$ is a compact set (§1.9(vii)), then $\phi$ is called a function of compact support. A test function is an infinitely differentiable function of compact support.

A sequence $\{\phi_{n}\}$ of test functions converges to a test function $\phi$ if the support of every $\phi_{n}$ is contained in a fixed compact set and as $n\to\infty$ the sequence $\{\phi_{n}^{(k)}\}$ converges uniformly on to $\phi^{(k)}$ for $k=0,1,2,\dots$ .

The linear space of all test functions with the above definition of convergence is called a test function space. We denote it by $\mathcal{D}(I)$ .

A mapping $\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}$ is a linear functional if

1.16.1 $\Lambda(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2})=\alpha_{1}\Lambda(\phi_{1})+% \alpha_{2}\Lambda(\phi_{2}),$

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Symbols:: $\phi(x)$ : test function and $\Lambda$ : mapping
Referenced by:: §1.16(i)
Permalink:: http://dlmf.nist.gov/1.16.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(i), §1.16 and Ch.1

where $\alpha_{1}$ and $\alpha_{2}$ are real or complex constants. $\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}$ is called a distribution, or generalized function, if it is a continuous linear functional on $\mathcal{D}(I)$ , that is, it is a linear functional and for every $\phi_{n}\to\phi$ in $\mathcal{D}(I)$ ,

1.16.2 $\lim_{n\to\infty}\Lambda(\phi_{n})=\Lambda(\phi).$

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Symbols:: : nonnegative integer, $\phi(x)$ : test function and $\Lambda$ : mapping
Referenced by:: §1.16(i)
Permalink:: http://dlmf.nist.gov/1.16.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(i), §1.16 and Ch.1

From here on we write $\left\langle\Lambda,\phi\right\rangle$ for $\Lambda(\phi)$ . The space of all distributions will be denoted by $\mathcal{D}^{*}(I)$ . A distribution $\Lambda$ is called regular if there is a locally integrable function on (i.e., a function on which is absolutely Lebesgue integrable on every compact subset of ) such that

1.16.3 $\left\langle\Lambda,\phi\right\rangle=\int_{I}f(x)\phi(x)\,\mathrm{d}x.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\int$ : integral, $\phi(x)$ : test function, : interval and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(i), §1.16 and Ch.1

We denote a regular distribution by $\Lambda_{f}$ , or simply , where is the function giving rise to the distribution. (If a distribution is not regular, it is called singular.) More generally, for $\alpha\colon[a,b]\to[-\infty,\infty]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure $\mu_{\alpha}$ (see §1.4(v)) can be considered as a distribution:

1.16.3_5 $\left\langle\mu_{\alpha},\phi\right\rangle=\int_{I}\phi(x)\,\mathrm{d}\alpha(x).$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\int$ : integral, $\phi(x)$ : test function and : interval
Referenced by:: §1.16(i), §1.16(ii), §1.16(iv), Erratum (V1.2.0) §1.16
Permalink:: http://dlmf.nist.gov/1.16.E3_5
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.16(i), §1.16 and Ch.1

Define

1.16.4 $\left\langle\Lambda_{1}+\Lambda_{2},\phi\right\rangle=\left\langle\Lambda_{1},% \phi\right\rangle+\left\langle\Lambda_{2},\phi\right\rangle,$

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Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\phi(x)$ : test function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(i), §1.16 and Ch.1

1.16.5 $\left\langle c\Lambda,\phi\right\rangle=c\left\langle\Lambda,\phi\right\rangle% =\left\langle\Lambda,c\phi\right\rangle,$

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Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\phi(x)$ : test function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(i), §1.16 and Ch.1

where is a constant. More generally, if $\alpha(x)$ is an infinitely differentiable function, then

1.16.6 $\left\langle\alpha\Lambda,\phi\right\rangle=\left\langle\Lambda,\alpha\phi% \right\rangle.$

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Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\phi(x)$ : test function and $\Lambda$ : mapping
Referenced by:: §1.16(vii)
Permalink:: http://dlmf.nist.gov/1.16.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(i), §1.16 and Ch.1

We say that a sequence of distributions $\{\Lambda_{n}\}$ converges to a distribution $\Lambda$ in $\mathcal{D}^{*}$ if

1.16.7 $\lim_{n\to\infty}\left\langle\Lambda_{n},\phi\right\rangle=\left\langle\Lambda% ,\phi\right\rangle$

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Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, : nonnegative integer, $\phi(x)$ : test function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(i), §1.16 and Ch.1

for all $\phi\in\mathcal{D}(I)$ .

§1.16(ii) Derivatives of a Distribution

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Keywords:: derivatives, distributional, distributional derivative, distributions, of distribution
Notes:: See Wong (1989, pp. 249–251).
Referenced by:: §1.16(vi), §1.16(vii)
Permalink:: http://dlmf.nist.gov/1.16.ii
Modification (effective with 1.2.0):: The sentence at the end of this subsection was modified.
Addition (effective with 1.2.0):: Equation (1.16.9_5), the sentence immediately above it and a sentence immediately below it were added.
See also:: Annotations for §1.16 and Ch.1

The derivative $\Lambda^{\prime}$ of a distribution is defined by

1.16.8 $\left\langle\Lambda^{\prime},\phi\right\rangle=-\left\langle\Lambda,\phi^{% \prime}\right\rangle,$ $\phi\in\mathcal{D}(I)$ .

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Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\phi(x)$ : test function, : interval, $\mathcal{D}(I)$ : test function space and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(ii), §1.16 and Ch.1

Similarly

1.16.9 $\left\langle\Lambda^{(k)},\phi\right\rangle=(-1)^{k}\left\langle\Lambda,\phi^{% (k)}\right\rangle,$ $k=1,2,\dots$ .

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Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, : integer, $\phi(x)$ : test function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(ii), §1.16 and Ch.1

If is a locally integrable function then its distributional derivative is $Df=\Lambda^{\prime}_{f}$ . In the situation of (1.16.3_5) we have

1.16.9_5 $\mu_{\alpha}=D\alpha.$

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Symbols:: : distributional derivative
Referenced by:: §1.16(ii), §1.16(iv), Erratum (V1.2.0) §1.16
Permalink:: http://dlmf.nist.gov/1.16.E9_5
Encodings:: TeX, pMML, png
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §1.16(ii), §1.16 and Ch.1

If the measure $\mu_{\alpha}$ is absolutely continuous with density (see §1.4(v)) then $D\alpha=\Lambda_{w}$ .

§1.16(iii) Dirac Delta Distribution

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Defines:: $\delta_{x}$ : Dirac delta distribution
Keywords:: Dirac delta distribution, distributions
Notes:: See Wong (1989, pp. 243–244, 252).
Referenced by:: §1.17(i), §1.4(v), §2.6(ii)
Permalink:: http://dlmf.nist.gov/1.16.iii
Addition (effective with 1.2.0):: A sentence at the end of this subsection indicating to see §1.17(i) was added.
See also:: Annotations for §1.16 and Ch.1

1.16.10 $\left\langle\delta,\phi\right\rangle=\phi(0),$ $\phi\in\mathcal{D}(I)$ ,

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Symbols:: $\delta_{x}$ : Dirac delta distribution, $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\phi(x)$ : test function, : interval and $\mathcal{D}(I)$ : test function space
Permalink:: http://dlmf.nist.gov/1.16.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iii), §1.16 and Ch.1

1.16.11 $\left\langle\delta_{x_{0}},\phi\right\rangle=\phi(x_{0}),$ $\phi\in\mathcal{D}(I)$ ,

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Symbols:: $\delta_{x}$ : Dirac delta distribution, $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\phi(x)$ : test function, : interval and $\mathcal{D}(I)$ : test function space
Permalink:: http://dlmf.nist.gov/1.16.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iii), §1.16 and Ch.1

1.16.12 $\left\langle{\delta_{x_{0}}}^{(n)},\phi\right\rangle=(-1)^{n}\phi^{(n)}(x_{0}),$ $\phi\in\mathcal{D}(I)$ .

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Symbols:: $\delta_{x}$ : Dirac delta distribution, $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, : nonnegative integer, $\phi(x)$ : test function, : interval and $\mathcal{D}(I)$ : test function space
Permalink:: http://dlmf.nist.gov/1.16.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iii), §1.16 and Ch.1

The Dirac delta distribution is singular. See also §1.17(i).

§1.16(iv) Heaviside Function

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Keywords:: Heaviside function, derivative, distributions
Notes:: See Wong (1989, pp. 252–254).
Referenced by:: §1.4(v), §2.6(iii)
Permalink:: http://dlmf.nist.gov/1.16.iv
Addition (effective with 1.2.0):: A sentence immediately following (1.16.16) were added.
See also:: Annotations for §1.16 and Ch.1

1.16.13 $H\left(x\right)=\begin{cases}1,&x>0,\\ 0,&x\leq 0.\end{cases}$

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Defines:: $H\left(\NVar{x}\right)$ : Heaviside function
Referenced by:: §1.14(iii), §1.16(viii), §18.40(ii)
Permalink:: http://dlmf.nist.gov/1.16.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

1.16.14 $H\left(x-x_{0}\right)=\begin{cases}1,&x>x_{0},\\ 0,&x\leq x_{0}.\end{cases}$

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Symbols:: $H\left(\NVar{x}\right)$ : Heaviside function
Referenced by:: §1.4(v)
Permalink:: http://dlmf.nist.gov/1.16.E14
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

1.16.15 $D\!H=\delta,$

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Symbols:: $H\left(\NVar{x}\right)$ : Heaviside function, $\delta_{x}$ : Dirac delta distribution and : distributional derivative
Referenced by:: §1.16(viii)
Permalink:: http://dlmf.nist.gov/1.16.E15
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

1.16.16 $D\!H\left(x-x_{0}\right)=\delta_{x_{0}}.$

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Symbols:: $H\left(\NVar{x}\right)$ : Heaviside function, $\delta_{x}$ : Dirac delta distribution and : distributional derivative
Referenced by:: §1.16(iv), §1.16(iv)
Permalink:: http://dlmf.nist.gov/1.16.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

Since $\delta_{x_{0}}$ is the Lebesgue–Stieltjes measure $\mu_{\alpha}$ corresponding to $\alpha(x)=H\left(x-x_{0}\right)$ (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of $\alpha$ .

Suppose is infinitely differentiable except at $x_{0}$ , where left and right derivatives of all orders exist, and

1.16.17 $\sigma_{n}=f^{(n)}(x_{0}+)-f^{(n)}(x_{0}-).$

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Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/1.16.E17
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

Then

1.16.18 $D^{m}f=f^{(m)}+\sigma_{0}{\delta_{x_{0}}}^{(m-1)}+\sigma_{1}{\delta_{x_{0}}}^{% (m-2)}+\dots+\sigma_{m-1}\delta_{x_{0}},$ $m=1,2,\dots$ .

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Symbols:: $\delta_{x}$ : Dirac delta distribution, : nonnegative integer and : distributional derivative
Permalink:: http://dlmf.nist.gov/1.16.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

For $\alpha>-1$ ,

1.16.19 $x^{\alpha}_{+}=x^{\alpha}H\left(x\right)=\begin{cases}x^{\alpha},&x>0,\\ 0,&x\leq 0.\end{cases}$

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Symbols:: $H\left(\NVar{x}\right)$ : Heaviside function
Permalink:: http://dlmf.nist.gov/1.16.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

For $\alpha>0$ ,

1.16.20 $Dx^{\alpha}_{+}=\alpha x_{+}^{\alpha-1}.$

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Symbols:: : distributional derivative
Permalink:: http://dlmf.nist.gov/1.16.E20
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

For $\alpha<-1$ and $\alpha$ not an integer, define

1.16.21 $x^{\alpha}_{+}=\frac{1}{(\alpha+1)_{n}}D^{n}x_{+}^{\alpha+n},$

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Symbols:: : nonnegative integer and : distributional derivative
Permalink:: http://dlmf.nist.gov/1.16.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

where is an integer such that $\alpha+n>-1$ . Similarly, we write

1.16.22 $\ln_{+}x=H\left(x\right)\ln x=\begin{cases}\ln x,&x>0,\\ 0,&x\leq 0,\end{cases}$

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Symbols:: $H\left(\NVar{x}\right)$ : Heaviside function and $\ln\NVar{z}$ : principal branch of logarithm function
Permalink:: http://dlmf.nist.gov/1.16.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

and define

1.16.23 $(-1)^{n}n!x_{+}^{-1-n}=D^{(n+1)}\ln_{+}x,$ $n=0,1,2,\dots$ .

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Symbols:: : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, : nonnegative integer and : distributional derivative
Permalink:: http://dlmf.nist.gov/1.16.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(iv), §1.16 and Ch.1

§1.16(v) Tempered Distributions

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Keywords:: convergence, distributions, tempered, tempered distributions, test function space, test functions
Notes:: See Wong (1989, pp. 261–265).
Permalink:: http://dlmf.nist.gov/1.16.v
See also:: Annotations for §1.16 and Ch.1

The space $\mathcal{T}(\mathbb{R})$ of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are $O\left({\left|x\right|}^{-N}\right)$ as $\left|x\right|\to\infty$ for all .

A sequence $\{\phi_{n}\}$ of functions in $\mathcal{T}$ is said to converge to a function $\phi\in\mathcal{T}$ as $n\to\infty$ if the sequence $\{\phi_{n}^{(k)}\}$ converges uniformly to $\phi^{(k)}$ on every finite interval and if the constants $c_{k,N}$ in the inequalities

1.16.24 $\left|x^{N}\phi_{n}^{(k)}\right|\leq c_{k,N}$

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Symbols:: : integer, : nonnegative integer, $\phi(x)$ : test function and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.16.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(v), §1.16 and Ch.1

do not depend on .

A tempered distribution is a continuous linear functional $\Lambda$ on $\mathcal{T}$ . (See the definition of a distribution in §1.16(i).) The set of tempered distributions is denoted by $\mathcal{T}^{*}$ .

A sequence of tempered distributions $\Lambda_{n}$ converges to $\Lambda$ in $\mathcal{T}^{*}$ if

1.16.25 $\lim_{n\to\infty}\left\langle\Lambda_{n},\phi\right\rangle=\left\langle\Lambda% ,\phi\right\rangle,$

ⓘ

Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, : nonnegative integer, $\Lambda$ : mapping and $\phi(x)$ : test function
Permalink:: http://dlmf.nist.gov/1.16.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(v), §1.16 and Ch.1

for all $\phi\in\mathcal{T}$ .

The derivatives of tempered distributions are defined in the same way as derivatives of distributions.

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

§1.16(vi) Distributions of Several Variables

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Keywords:: distributions, several variables, tempered distributions
Notes:: See Wong (1989, pp. 265–274).
Permalink:: http://dlmf.nist.gov/1.16.vi
See also:: Annotations for §1.16 and Ch.1

Let $\mathcal{D}({\mathbb{R}}^{n})=\mathcal{D}_{n}$ be the set of all infinitely differentiable functions in variables, $\phi(x_{1},x_{2},\dots,x_{n})$ , with compact support in ${\mathbb{R}}^{n}$ . If $k=(k_{1},\dots,k_{n})$ is a multi-index and $x=(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}$ , then we write $x^{k}=x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}$ and $\phi^{(k)}(x)=\,{\partial}^{k}\phi/(\,\partial x_{1}^{k_{1}}\cdots\,\partial x% _{n}^{k_{n}})$ . A sequence $\{\phi_{m}\}$ of functions in $\mathcal{D}_{n}$ converges to a function $\phi\in\mathcal{D}_{n}$ if the supports of $\phi_{m}$ lie in a fixed compact subset of ${\mathbb{R}}^{n}$ and $\phi_{m}^{(k)}$ converges uniformly to $\phi^{(k)}$ in for every multi-index $k=(k_{1},k_{2},\dots,k_{n})$ . A distribution in ${\mathbb{R}}^{n}$ is a continuous linear functional on $\mathcal{D}_{n}$ .

The partial derivatives of distributions in ${\mathbb{R}}^{n}$ can be defined as in §1.16(ii). A locally integrable function $f(x)=f(x_{1},x_{2},\dots,x_{n})$ gives rise to a distribution $\Lambda_{f}$ defined by

1.16.26 $\left\langle\Lambda_{f},\phi\right\rangle=\int_{{\mathbb{R}}^{n}}f(x)\phi(x)\,% \mathrm{d}x,$ $\phi\in\mathcal{D}_{n}$ .

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\int$ : integral, $\mathbb{R}$ : real line, : nonnegative integer, $\mathcal{D}_{n}$ : space of test functions, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(vi), §1.16 and Ch.1

The distributional derivative $D^{k}f$ of is defined by

1.16.27 $\left\langle D^{k}f,\phi\right\rangle=(-1)^{\left|k\right|}\int_{{\mathbb{R}}^% {n}}f(x)\phi^{(k)}(x)\,\mathrm{d}x,$ $\phi\in\mathcal{D}_{n}$ ,

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\int$ : integral, $\mathbb{R}$ : real line, : integer, : nonnegative integer, $\mathcal{D}_{n}$ : space of test functions, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function, : distributional derivative and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.16.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(vi), §1.16 and Ch.1

where is a multi-index and $\left|k\right|=k_{1}+k_{2}+\dots+k_{n}$ .

For tempered distributions the space of test functions $\mathcal{T}_{n}$ is the set of all infinitely-differentiable functions $\phi$ of variables that satisfy

1.16.28 $\left|x^{m}\phi^{(k)}(x)\right|\leq c_{m,k},$ $x\in{\mathbb{R}}^{n}$ .

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Symbols:: $\in$ : element of, $\mathbb{R}$ : real line, : integer, : nonnegative integer, : nonnegative integer, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.16.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(vi), §1.16 and Ch.1

Here $m=(m_{1},m_{2},\dots,m_{n})$ and $k=(k_{1},k_{2},\dots,k_{n})$ are multi-indices, and $c_{m,k}$ are constants. Tempered distributions are continuous linear functionals on this space of test functions. The space of tempered distributions is denoted by $\mathcal{T}^{*}_{n}$ .

§1.16(vii) Fourier Transforms of Tempered Distributions

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Keywords:

Fourier transform, Fourier transforms, distributions, several variables, tempered distributions

Notes:

See Wong (1989, pp. 274–279).

Referenced by:

§1.14(i), Erratum (V1.0.15) for Subsection 1.16(vii)

Permalink:

http://dlmf.nist.gov/1.16.vii

Notational Change and Clarification (effective with 1.0.15):

Several changes have been made in this subsection to

(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 is a function of one real variable, $\phi$ is a test function of variables associated with tempered distributions, and is a tempered distribution (see (1.14.1), (1.16.29) and (1.16.35));
(ii)

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

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

See also:

Annotations for §1.16 and Ch.1

Suppose $\phi$ is a test function in $\mathcal{T}_{n}$ . Then its Fourier transform is

1.16.29 $\mathscr{F}(\phi)(\mathbf{x})=\mathscr{F}\phi(\mathbf{x})=\frac{1}{(2\pi)^{n/2% }}\int_{{\mathbb{R}}^{n}}\phi(\mathbf{t}){\mathrm{e}}^{\mathrm{i}\mathbf{x}% \cdot\mathbf{t}}\,\mathrm{d}\mathbf{t},$

ⓘ

Defines:: $\mathscr{F}(\phi)$ : Fourier transform of a test function (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{R}$ : real line, : nonnegative integer and $\phi(x_{1},x_{2},\dots,x_{n})$ : test function
Referenced by:: (1.14.1), (1.16.33), (1.16.34), (1.16.35), item (i), item (i)
Permalink:: http://dlmf.nist.gov/1.16.E29
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.15):: Previously this equation was written as
$F(\mathbf{x})=F=\frac{1}{(2\pi)^{n/2}}\int_{{\mathbb{R}}^{n}}\phi(\mathbf{t}){% \mathrm{e}}^{\mathrm{i}\mathbf{x}\cdot\mathbf{t}}\,\mathrm{d}\mathbf{t}.$
The notation used in §1.14(i) is extended to a test function of several variables.
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

where $\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$ and $\mathbf{x}\cdot\mathbf{t}=x_{1}t_{1}+\dots+x_{n}t_{n}$ . $\mathscr{F}\phi(\mathbf{x})$ is also in $\mathcal{T}_{n}$ .

Let

1.16.30 $\mathbf{D}=\left(\frac{1}{\mathrm{i}}\frac{\partial}{\partial x_{1}},\frac{1}{% \mathrm{i}}\frac{\partial}{\partial x_{2}},\ldots,\frac{1}{\mathrm{i}}\frac{% \partial}{\partial x_{n}}\right).$

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Defines:: $\mathbf{D}$ : vector differential operator (locally)
Symbols:: $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : nonnegative integer, : distributional derivative and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Referenced by:: (1.16.32), item (ii), item (ii)
Permalink:: http://dlmf.nist.gov/1.16.E30
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.15):: Previously this equation was completely different:
$D_{\boldsymbol{{\alpha}}}={\mathrm{i}}^{-\left|\boldsymbol{{\alpha}}\right|}D^% {\boldsymbol{{\alpha}}}=\left(\frac{1}{\mathrm{i}}\frac{\partial}{\partial x_{% 1}}\right)^{\alpha_{1}}\cdots\left(\frac{1}{\mathrm{i}}\frac{\partial}{% \partial x_{n}}\right)^{\alpha_{n}}.$
The change was made to clarify the meaning of (1.16.32).
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

For a multi-index $\boldsymbol{{\alpha}}=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$ , define

1.16.31 $P(\mathbf{x})=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\mathbf{x}^% {\boldsymbol{{\alpha}}}=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}x% _{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}},$

ⓘ

Symbols:: : nonnegative integer and : polynomial of several variables
Permalink:: http://dlmf.nist.gov/1.16.E31
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

and

1.16.32 $P(\mathbf{D})=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\mathbf{D}^% {\alpha}=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}\left(\frac{1}{% \mathrm{i}}\frac{\partial}{\partial x_{1}}\right)^{\alpha_{1}}\dots\left(\frac% {1}{\mathrm{i}}\frac{\partial}{\partial x_{n}}\right)^{\alpha_{n}}.$

ⓘ

Symbols:: $\mathrm{i}$ : imaginary unit, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , : nonnegative integer, $\mathbf{D}$ : vector differential operator, : polynomial of several variables and : distributional derivative
Referenced by:: (1.16.30), (1.16.33), (1.16.34), (1.16.36), (1.16.37), item (iii), item (iii)
Permalink:: http://dlmf.nist.gov/1.16.E32
Encodings:: TeX, pMML, png
Clarification (effective with 1.0.15):: Previously this equation was written
$P(D)=\sum_{\boldsymbol{{\alpha}}}c_{\boldsymbol{{\alpha}}}D_{\boldsymbol{{% \alpha}}}$
where $D_{\boldsymbol{{\alpha}}}$ was defined in (1.16.30). The change was made to clarify the meaning of this equation.
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

Here $\boldsymbol{{\alpha}}$ ranges over a finite set of multi-indices, $P(\mathbf{x})$ is a multivariate polynomial, and $P(\mathbf{D})$ is a partial differential operator. Then

1.16.33 $\mathscr{F}(P(\mathbf{D})\phi)(\mathbf{x})=P(-\mathbf{x})\mathscr{F}\phi(% \mathbf{x}),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{R}$ : real line, : nonnegative integer, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function, $\mathscr{F}(\phi)$ : Fourier transform of a test function, $\mathbf{D}$ : vector differential operator, : polynomial of several variables and : distributional derivative
Referenced by:: item (iv), §1.16(vii), item (iv)
Permalink:: http://dlmf.nist.gov/1.16.E33
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.15):: Previously this equation was written as
$\frac{1}{(2\pi)^{n/2}}\int_{{\mathbb{R}}^{n}}(P(D)\phi)(\mathbf{t}){\mathrm{e}% }^{\mathrm{i}\mathbf{x}\cdot\mathbf{t}}\,\mathrm{d}\mathbf{t}=P(-\mathbf{x})F(% \mathbf{x}).$
It has been rewritten using the notation of (1.16.29) and (1.16.32).
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

and

1.16.34 $\mathscr{F}(P\phi)(\mathbf{x})=P(\mathbf{D})\mathscr{F}\phi(\mathbf{x}).$

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{R}$ : real line, : nonnegative integer, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function, $\mathscr{F}(\phi)$ : Fourier transform of a test function, $\mathbf{D}$ : vector differential operator, : polynomial of several variables and : distributional derivative
Referenced by:: item (iv), §1.16(vii), item (iv)
Permalink:: http://dlmf.nist.gov/1.16.E34
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.15):: Previously this equation was written as
$\frac{1}{(2\pi)^{n/2}}\int_{{\mathbb{R}}^{n}}P(\mathbf{t})\phi(\mathbf{t}){% \mathrm{e}}^{\mathrm{i}\mathbf{x}\cdot\mathbf{t}}\,\mathrm{d}\mathbf{t}=P(D)F(% \mathbf{x}).$
It has been rewritten using the notation of (1.16.29) and (1.16.32).
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

If $u\in\mathcal{T}^{*}_{n}$ is a tempered distribution, then its Fourier transform $\mathscr{F}\left(u\right)$ is defined by

1.16.35 $\left\langle\mathscr{F}\left(u\right),\phi\right\rangle=\left\langle u,% \mathscr{F}(\phi)\right\rangle,$ $\phi\in\mathcal{T}_{n}$ .

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Defines:: $\mathscr{F}\left(\NVar{u}\right)$ : Fourier transform of a tempered distribution
Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, : nonnegative integer, $\mathcal{T}$ : space of test functions, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function and $\mathscr{F}(\phi)$ : Fourier transform of a test function
Referenced by:: (1.16.36), (1.16.37), item (i), §1.16(viii), item (i)
Permalink:: http://dlmf.nist.gov/1.16.E35
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.15):: Previously this equation was written as
$\left\langle\mathscr{F}\left(u\right),\phi\right\rangle=\left\langle u,F\right\rangle,$ $\phi\in\mathcal{T}_{n}$ ,
where was the local definition of the Fourier transform in (1.16.29).
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

The Fourier transform $\mathscr{F}\left(u\right)$ of a tempered distribution is again a tempered distribution, and

1.16.36 $\left\langle\mathscr{F}\left(P(\mathbf{D})u\right),\phi\right\rangle=\left% \langle P_{-}\mathscr{F}\left(u\right),\phi\right\rangle=\left\langle\mathscr{% F}\left(u\right),P_{-}\phi\right\rangle,$

ⓘ

Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\mathscr{F}\left(\NVar{u}\right)$ : Fourier transform of a tempered distribution, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function, $\mathbf{D}$ : vector differential operator, : polynomial of several variables and : distributional derivative
Referenced by:: item (iv), §1.16(vii), §1.16(viii), item (iv)
Permalink:: http://dlmf.nist.gov/1.16.E36
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.15):: Previously this equation was written as
$\mathcal{F}(P(D)u)=P(-\mathbf{x})\mathcal{F}(u).$
It has been rewritten using the notation of (1.16.32) and (1.16.35).
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

1.16.37 $\left\langle\mathscr{F}\left(Pu\right),\phi\right\rangle=\left\langle P(% \mathbf{D})\mathscr{F}\left(u\right),\phi\right\rangle,$

ⓘ

Symbols:: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\mathscr{F}\left(\NVar{u}\right)$ : Fourier transform of a tempered distribution, $\phi(x_{1},x_{2},\dots,x_{n})$ : test function, $\mathbf{D}$ : vector differential operator, : polynomial of several variables and : distributional derivative
Referenced by:: item (iv), §1.16(vii), item (iv)
Permalink:: http://dlmf.nist.gov/1.16.E37
Encodings:: TeX, pMML, png
Notational Change (effective with 1.0.15):: Previously this equation was written as
$\mathcal{F}(Pu)=P(D)\mathcal{F}(u).$
It has been rewritten using the notation of (1.16.32) and (1.16.35).
See also:: Annotations for §1.16(vii), §1.16 and Ch.1

in which $P_{-}(\mathbf{x})=P(-\mathbf{x})$ ; compare (1.16.33) and (1.16.34). In (1.16.36) and (1.16.37) the derivatives in $P(\mathbf{D})$ are understood to be in the sense of distributions, as defined in §1.16(ii) and we also use the convention (1.16.6).

§1.16(viii) Fourier Transforms of Special Distributions

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Keywords:: Dirac delta distribution, Fourier transform, Fourier transforms, Heaviside function, distributions, sign function, special distributions
Notes:: See Wong (1989, pp. 274–279).
Referenced by:: §1.16, Erratum (V1.0.15) for Subsection 1.16(viii), Erratum (V1.0.15) for Subsection 1.16(viii)
Permalink:: http://dlmf.nist.gov/1.16.viii
Addition (effective with 1.0.15):: This entire subsection is new.
Suggested 2017-03-07 by Roderick Wong
See also:: Annotations for §1.16 and Ch.1

We use the notation of the previous subsection and take and $u=\delta$ in (1.16.35). We obtain

1.16.38 $\left\langle\mathscr{F}\left(\delta\right),\phi\right\rangle=\left\langle% \delta,\mathscr{F}(\phi)\right\rangle=\left\langle\delta,\frac{1}{\sqrt{2\pi}}% \int^{\infty}_{-\infty}\phi(t){\mathrm{e}}^{\mathrm{i}xt}\,\mathrm{d}t\right% \rangle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\phi(t)\,\mathrm{d}t=\frac% {1}{\sqrt{2\pi}}\left\langle 1,\phi\right\rangle,$ $\phi\in\mathcal{T}$ .

As distributions, the last equation reads

1.16.39 $\mathscr{F}\left(\delta\right)=\frac{1}{\sqrt{2\pi}},$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\delta_{x}$ : Dirac delta distribution and $\mathscr{F}\left(\NVar{u}\right)$ : Fourier transform of a tempered distribution
Permalink:: http://dlmf.nist.gov/1.16.E39
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(viii), §1.16 and Ch.1

which is often written conventionally as

1.16.40 $\int^{\infty}_{-\infty}\delta\left(t\right){\mathrm{e}}^{\mathrm{i}xt}\,% \mathrm{d}t=1;$

ⓘ

Symbols:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $\int$ : integral
Permalink:: http://dlmf.nist.gov/1.16.E40
Encodings:: TeX, pMML, png
See also:: Annotations for §1.16(viii), §1.16 and Ch.1

§1.16(ix) References for Section 1.16

ⓘ

Referenced by:: Erratum (V1.2.0) §1.16
Permalink:: http://dlmf.nist.gov/1.16.ix
Addition (effective with 1.2.0):: This subsection was added.
See also:: Annotations for §1.16 and Ch.1

See Hildebrandt (1938) and Chihara (1978, Chapter II) for Stieltjes measures which are used in §18.39(iii); see also Shohat and Tamarkin (1970, Chapter II). Friedman (1990) gives an overview of generalized functions and their relation to distributions. See also Lighthill (1958), and Zemanian (1987).

1.15 Summability Methods 1.17 Integral and Series Representations of the Dirac Delta

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