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
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 ,
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.)
where is a constant. More generally, if is an infinitely differentiable function, then
We say that a sequence of distributions converges to a distribution in if
for all .
The derivative of a distribution is defined by
For any locally integrable function , its distributional derivative is .
The Dirac delta distribution is singular.
Suppose is infinitely differentiable except at , where left and right derivatives of all orders exist, and
For and not an integer, define
where is an integer such that . Similarly, we write
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
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).
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
The distributional derivative of is defined by
where is a multi-index and .
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 .
Suppose is a test function in . Then its Fourier transform is
where and . is also in .
For a multi-index , define
Here ranges over a finite set of multi-indices, is a multivariate polynomial, and is a partial differential operator. Then
If is a tempered distribution, then its Fourier transform is defined by
The Fourier transform of a tempered distribution is again a tempered distribution, and
in which ; compare (1.16.33) and (1.16.34). In (1.16.36) and (1.16.37) the derivatives in 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 and in (1.16.35). We obtain
As distributions, the last equation reads
which is often written conventionally as
see also (1.17.2).
Since , we have
in which . 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 , as distributions, the result in this equation can be stated as
and conventionally it is expressed as
see also (1.17.12).
It is easily verified that
and from (1.16.15) we find
where is the Heaviside function defined in (1.16.13), and the derivatives are to be understood in the sense of distributions. Then
and from (1.16.36) with , , and , we have also