Distribution (mathematics)
In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. They allow to extend the concept of derivative to all continuous functions and beyond and are used to formulate generalized solutions of partial differential equations. They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions are distributions, such as the Dirac delta distribution.
Basic idea
The basic idea is as follows. If f : R -> R is an integrable function, and φ : R -> R is a smooth ( = infinitely often differentiable) function which is identically zero except on some bounded set, then ∫fφdx is a real number which linearly and continuously depends on φ. One can therefore think of the function f as a continuous linear functional on the space which consists of all the "test functions" φ. Similarly, if P is a probability distribution on the reals and φ is a test function, then ∫φdP is a real number that continuously and linearly depends on φ: probability distributions can thus also be viewed as continuous linear functionals on the space of test functions. This notion of "continuous linear functional on the space of test functions" is therefore used as the definition of a distribution.
Such distributions may be multiplied with real numbers and can by added together, but it is in general not possible to define a multiplication for distributions. Complex-valued distributions are also not defined.
To define the derivative of a distribution, we first consider the case of a differentiable and integrable function f : R -> R. If φ is a test function, then we have
- ∫f 'φ dx = &emdash;∫ fφ' dx
using integration by parts (note that φ is zero outside of a bounded set and that therefore no boundary values have to be taken into account). This suggests that if Ψ is a distribution, we should define its derivative Ψ' as the linear functional which sends the test function φ to -Ψ(φ'). It turns out that this is the proper definition; it extends the ordinary definition of derivative, every distribution becomes differentiable and the usual properties of derivatives hold.
The Dirac delta is the distribution which sends the test function φ to φ(0). It is the derivative of the function f(x) = 0 if x < 0 and f(x) = 1 if x ≥ 0. The derivative of the Dirac delta is the distribution which sends the test function φ to -φ'(0). This latter distribution is our first example of a distribution which is neither a function nor a probability distribution.
Formal definition
In the sequel, distributions on an open subset U of Rn will be formally defined. (With minor modifications, they can also be defined on any smooth manifold.) First, the space D(U) of test functions on U needs to be explained. A function φ : U -> R is said to have compact support if there exists a compact subset K of U such that φ(x) = 0 for all x in U \ K. The elements of D(U) are the infinitely often differentiable functions φ : U -> R with compact support. This is a real vector space. We turn it into a topological vector space by requiring that a sequence (or net) (φk) converges to 0 if and only if there exists a compact subset K of U such that all φk are identically zero outside K, and for every ε > 0 and natural number d ≥ 0 there exists a natural number k0 such that for all k ≥ k0 the absolute value of all d-th derivatives of φk is smaller than ε. With this definition, D(U) becomes a complete topological vector space (in fact, a so-called LF-space).
The dual space of the topological vector space D(U), consisting of all continuous linear functionals Ψ : D(U) -> R, is the space of all distributions on U; it is also a complete topological vector space and is denoted by D'(U).
The function f : U -> R is called locally integrable if f is Lebesgue integrable over every compact subset K of U. This is a large class of functions which includes all continuous functions. The topology on D(U) is defined in such a fashion that any locally integrable function f yields a continuous linear functional on D(U) whose value on the test function φ is given by the Lebesgue integral ∫U fφ dx. Two locally integrable functions f and g yield the same element of D(U) if and only if they are equal almost everywhere. Similarly, every Radon measure μ on U (which includes the probability distributions) defines an element of D'(U) whose value on the test function φ is ∫φ dμ.
As mentioned above, integration by parts suggests that the derivative dΨ/dx of the distribution Ψ in direction x should be defined using the formula
- dΨ / dx (φ) = -Ψ(dφ / dx)
for all test functions φ. In this way, every distribution is infinitely often differentiable, and the derivative is a continuous linear operator on D'(U).
Tempered distributions
By using a larger space of test functions, one can define the tempered distributions, a subspace of D'(Rn). These distributions are useful if one studies the Fourier transform in generality: all tempered distributions have a Fourier transform, but not all distributions have one.
The space of test functions employed here is the space of all infinitely differentiable rapidly decreasing functions, where φ : Rn -> R is called rapidly decreasing if any derivative of φ, multiplied with any power of |x|, converges towards 0 for |x| to ∞. These functions form a complete topological vector space if we define convergence as follows: a sequence (or net) (φk) converges to 0 if and only if for every ε > 0 and natural number d ≥ 0 there exists a natural number k0 such that for all k ≥ k0 the absolute value of all d-th derivatives of φk is smaller than ε.
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded locally integrable functions.