Convolution of functions

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

The function $h$ defined by $$h(x)=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(x-y)g(y)dy=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(y)g(x-y)dy;$$ it is denoted by the symbol $f\ast g$. The function $f\ast g$ is defined almost everywhere and also belongs to $L(-\mathrm{\infty },+\mathrm{\infty })$.

Properties[edit]

The convolution has the basic properties of multiplication, namely, $$f\ast g=g\ast f,$$ $$({\alpha }_1{f}_1+{\alpha }_2{f}_2)\ast g={\alpha }_1(f_1\ast g)+{\alpha }_2(f_2\ast g),{\alpha }_1,{\alpha }_2\in \mathbb{R},$$ $$(f\ast g)\ast h=f\ast (g\ast h)$$

for any three functions in $L(-\mathrm{\infty },\mathrm{\infty })$. Therefore, $L(-\mathrm{\infty },\mathrm{\infty })$ 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 $$\Vert f\Vert =\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}|f(x)|dx,$$ is a Banach algebra (for this norm $\Vert f\ast g\Vert \le \Vert f\Vert \cdot \Vert g\Vert$).

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

$$F[f\ast g]=\sqrt{2\pi }F[f]F[g],$$

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

$$\begin{array}{}\text{(*)}& f(x)=g(x)+\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}K(x-y)f(y)dy,\end{array}$$

where

$$g(x)\in L_2(-\mathrm{\infty },\mathrm{\infty }),K(x)\in L(-\mathrm{\infty },\mathrm{\infty }),$$

$$\underset{x}{\sup }|F[K](x)|\le \frac{1}{\sqrt{2\pi }},$$

then, under the assumption that $f\in L(-\mathrm{\infty },\mathrm{\infty })$, by applying the Fourier transformation to (*) one obtains

$$F[f]=F[g]+\sqrt{2\pi }F[f]F[K],$$

hence

$$F[f]=\frac{F[g]}{1-\sqrt{2\pi }F[K]},$$

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

$$f(x)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{F[g](\zeta )e^{-i\zeta x}}{1-\sqrt{2\pi }F[K](\zeta )}d\zeta .$$

The properties of a convolution of functions have important applications in probability theory. If $f$ and $g$ are the probability densities of independent random variables $X$ and $Y$, respectively, then $(f\ast g)$ is the probability density of the random variable $X+Y$.

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

$$⟨f\ast g,\varphi ⟩=⟨f(x)\times g(y),\varphi (x+y)⟩,$$

where $f(x)\times g(y)$ is the direct product of $f$ and $g$, that is, the functional on the space of test functions of two independent variables given by

$$⟨f(x)\times g(y),u(x,y)⟩=<f(x),<g(y),u(x,y)\gg$$

for every infinitely-differentiable function $u(x,y)$ 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:

$$D^{\alpha }(f\ast g)=D^{\alpha }f\ast g=f\ast D^{\alpha }g,$$

where $D$ is the differentiation operator and $\alpha$ is any multi-index,

$$(D^{\alpha }\delta )\ast f=D^{\alpha }f,$$

in particular, $\delta \ast f=f$, where $\delta$ denotes the delta-function. Also, if $f_n$, $n=1,2\dots$ are generalized functions such that $f_n\to f_0$, and if there is a compact set $K$ such that

$$K\supset \mathrm{supp}{f}_n,n=1,2\dots$$

then

$$f_n\ast g\to f_0\ast g.$$

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

$$F[f\ast g]=\sqrt{2\pi }F[f]F[g].$$

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

$$U(x,t)=\mu (x)\ast \frac{1}{2\sqrt{\pi t}}{e}^{-x^2/4t},$$

is a solution to the thermal-conductance equation for an infinite bar, where the initial temperature $\mu$ 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 $x$ and $y$ must be regarded as vectors from ${\mathbf{R}}^n$ and not as real numbers.

References[edit]