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

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

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

$$\tag{*} f (x) \ = \ g (x) + \int\limits _ {- \infty} ^ \infty K (x - y) f (y) \ dy,$$

where

$$g (x) \ \in \ L _{2} (- \infty ,\ \infty ),\ \ K (x) \ \in \ L (- \infty ,\ \infty ),$$

$$\mathop{\rm sup} _{x} \ | F [K] (x) | \ \leq \ \frac{1}{\sqrt {2 \pi}} ,$$

then, under the assumption that $f \in L (- \infty ,\ \infty )$, 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

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

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 $(f * g)$ 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 $\phi$ is a test function, then $f * g$ is defined by

$$\langle f * g,\ \phi \rangle \ = \ \langle f (x) \times g (y),\ \phi (x + y) \rangle,$$

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

$$\langle f (x) \times g (y),\ u (x,\ y) \rangle \ = \ < f (x),\ < g (y),\ u (x,\ y) \gg$$

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:

$$D ^ \alpha (f * g) \ = \ D ^ \alpha f * g \ = \ f * D ^ \alpha g,$$

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

$$(D ^ \alpha \delta ) * f \ = \ D ^ \alpha f,$$

in particular, $\delta * f = f$, where $\delta$ denotes the delta-function. Also, if $f _{n}$, $n = 1,\ 2 \dots$ are generalized functions such that $f _{n} \rightarrow f _{0}$, and if there is a compact set $K$ such that

$$K \ \supset \ \mathop{\rm supp}\nolimits \ f _{n} ,\ \ n = 1,\ 2 \dots$$

then

$$f _{n} * g \ \rightarrow \ f _{0} * g.$$

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 $f * g$, and again

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

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

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

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

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