# §1.16 Distributions

## §1.16(i) Test Functions

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

A sequence of test functions converges to a test function if the support of every is contained in a fixed compact set and as the sequence converges uniformly on to for .

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

A mapping on is a linear functional if it takes complex values and

1.16.1

where and are real or complex constants. is called a distribution if it is a continuous linear functional on , that is, it is a linear functional and for every in ,

1.16.2

From here on we write for . The space of all distributions will be denoted by . A distribution is called regular if there is a function on , which is absolutely integrable on every compact subset of , such that

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

Define

1.16.4
1.16.5

where is a constant. More generally, if is an infinitely differentiable function, then

1.16.6

We say that a sequence of distributions converges to a distribution in if

for all .

## §1.16(ii) Derivatives of a Distribution

The derivative of a distribution is defined by

Similarly

1.16.9.

For any locally integrable function , its distributional derivative is .

## §1.16(iv) Heaviside Function

1.16.13
1.16.14

Suppose is infinitely differentiable except at , where left and right derivatives of all orders exist, and

1.16.17

Then

For ,

1.16.19

For ,

1.16.20

For and not an integer, define

where is an integer such that . Similarly, we write

and define

## §1.16(v) Tempered Distributions

The space of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are as for all .

A sequence of functions in is said to converge to a function as if the sequence converges uniformly to on every finite interval and if the constants in the inequalities

1.16.24

do not depend on .

A tempered distribution is a continuous linear functional on . (See the definition of a distribution in §1.16(i).) The set of tempered distributions is denoted by .

A sequence of tempered distributions converges to in if

for all .

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 be the set of all infinitely differentiable functions in variables, , with compact support in . If is a multi-index and , then we write and . A sequence of functions in converges to a function if the supports of lie in a fixed compact subset of and converges uniformly to in for every multi-index . A distribution in is a continuous linear functional on .

The partial derivatives of distributions in can be defined as in §1.16(ii). A locally integrable function gives rise to a distribution defined by

For tempered distributions the space of test functions is the set of all infinitely-differentiable functions of variables that satisfy

Here and are multi-indices, and are constants. Tempered distributions are continuous linear functionals on this space of test functions. The space of tempered distributions is denoted by .

## §1.16(vii) Fourier Transforms of Distributions

Suppose is a test function in . Then its Fourier transform is

where and . is also in . For a multi-index , set and

1.16.30
1.16.31

and

1.16.32

Then

and

If is a tempered distribution, then its Fourier transform is defined by

where is given by (1.16.29). The Fourier transform of a tempered distribution is again a tempered distribution, and

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