Definition of Convolution
Definition
Let’s assume that two functions $f$ and $g$ defined in $\mathbb{R}$ are given. If the integral below exists, it is called the convolution of the two functions $f$ and $g$, and is denoted by $f \ast g$.
$$ f \ast g(x):=\int _{-\infty} ^{\infty} f(y)g(x-y)dy $$
If $f$ and $g$ are discrete functions, they are defined as follows.
$$ (f \ast g)(m)=\sum \limits_{n}f(n)g(m-n) $$
Explanation
Although there is a translation as convolution, the term convolution is more commonly used. Generally, the above definition is learned as convolution, but more generally, it is a convolution regarding Fourier transform which is an integral transform. It is used in various fields because it has many good properties such as commutative law, distributive law, etc.
In the case of discrete convolution, it is defined a bit differently in analytic number theory.
The conditions for convolution to be defined are as follows:
(a)
If $f\in L^{1}$ and $|g|<M$, then
$$ \left| \int f(y)g(x-y)dy \right| \le \int \left| f(y)g(x-y) \right|dy \le M\int \left| f(y) \right|dy \lt \infty $$
(b)
If $\left| f \right| \le M$ and $g\in L^{1}$, then
$$ \left| \int f(y)g(x-y)dy \right| \le \int \left| f(y) g(x-y) \right|dy \le M\int \left| g(x-y) \right|dy \lt \infty $$
(c)
Let’s assume $f,g\in L^{2}$ and $\tilde{g}_{x}(y)=g(x-y)$. Then $\tilde{g}_{x}\in L^{2}$ and $\left\| g \right\|_{2}=\left\| \tilde{g}_{x} \right\|_{2}$, and by the Cauchy-Schwarz inequality,
$$ \begin{align*} \left| \int f(y)g(x-y)dy \right| &= \left| \int f(y)\tilde{g}_{x}(y)dy \right| \\ & = \left| \left\langle f,\tilde{g}_{x} \right\rangle \right| \\ &\le \left\| f \right\|_{2} \left\| \tilde{g}_{x} \right\|_{2} \\ &<\infty \end{align*} $$
(d)
If $f$ is bounded except for the closed interval $[a,b]$ where it is $0$ and when $g$ is piecewise continuous,
$$ \int _{-\infty} ^{\infty} f(y)g(x-y)dy=\int _{a}^{b}f(y)g(x-y)dy<\infty $$