Distribution (mathematics) - Basic Idea

Basic Idea

Distributions are a class of linear functionals that map a set of test functions (conventional and well-behaved functions) onto the set of real numbers. In the simplest case, the set of test functions considered is D(R), which is the set of functions from R to R having two properties:

• The function is smooth (infinitely differentiable);
• The function has compact support (is identically zero outside some bounded interval).

Then, a distribution d is a linear mapping from D(R) to R. Instead of writing d(φ), where φ is a test function in D(R), it is conventional to write . A simple example of a distribution is the Dirac delta δ, defined by

There are straightforward mappings from both locally integrable functions and probability distributions to corresponding distributions, as discussed below. However, not all distributions can be formed in this manner.

Suppose that

is a locally integrable function, and let

be a test function in D(R). We can then define a corresponding distribution by:

This integral is a real number which linearly and continuously depends on . This suggests the requirement that a distribution should be linear and continuous over the space of test functions D(R), which completes the definition. In a conventional abuse of notation, may be used to represent both the original function and the distribution derived from it.

Similarly, if is a probability distribution on the reals and is a test function, then a corresponding distribution may be defined by:

Again, this integral continuously and linearly depends on, so that is in fact a distribution.

Such distributions may be multiplied with real numbers and can be added together, so they form a real vector space. In general it is not possible to define a multiplication for distributions, but distributions may be multiplied with infinitely differentiable functions.

It's desirable to choose a definition for the derivative of a distribution which, at least for distributions derived from locally integrable functions, has the property that . If is a test function, we can show that

using integration by parts and noting that, since is zero outside of a bounded set. This suggests that if is a distribution, we should define its derivative by

It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes infinitely differentiable and the usual properties of derivatives hold.

Example: Recall that the Dirac delta (so-called Dirac delta function) is the distribution defined by

It is the derivative of the distribution corresponding to the Heaviside step function : For any test function ,

so . Note, because of compact support. Similarly, the derivative of the Dirac delta is the distribution

This latter distribution is our first example of a distribution which is derived from neither a function nor a probability distribution.