§1.16 Distributions

Let $\varphi$ be a function defined on an open interval $I=(a,b)$, which can be infinite. The closure of the set of points where $\varphi \ne 0$ is called the support of $\varphi$. If the support of $\varphi$ is a compact set (§1.9(vii)), then $\varphi$ is called a function of compact support. A test function is an infinitely differentiable function of compact support.

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

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

A mapping $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is a linear functional if

where ${\alpha }_1$ and ${\alpha }_2$ are real or complex constants. $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$,

From here on we write $⟨\mathrm{\Lambda },\varphi ⟩$ for $\mathrm{\Lambda }(\varphi )$. The space of all distributions will be denoted by ${𝒟}^{\ast }(I)$. A distribution $\mathrm{\Lambda }$ is called regular if there is a locally integrable function $f$ on $I$ (i.e., a function $f$ on $I$ which is absolutely Lebesgue integrable on every compact subset of $I$) such that

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

Define

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

We say that a sequence of distributions $\{{\mathrm{\Lambda }}_n\}$ converges to a distribution $\mathrm{\Lambda }$ in ${𝒟}^{\ast }$ if

for all $\varphi \in 𝒟(I)$.

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

Similarly

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

If the measure ${\mu }_{\alpha }$ is absolutely continuous with density $w$ (see §1.4(v)) then $𝐷\alpha ={\mathrm{\Lambda }}_w$.

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

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 $f(x)$ is infinitely differentiable except at $x_0$, where left and right derivatives of all orders exist, and

Then

For $\alpha >-1$,

For $\alpha >0$,

For and $\alpha$ not an integer, define

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

and define

The space $𝒯(ℝ)$ 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 $\left|x\right|\to \mathrm{\infty }$ for all $N$.

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

do not depend on $n$.

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

A sequence of tempered distributions ${\mathrm{\Lambda }}_n$ converges to $\mathrm{\Lambda }$ in ${𝒯}^{\ast }$ if

for all $\varphi \in 𝒯$.

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).

Let $𝒟({ℝ}^n)={𝒟}_n$ be the set of all infinitely differentiable functions in $n$ variables, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$, with compact support in ${ℝ}^n$. If $k=(k_1,\mathrm{\dots },k_n)$ is a multi-index and $x=(x_1,\mathrm{\dots },x_n)\in {ℝ}^n$, then we write $x^k=x_1^{k_1}\mathrm{\cdots }{x}_n^{k_n}$ and ${\varphi }^{(k)}(x)={\partial }^k\varphi /(\partial x_1^{k_1}\mathrm{\cdots }\partial x_n^{k_n})$. A sequence $\{{\varphi }_m\}$ of functions in ${𝒟}_n$ converges to a function $\varphi \in {𝒟}_n$ if the supports of ${\varphi }_m$ lie in a fixed compact subset $K$ of ${ℝ}^n$ and ${\varphi }_m^{(k)}$ converges uniformly to ${\varphi }^{(k)}$ in $K$ for every multi-index $k=(k_1,k_2,\mathrm{\dots },k_n)$. A distribution in ${ℝ}^n$ is a continuous linear functional on ${𝒟}_n$.

The partial derivatives of distributions in ${ℝ}^n$ can be defined as in §1.16(ii). A locally integrable function $f(x)=f(x_1,x_2,\mathrm{\dots },x_n)$ gives rise to a distribution ${\mathrm{\Lambda }}_f$ defined by

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

where $k$ is a multi-index and $\left|k\right|=k_1+k_2+\mathrm{\cdots }+k_n$.

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

Here $m=(m_1,m_2,\mathrm{\dots },m_n)$ and $k=(k_1,k_2,\mathrm{\dots },k_n)$ 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 ${𝒯}_n^{\ast }$.

Suppose $\varphi$ is a test function in ${𝒯}_n$. Then its Fourier transform is

where $𝐱=(x_1,x_2,\mathrm{\dots },x_n)$ and $𝐱\cdot 𝐭=x_1{t}_1+\mathrm{\cdots }+x_n{t}_n$. $ℱ\varphi (𝐱)$ is also in ${𝒯}_n$.

Let

For a multi-index $𝜶=({\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n)$, define

and

Here $𝜶$ ranges over a finite set of multi-indices, $P(𝐱)$ is a multivariate polynomial, and $P(𝐃)$ is a partial differential operator. Then

and

If $u\in {𝒯}_n^{\ast }$ is a tempered distribution, then its Fourier transform $ℱ\left(u\right)$ is defined by

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

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

We use the notation of the previous subsection and take $n=1$ and $u=\delta$ in (1.16.35). We obtain

As distributions, the last equation reads

which is often written conventionally as

see also (1.17.2).

Since $\sqrt{2\pi }ℱ\left(\delta \right)=1$, we have

in which ${\varphi }_{-}(x)=\varphi (-x)$. The second to last equality follows from the Fourier integral formula (1.17.8). Since the quantity on the extreme right of (1.16.41) is equal to $\sqrt{2\pi }⟨\delta ,\varphi ⟩$, as distributions, the result in this equation can be stated as

and conventionally it is expressed as

see also (1.17.12).

The Fourier transform of $H\left(x\right)$ now follows from (1.16.42) and (1.16.48). Indeed, we have

that is