# §1.16 Distributions

## §1.16(i) Test Functions

Let $\phi$ be a function defined on an open interval $I=(a,b)$, 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 $K$ and as $n\to\infty$ the sequence $\{\phi_{n}^{(k)}\}$ converges uniformly on $K$ 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 $\Lambda$ on $\mathcal{D}(I)$ is a linear functional if it takes complex values and

 1.16.1 $\Lambda(\alpha_{1}\phi_{1}+\alpha_{2}\phi_{2})=\alpha_{1}\Lambda(\phi_{1})+% \alpha_{2}\Lambda(\phi_{2}),$ Symbols: $\phi(x)$: function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E1 Encodings: TeX, pMML, png See also: Annotations for 1.16(i)

where $\alpha_{1}$ and $\alpha_{2}$ are real or complex constants. $\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}$ is called a distribution 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 $\lim_{n\to\infty}\Lambda(\phi_{n})=\Lambda(\phi).$ Symbols: $n$: nonnegative integer, $\phi(x)$: function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E2 Encodings: TeX, pMML, png See also: Annotations for 1.16(i)

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 function $f$ on $I$, which is absolutely integrable on every compact subset of $I$, such that

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

We denote a regular distribution by $\Lambda_{f}$, or simply $f$, where $f$ is the function giving rise to the distribution. (If a distribution is not regular, it is called singular.)

Define

 1.16.4 $\left\langle\Lambda_{1}+\Lambda_{2},\phi\right\rangle=\left\langle\Lambda_{1},% \phi\right\rangle+\left\langle\Lambda_{2},\phi\right\rangle,$ Symbols: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $\phi(x)$: function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E4 Encodings: TeX, pMML, png See also: Annotations for 1.16(i)
 1.16.5 $\left\langle c\Lambda,\phi\right\rangle=c\left\langle\Lambda,\phi\right\rangle% =\left\langle\Lambda,c\phi\right\rangle,$ Symbols: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $\phi(x)$: function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E5 Encodings: TeX, pMML, png See also: Annotations for 1.16(i)

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

 1.16.6 $\left\langle\alpha\Lambda,\phi\right\rangle=\left\langle\Lambda,\alpha\phi% \right\rangle.$ Symbols: $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$: inner-product of distribution, $\phi(x)$: function and $\Lambda$: mapping Permalink: http://dlmf.nist.gov/1.16.E6 Encodings: TeX, pMML, png See also: Annotations for 1.16(i)

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

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

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

## §1.16(ii) Derivatives of a Distribution

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

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

Similarly

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

For any locally integrable function $f$, its distributional derivative is $Df=\Lambda^{\prime}_{f}$.

## §1.16(iii) Dirac Delta Distribution

 1.16.10 $\displaystyle\left\langle\mathop{\delta\/}\nolimits,\phi\right\rangle$ $\displaystyle=\phi(0),$ $\phi\in\mathcal{D}(I)$, 1.16.11 $\displaystyle\left\langle\delta_{x_{0}},\phi\right\rangle$ $\displaystyle=\phi(x_{0}),$ $\phi\in\mathcal{D}(I)$, 1.16.12 $\displaystyle\left\langle{\delta_{x_{0}}^{(n)}},\phi\right\rangle$ $\displaystyle=(-1)^{n}\phi^{(n)}(x_{0}),$ $\phi\in\mathcal{D}(I)$.

The Dirac delta distribution is singular.

## §1.16(iv) Heaviside Function

 1.16.13 $\displaystyle\mathop{H\/}\nolimits\!\left(x\right)$ $\displaystyle=\begin{cases}1,&x>0,\\ 0,&x\leq 0.\end{cases}$ Defines: $\mathop{H\/}\nolimits\!\left(\NVar{x}\right)$: Heaviside function Referenced by: §1.14(iii) Permalink: http://dlmf.nist.gov/1.16.E13 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv) 1.16.14 $\displaystyle\mathop{H\/}\nolimits\!\left(x-x_{0}\right)$ $\displaystyle=\begin{cases}1,&x>x_{0},\\ 0,&x\leq x_{0}.\end{cases}$ Symbols: $\mathop{H\/}\nolimits\!\left(\NVar{x}\right)$: Heaviside function Permalink: http://dlmf.nist.gov/1.16.E14 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv) 1.16.15 $\displaystyle D\!\mathop{H\/}\nolimits$ $\displaystyle=\mathop{\delta\/}\nolimits,$ 1.16.16 $\displaystyle D\!\mathop{H\/}\nolimits\!\left(x-x_{0}\right)$ $\displaystyle=\delta_{x_{0}}.$ Symbols: $\mathop{H\/}\nolimits\!\left(\NVar{x}\right)$: Heaviside function and $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E16 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv)

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

 1.16.17 $\sigma_{n}=f^{(n)}(x_{0}+)-f^{(n)}(x_{0}-).$ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.16.E17 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv)

Then

 1.16.18 $D^{m}f=f^{(m)}+\sigma_{0}{\delta_{x_{0}}^{(m-1)}}+\sigma_{1}{\delta_{x_{0}}^{(% m-2)}}+\dots+\sigma_{m-1}\delta_{x_{0}},$ $m=1,2,\dots$. Symbols: $m$: nonnegative integer and $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E18 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv)

For $\alpha>-1$,

 1.16.19 $x^{\alpha}_{+}=x^{\alpha}\mathop{H\/}\nolimits\!\left(x\right)=\begin{cases}x^% {\alpha},&x>0,\\ 0,&x\leq 0.\end{cases}$ Symbols: $\mathop{H\/}\nolimits\!\left(\NVar{x}\right)$: Heaviside function Permalink: http://dlmf.nist.gov/1.16.E19 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv)

For $\alpha>0$,

 1.16.20 $Dx^{\alpha}_{+}=\alpha x_{+}^{\alpha-1}.$ Symbols: $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E20 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv)

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

 1.16.21 $x^{\alpha}_{+}=\frac{1}{(\alpha+1)_{n}}D^{n}x_{+}^{\alpha+n},$ Symbols: $n$: nonnegative integer and $Df$: distributional derivative Permalink: http://dlmf.nist.gov/1.16.E21 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv)

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

 1.16.22 $\mathop{\ln\/}\nolimits_{+}x=\mathop{H\/}\nolimits\!\left(x\right)\mathop{\ln% \/}\nolimits x=\begin{cases}\mathop{\ln\/}\nolimits x,&x>0,\\ 0,&x\leq 0,\end{cases}$ Symbols: $\mathop{H\/}\nolimits\!\left(\NVar{x}\right)$: Heaviside function and $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function Permalink: http://dlmf.nist.gov/1.16.E22 Encodings: TeX, pMML, png See also: Annotations for 1.16(iv)

and define

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

## §1.16(v) Tempered Distributions

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 $\mathop{O\/}\nolimits\!\left(|x|^{-N}\right)$ as $|x|\to\infty$ for all $N$.

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 $|x^{N}\phi_{n}^{(k)}|\leq c_{k,N}$ Symbols: $k$: integer, $n$: nonnegative integer and $\phi(x)$: function Permalink: http://dlmf.nist.gov/1.16.E24 Encodings: TeX, pMML, png See also: Annotations for 1.16(v)

do not depend on $n$.

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

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

Let $\mathcal{D}({\mathbb{R}^{n}})=\mathcal{D}_{n}$ be the set of all infinitely differentiable functions in $n$ 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 $K$ of ${\mathbb{R}^{n}}$ and $\phi_{m}^{(k)}$ converges uniformly to $\phi^{(k)}$ in $K$ 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 $\left\langle\Lambda_{f},\phi\right\rangle=\int_{{\mathbb{R}^{n}}}f(x)\phi(x)% \mathrm{d}x,$ $\phi\in\mathcal{D}_{n}$.

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

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

where $k$ 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 $n$ variables that satisfy

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

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 Distributions

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

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

where $\mathbf{x}=(x_{1},x_{2},\dots,x_{n})$ and $\mathbf{x}\cdot\mathbf{t}=x_{1}t_{1}+\dots+x_{n}t_{n}$. $F(\mathbf{x})$ is also in $\mathcal{T}_{n}$. For a multi-index $\boldsymbol{{\alpha}}=(\alpha_{1},\alpha_{2},\dots,\alpha_{n})$, set $\left|\alpha\right|=\alpha_{1}+\alpha_{2}+\dots+\alpha_{n}$ and

 1.16.30 $D_{\boldsymbol{{\alpha}}}={\mathrm{i}^{-|\boldsymbol{{\alpha}}|}}D^{% \boldsymbol{{\alpha}}}=\left(\frac{1}{\mathrm{i}}\frac{\partial}{\partial x_{1% }}\right)^{\alpha_{1}}\cdots\left(\frac{1}{\mathrm{i}}\frac{\partial}{\partial x% _{n}}\right)^{\alpha_{n}},$ Defines: $D_{\boldsymbol{{\alpha}}}$: differential operator (locally) Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative of $f$ with respect to $x$, $\partial\NVar{x}$: partial differential of $x$ and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.16.E30 Encodings: TeX, pMML, png See also: Annotations for 1.16(vii)
 1.16.31 $P(\mathbf{x})=P=\sum c_{\boldsymbol{{\alpha}}}\mathbf{x}^{\boldsymbol{{\alpha}% }}=\sum c_{\boldsymbol{{\alpha}}}x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}},$ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/1.16.E31 Encodings: TeX, pMML, png See also: Annotations for 1.16(vii)

and

 1.16.32 $P(D)=\sum c_{\boldsymbol{{\alpha}}}D_{\boldsymbol{{\alpha}}}.$ Symbols: $D_{\boldsymbol{{\alpha}}}$: differential operator Permalink: http://dlmf.nist.gov/1.16.E32 Encodings: TeX, pMML, png See also: Annotations for 1.16(vii)

Then

 1.16.33 $\frac{1}{(2\pi)^{n/2}}\int_{{\mathbb{R}^{n}}}(P(D)\phi)(\mathbf{t})e^{i\mathbf% {x}\cdot\mathbf{t}}\mathrm{d}\mathbf{t}=P(-\mathbf{x})F(\mathbf{x}),$

and

 1.16.34 $\frac{1}{(2\pi)^{n/2}}\int_{{\mathbb{R}^{n}}}P(\mathbf{t})\phi(\mathbf{t})e^{% \mathrm{i}\mathbf{x}\cdot\mathbf{t}}\mathrm{d}\mathbf{t}=P(D)F(\mathbf{x}).$

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

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

where $F$ is given by (1.16.29). The Fourier transform $\mathcal{F}(u)$ of a tempered distribution is again a tempered distribution, and

 1.16.36 $\displaystyle\mathcal{F}(P(D)u)$ $\displaystyle=P(-\mathbf{x})\mathcal{F}(u),$ Symbols: $\mathcal{F}(u)$: Fourier transform of distribution $u$ and $D_{\boldsymbol{{\alpha}}}$: differential operator Referenced by: §1.16(vii) Permalink: http://dlmf.nist.gov/1.16.E36 Encodings: TeX, pMML, png See also: Annotations for 1.16(vii) 1.16.37 $\displaystyle\mathcal{F}(Pu)$ $\displaystyle=P(D)\mathcal{F}(u).$ Symbols: $\mathcal{F}(u)$: Fourier transform of distribution $u$ and $D_{\boldsymbol{{\alpha}}}$: differential operator Referenced by: §1.16(vii) Permalink: http://dlmf.nist.gov/1.16.E37 Encodings: TeX, pMML, png See also: Annotations for 1.16(vii)

In (1.16.36) and (1.16.37) the derivatives in $P(D)$ are understood to be in the sense of distributions.