§1.16 Distributions

Keywords:: distributions
Referenced by:: §2.6(i)
Permalink:: http://dlmf.nist.gov/1.16

§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 Distributions

§1.16(i) Test Functions

Notes:: See Wong (1989, pp. 241–248).
Defines:: $\left\langle\Lambda,\phi\right\rangle$ : distribution
Keywords:: convergence, distributions, functions, linear functional, support, test functions
Referenced by:: §1.16(v)
Permalink:: http://dlmf.nist.gov/1.16.i

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

where $\alpha _{1}$ and $\alpha _{2}$ are real or complex constants. $\Lambda:\mathcal{D}(I)\rightarrow\Complex$ 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

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 $\left\langle\Lambda,\phi\right\rangle=\int _{I}f(x)\phi(x)dx.$

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\int$ : integral, $\phi(x)$ : function, $I$ : interval and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E3
Encodings:: TeX, pMML, png

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:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\phi(x)$ : function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E4
Encodings:: TeX, pMML, png

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:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\phi(x)$ : function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E5
Encodings:: TeX, pMML, png

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:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\phi(x)$ : function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E6
Encodings:: TeX, pMML, png

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

1.16.7 $\lim _{{n\to\infty}}\left\langle\Lambda _{n},\phi\right\rangle=\left\langle\Lambda,\phi\right\rangle$

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $n$ : nonnegative integer, $\phi(x)$ : function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E7
Encodings:: TeX, pMML, png

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

§1.16(ii) Derivatives of a Distribution

Notes:: See Wong (1989, pp. 249–251).
Keywords:: derivatives, distributional derivative, distributions
Referenced by:: §1.16(vi)
Permalink:: http://dlmf.nist.gov/1.16.ii

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

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

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\phi(x)$ : function, $I$ : interval, $\mathcal{D}(I)$ : test function space and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E8
Encodings:: TeX, pMML, png

Similarly

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

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $k$ : integer, $\phi(x)$ : function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E9
Encodings:: TeX, pMML, png

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

§1.16(iii) Dirac Delta Distribution

Notes:: See Wong (1989, pp. 243–244, 252).
Keywords:: Dirac delta, distributions
Referenced by:: §1.17(i), §2.6(ii)
Permalink:: http://dlmf.nist.gov/1.16.iii

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

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\phi(x)$ : function, $I$ : interval and $\mathcal{D}(I)$ : test function space
Permalink:: http://dlmf.nist.gov/1.16.E10
Encodings:: TeX, pMML, png

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

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\phi(x)$ : function, $I$ : interval and $\mathcal{D}(I)$ : test function space
Permalink:: http://dlmf.nist.gov/1.16.E11
Encodings:: TeX, pMML, png

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

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $n$ : nonnegative integer, $\phi(x)$ : function, $I$ : interval and $\mathcal{D}(I)$ : test function space
Permalink:: http://dlmf.nist.gov/1.16.E12
Encodings:: TeX, pMML, png

The Dirac delta distribution is singular.

§1.16(iv) Heaviside Function

Notes:: See Wong (1989, pp. 252–254).
Keywords:: Heaviside function, distributions
Referenced by:: §2.6(iii)
Permalink:: http://dlmf.nist.gov/1.16.iv

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

Defines:: $\mathop{H\/}\nolimits\!\left(x\right)$ : Heaviside function
Referenced by:: ¶ ‣ §1.14(iii)
Permalink:: http://dlmf.nist.gov/1.16.E13
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{H\/}\nolimits\!\left(x\right)$ : Heaviside function
Permalink:: http://dlmf.nist.gov/1.16.E14
Encodings:: TeX, pMML, png

1.16.15 $D\!\mathop{H\/}\nolimits=\delta,$

Symbols:: $\mathop{H\/}\nolimits\!\left(x\right)$ : Heaviside function and $Df$ : distributional derivative
Permalink:: http://dlmf.nist.gov/1.16.E15
Encodings:: TeX, pMML, png

1.16.16 $D\!\mathop{H\/}\nolimits\!\left(x-x_{0}\right)=\delta _{{x_{0}}}.$

Symbols:: $\mathop{H\/}\nolimits\!\left(x\right)$ : Heaviside function and $Df$ : distributional derivative
Permalink:: http://dlmf.nist.gov/1.16.E16
Encodings:: TeX, pMML, png

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

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

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:: $\mathop{H\/}\nolimits\!\left(x\right)$ : Heaviside function
Permalink:: http://dlmf.nist.gov/1.16.E19
Encodings:: TeX, pMML, png

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

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

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:: $\mathop{H\/}\nolimits\!\left(x\right)$ : Heaviside function and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
Permalink:: http://dlmf.nist.gov/1.16.E22
Encodings:: TeX, pMML, png

and define

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

Symbols:: $!$ : $n!$ : factorial, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : nonnegative integer and $Df$ : distributional derivative
Permalink:: http://dlmf.nist.gov/1.16.E23
Encodings:: TeX, pMML, png

§1.16(v) Tempered Distributions

Notes:: See Wong (1989, pp. 261–265).
Keywords:: convergence, distributions, tempered distributions
Permalink:: http://dlmf.nist.gov/1.16.v

The space $\mathcal{T}(\Real)$ 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

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 $\lim _{{n\to\infty}}\left\langle\Lambda _{n},\phi\right\rangle=\left\langle\Lambda,\phi\right\rangle,$

Symbols:: $\left\langle\Lambda,\phi\right\rangle$ : distribution, $n$ : nonnegative integer, $\phi(x)$ : function and $\Lambda$ : mapping
Permalink:: http://dlmf.nist.gov/1.16.E25
Encodings:: TeX, pMML, png

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

Notes:: See Wong (1989, pp. 265–274).
Keywords:: distributions
Permalink:: http://dlmf.nist.gov/1.16.vi

Let $\mathcal{D}(\Real^{n})=\mathcal{D}_{n}$ be the set of all infinitely differentiable functions in $n$ variables, $\phi(x_{1},x_{2},\dots,x_{n})$ , with compact support in $\Real^{n}$ . If $k=(k_{1},\dots,k_{n})$ is a multi-index and $x=(x_{1},\dots,x_{n})\in\Real^{n}$ , 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 $\Real^{n}$ 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 $\Real^{n}$ is a continuous linear functional on $\mathcal{D}_{n}$ .

The partial derivatives of distributions in $\Real^{n}$ 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 $\left\langle\Lambda _{f},\phi\right\rangle=\int _{{\Real^{n}}}f(x)\phi(x)dx,$ $\phi\in\mathcal{D}_{n}$ .

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\int$ : integral, $\Real$ : real line (excluding infinity), $n$ : nonnegative integer, $\phi(x)$ : function, $\Lambda$ : mapping and $\mathcal{D}_{n}$ : set of differentiable functions
Permalink:: http://dlmf.nist.gov/1.16.E26
Encodings:: TeX, pMML, png

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

1.16.27 $\left\langle D^{{k}}f,\phi\right\rangle=(-1)^{{\left|k\right|}}\int _{{\Real^{n}}}f(x)\phi^{{(k)}}(x)dx,$ $\phi\in\mathcal{D}_{n}$ ,

Symbols:: $dx$ : differential of $x$ , $\left\langle\Lambda,\phi\right\rangle$ : distribution, $\in$ : element of, $\int$ : integral, $\Real$ : real line (excluding infinity), $k$ : integer, $n$ : nonnegative integer, $\phi(x)$ : function, $Df$ : distributional derivative and $\mathcal{D}_{n}$ : set of differentiable functions
Permalink:: http://dlmf.nist.gov/1.16.E27
Encodings:: TeX, pMML, png

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

Symbols:: $\in$ : element of, $\Real$ : real line (excluding infinity), $k$ : integer, $m$ : nonnegative integer, $n$ : nonnegative integer and $\phi(x)$ : function
Permalink:: http://dlmf.nist.gov/1.16.E28
Encodings:: TeX, pMML, png

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

Notes:: See Wong (1989, pp. 274–279).
Keywords:: Fourier transform, distributions, tempered distributions
Permalink:: http://dlmf.nist.gov/1.16.vii

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

1.16.29 $F(\mathbf{x})=F=\frac{1}{(2\pi)^{{n/2}}}\int _{{\Real^{n}}}\phi(\mathbf{t})e^{{i\mathbf{x}\cdot\mathbf{t}}}d\mathbf{t},$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\Real$ : real line (excluding infinity), $n$ : nonnegative integer, $F(x)$ : Fourier transform of $f(t)$ and $\phi(x)$ : function
Referenced by:: §1.16(vii)
Permalink:: http://dlmf.nist.gov/1.16.E29
Encodings:: TeX, pMML, png

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 $D_{{\boldsymbol{{\alpha}}}}=i^{{-|\boldsymbol{{\alpha}}|}}D^{{\boldsymbol{{\alpha}}}}=\left(\frac{1}{i}\frac{\partial}{\partial x_{1}}\right)^{{\alpha _{1}}}\cdots\left(\frac{1}{i}\frac{\partial}{\partial x_{n}}\right)^{{\alpha _{n}}},$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $n$ : nonnegative integer and $D_{{\boldsymbol{{\alpha}}}}$ : differential operator
Permalink:: http://dlmf.nist.gov/1.16.E30
Encodings:: TeX, pMML, png

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