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, 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 often differentiable functions.
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
The Dirac delta (so-called Dirac delta function) is the distribution which sends the test function φ to φ(0). It is the derivative of the Heaviside step function H(x) = 0 if x < 0 and H(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.
In the sequel, real-valued distributions on an open subset U of Rn will be formally defined. (With minor modifications, one can also define complex-valued distributions, and one can replace Rn by 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 S : D(U) -> R, is the space of all distributions on U; it is a vector space and is denoted by D'(U).
The function f : U -> R is called locally integrable if it 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 dS/dx of the distribution S in direction x should be defined using the formula
The space D'(U) is turned into a locally convex[?] topological vector space by defining that the sequence (Sk) converges towards 0 if and only if Sk(φ) → 0 for all test functions φ. This is the case if and only if Sk converges uniformly to 0 on all bounded subsets of D(U). (A subset of E of D(U) is bounded if there exists a compact subset K of U and numbers dn such that every φ in E has its support in K and has its n-th derivatives bounded by dn.) With respect to this topology, differentiation of distributions is a continuous operator; this is an important and desirable property that is not shared by most other notions of differentiation. Furthermore, the test functions (which can itself be viewed as distributions) are dense in D'(U) with respect to this topology.
If ψ : U -> R is an infinitely often differentiable function and S is a distribution on U, we define the product Sψ by (Sψ)(φ) = S(ψφ) for all test functions φ. The ordinary product rule of calculus remains valid.
We say that a distribution S has compact support if there is a compact subset K of U such that for every test function φ whose support is completely outside of K, we have S(φ) = 0. Alternatively, one may define distributions with compact support as continuous linear functionals on the space C∞(U); the topology on C∞(U) is defined such that φk converges to 0 if and only if all derivatives of φk converge uniformly to 0 on every compact subset of U.
If both S and T are distributions on Rn and one of them has compact support, then one can define a new distribution, the convolution S*T of S and T, as follows: if φ is a test function in D(Rn) and x, y elements of Rn, write φx(y) = x + y, ψ(x) = T(φx) and (S*T)(φ) = S(ψ). This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense:
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, the so-called Schwartz-space, 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| → ∞. These functions form a complete topological vector space if we define the topology as above for the space C∞(Rn).
The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow growing) locally integrable functions; all distributions with compact support and all square-integrable functions can be viewed as tempered distributions.
To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform F yields then an automorphism of Schwartz-space, and we can define the Fourier transform of the tempered distribution S by (FS)(φ) = S(Fφ) for every test function φ. FS is thus again a tempered distribution. The Fourier transform is a continuous, linear, bijective operator from the space of tempered distributions to itself. This operation is compatible with differentiation in the sense that