Distribution (mathematics) - Convolution

Convolution

Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions.

Convolution of a test function with a distribution

If ƒ ∈ D(Rn) is a compactly supported smooth test function, then convolution with ƒ defines an operator

defined by C_ƒg = ƒ∗g, which is linear (and continuous with respect to the LF space topology on D(Rn).)

Convolution of ƒ with a distribution S ∈ D′(Rn) can be defined by taking the transpose of C_ƒ relative to the duality pairing of D(Rn) with the space D′(Rn) of distributions (Trèves 1967, Chapter 27). If ƒ, g, 'φ' ∈ D(Rn), then by Fubini's theorem

where . Extending by continuity, the convolution of ƒ with a distribution S is defined by

for all test functions 'φ' ∈ D(Rn).

An alternative way to define the convolution of a function ƒ and a distribution S is to use the translation operator τ_x defined on test functions by

and extended by the transpose to distributions in the obvious way (Rudin 1991, §6.29). The convolution of the compactly supported function ƒ and the distribution S is then the function defined for each x ∈ Rn by

It can be shown that the convolution of a compactly supported function and a distribution is a smooth function. If the distribution S has compact support as well, then ƒ∗S is a compactly supported function, and the Titchmarsh convolution theorem (Hörmander 1983, Theorem 4.3.3) implies that

where ch denotes the convex hull.

Distribution of compact support

It is also possible to define the convolution of two distributions S and T on Rn, provided one of them has compact support. Informally, in order to define S∗T where T has compact support, the idea is to extend the definition of the convolution ∗ to a linear operation on distributions so that the associativity formula

continues to hold for all test-functions 'φ'. Hörmander (1983, §IV.2) proves the uniqueness of such an extension.

It is also possible to provide a more explicit characterization of the convolution of distributions (Trèves 1967, Chapter 27). Suppose that it is T that has compact support. For any test function 'φ' in D(Rn), consider the function

It can be readily shown that this defines a smooth function of x, which moreover has compact support. The convolution of S and T is defined by

This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense:

This definition of convolution remains valid under less restrictive assumptions about S and T; see for instance Gel'fand & Shilov (1966–1968, v. 1, pp. 103–104) and Benedetto (1997, Definition 2.5.8).