Section 6.1 Convolution
This chapter will establish the convergence of classical Fourier series in \(L^2\text{.}\) To do so, we’ll use a technique that is one of the basic tools of analysis, the convolution of two functions.
Definition 6.1.1.
Let \(f\) and \(g\) be integrable functions on \([a,b]\subset \R\text{.}\) The convolution (or convolution product) of \(f\) and \(g\) is the function \(f * g: [a,b] \to \R\) defined by
\begin{equation*}
(g * f)(y) = \int_a^b f(x) g(y - x) \, dx.
\end{equation*}
Let’s look at an example that illustrates something of what a convolution can do. Suppose that we’re working on a symmetric interval \([-a,a]\text{,}\) and for a constant \(\delta > 0\text{,}\) define a function \(g_\delta\) by
\begin{equation*}
g_\delta(y) = \begin{cases} \frac{1}{2 \delta} \amp y \in (-\delta, \delta) \\ 0 \amp y\in [-a, -\delta]\cup [\delta, a] \end{cases}.
\end{equation*}
Notice that \(g_\delta\) has integral 1 on \([-a,a]\text{.}\) Extend \(g_\delta\) to be periodic with period \(2a\text{.}\) What will convolving \(f\) with \(g\) do? Notice that
\begin{align*}
(g*f)(y) \amp= \int_{-a}^a f(x) g(y-x)\, dx\\
\amp= \int_{y - \delta}^{y+\delta}f(x) \frac{1}{2\delta}\, dx\\
\amp= \frac{1}{2\delta} \int_{y-\delta}^{y + \delta} f(x)\, dx.
\end{align*}
That is, at each \(y\text{,}\) the value of \(g*f\) is the average value of \(f\) on the \(\delta\)-neighborhood near \(y\text{.}\) This has the effect of smoothing the graph of \(f\text{.}\) (Indeed, this smoothing is one of the primary applications of convolution in many fields, including signals and circuits.)
Natural choices for smoothing functions are similar to the
\(g\) that we defined above - we’d like the integral to be 1, and we typically want to be able to concentrate the area of
\(g\) very close to
\(0\text{,}\) so that
\(g(x)\) is close to 0 in value away from
\(x=0\) and
\(g*f(y)\) is very close to
\(f(y)\text{.}\) (Here, one might consider investigating
mollifiers.)
Construction of functions via convolution is a frequent approach to solving problems in the fields of harmonic and complex analysis. (See, for example, the solution to the
Dirichlet problem on the disk via convolution of a function with the
Poisson kernel.)
