Convolution of functions

$f$ and $g$ belonging to $L(-\infty, +\infty)$

The function $h$ defined by \begin{equation} h(x) = \int\limits_{-\infty}^{+\infty}f(x-y)g(y)\,dy = \int\limits_{-\infty}^{+\infty}f(y)g(x-y)\,dy; \end{equation} it is denoted by the symbol $f*g$. The function $f*g$ is defined almost everywhere and also belongs to $L(-\infty, +\infty)$.

Properties

The convolution has the basic properties of multiplication, namely, \begin{equation} f*g = g*f, \end{equation} \begin{equation} (\alpha_1f_1 + \alpha_2f_2)*g = \alpha_1(f_1*g) + \alpha_2(f_2*g), \quad \alpha_1, \alpha_2 \in \mathbb{R}, \end{equation} \begin{equation} (f*g)*h = f*(g*h) \end{equation}

for any three functions in $L(-\infty, \infty)$. Therefore, $L(-\infty, \infty)$ with the usual operations of addition and of multiplication by a scalar, with the operation of convolution as the multiplication of elements, and with the norm \begin{equation} \|f\| = \int\limits_{-\infty}^{\infty}|f(x)|\, dx, \end{equation} is a Banach algebra (for this norm $\|f*g\|\leq \|f\|\cdot \|g\|$).

If $F[f]$ denotes the Fourier transform of $f$, then

and this is used in solving a number of applied problems.

Thus, if a problem has been reduced to an integral equation of the form

(*)

where

then, under the assumption that , by applying the Fourier transformation to (*) one obtains

hence

and the inverse Fourier transformation yields the solution to (*) as

The properties of a convolution of functions have important applications in probability theory. If and are the probability densities of independent random variables and , respectively, then is the probability density of the random variable .

The convolution operation can be extended to generalized functions (cf. Generalized function). If and are generalized functions such that at least one of them has compact support, and if is a test function, then is defined by

where is the direct product of and , that is, the functional on the space of test functions of two independent variables given by

for every infinitely-differentiable function of compact support.

The convolution of generalized functions also has the commutativity property and is linear in each argument; it is associative if at least two of the three generalized functions have compact supports. The following equalities hold:

where is the differentiation operator and is any multi-index,

in particular, , where denotes the delta-function. Also, if , are generalized functions such that , and if there is a compact set such that

then

Finally, if is a generalized function of compact support and is a generalized function of slow growth, then the Fourier transformation can be applied to , and again

The convolution of generalized functions is widely used in solving boundary value problems for partial differential equations. Thus, the Poisson integral, written in the form

is a solution to the thermal-conductance equation for an infinite bar, where the initial temperature can be not only an ordinary function but also a generalized one.

Both for ordinary and generalized functions the concept of a convolution carries over in a natural way to functions of several variables; then in the above and must be regarded as vectors from and not as real numbers.

References

[1]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101
[2]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , 1–5 , Acad. Press (1964) (Translated from Russian) MR435831 Zbl 0115.33101
[3]	E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) MR0942661 Zbl 0017.40404 Zbl 63.0367.05

References

[a1]	W. Kecs, "The convolution product and some applications" , Reidel & Ed. Academici (1982) MR0690953 Zbl 0512.46041