1.15 Summability Methods1.17 Integral and Series Representations of the Dirac Delta

§1.16 Distributions

Contents

§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}),

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).

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.

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

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.

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(iv) Heaviside Function

1.16.13 \mathop{H\/}\nolimits\!\left(x\right)=\begin{cases}1,&x>0,\\
0,&x\leq 0.\end{cases}
1.16.14 \mathop{H\/}\nolimits\!\left(x-x_{0}\right)=\begin{cases}1,&x>x_{0},\\
0,&x\leq x_{0}.\end{cases}
1.16.15 D\!\mathop{H\/}\nolimits=\delta,
1.16.16 D\!\mathop{H\/}\nolimits\!\left(x-x_{0}\right)=\delta _{{x_{0}}}.

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}-).

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.

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}

For \alpha>0,

1.16.20 Dx^{\alpha}_{{+}}=\alpha x_{{+}}^{{\alpha-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}},

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}

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}(\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}}

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}(\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}.

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}.

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 _{{\Real^{n}}}\phi(\mathbf{t})e^{{i\mathbf{x}\cdot\mathbf{t}}}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}}}}=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}}},
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}}},

and

1.16.32 P(D)=\sum c_{{\boldsymbol{{\alpha}}}}D_{{\boldsymbol{{\alpha}}}}.

Then

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

and

1.16.34 \frac{1}{(2\pi)^{{n/2}}}\int _{{\Real^{n}}}P(\mathbf{t})\phi(\mathbf{t})e^{{i\mathbf{x}\cdot\mathbf{t}}}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

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