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 , set 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