How Is the Convolution of an L1 and an L2 Function Defined?

For functions on $\mathbb{R}^n$ with Lebesgue measure, the convolution of an $L^1$ function and an $L^2$ function is defined by an integral, but generally only for almost every input $x$. The result is an $L^2$ function, understood up to equality almost everywhere.

The key estimate is Young’s convolution inequality: $\|f*g\|_2 \leq \|f\|_1\|g\|_2$. It both guarantees that the convolution is well behaved and controls its size.

Definition

Let $f\in L^1(\mathbb{R}^n)$ and $g\in L^2(\mathbb{R}^n)$. Using the standard convention, define

$$(f*g)(x)=\int_{\mathbb{R}^n} f(y)g(x-y)\,dy.$$

This formula is meaningful for almost every $x$, not necessarily for every $x$. The spaces $L^1$ and $L^2$ contain equivalence classes of functions: changing a function on a set of measure zero does not change the element of the space. Accordingly, the convolution is also an equivalence class of functions in $L^2$.

A common alternative writes the formula as $\int f(x-y)g(y)\,dy$. A change of variables shows that these conventions agree almost everywhere on $\mathbb{R}^n$.

Why the integral exists almost everywhere

The integrand need not be integrable for every fixed $x$. However, the integral of its absolute value is finite for almost every $x$. To see this, take any bounded measurable set $K\subset\mathbb{R}^n$. Tonelli’s theorem and the Cauchy–Schwarz inequality give

$$\begin{aligned} \int_K\int_{\mathbb{R}^n}|f(y)|\,|g(x-y)|\,dy\,dx &=\int_{\mathbb{R}^n}|f(y)|\int_K|g(x-y)|\,dx\,dy\\ &\leq \int_{\mathbb{R}^n}|f(y)|\,|K|^{1/2}\|g\|_2\,dy\\ &=|K|^{1/2}\|f\|_1\|g\|_2<\infty. \end{aligned}$$

Here $|K|$ denotes the Lebesgue measure of $K$. Therefore the inner integral is finite for almost every $x$ in $K$. Covering $\mathbb{R}^n$ by countably many bounded sets shows that the convolution integral is absolutely convergent for almost every $x$ in all of $\mathbb{R}^n$.

Why the convolution belongs to L2

For a fixed $y$, the function $x\mapsto g(x-y)$ is a translate of $g$. Translation preserves the $L^2$ norm, so $\|g(\cdot-y)\|_2=\|g\|_2$. Applying the integral form of the triangle inequality in $L^2$ (Minkowski’s inequality) gives

$$\begin{aligned} \|f*g\|_2 &=\left\|\int_{\mathbb{R}^n} f(y)g(\cdot-y)\,dy\right\|_2\\ &\leq \int_{\mathbb{R}^n}|f(y)|\,\|g(\cdot-y)\|_2\,dy\\ &=\|f\|_1\|g\|_2. \end{aligned}$$

This is the $L^1*L^2\to L^2$ case of Young’s inequality. One rigorous way to read the integral in this argument is as an integral taking values in the Banach space $L^2$: the integrand has norm $|f(y)|\|g\|_2$, which is integrable in $y$. This produces an $L^2$ function and agrees almost everywhere with the pointwise integral above.

Worked example

On $\mathbb{R}$, take $f=g=\mathbf{1}_{[0,1]}$, where $\mathbf{1}_{[0,1]}$ is 1 on $[0,1]$ and 0 elsewhere. Both functions belong to $L^1$ and $L^2$, and each has $L^1$ norm 1 and $L^2$ norm 1.

For each $x$, the convolution is the length of the overlap of the intervals $[0,1]$ and $[x-1,x]$. Thus, up to endpoint values, which do not affect the $L^2$ function,

$$(f*g)(x)= \begin{cases} x, & 0<x<1,\\ 2-x, & 1\leq x<2,\\ 0, & \text{otherwise}. \end{cases}$$

Its squared $L^2$ norm is

$$\|f*g\|_2^2=\int_0^1 x^2\,dx+\int_1^2(2-x)^2\,dx=\frac{2}{3},$$

so $\|f*g\|_2=\sqrt{2/3}$. Young’s inequality predicts only that this norm is at most $1\cdot1=1$, which the example satisfies.

Common mistakes

• Assuming the integral exists for every $x$. The standard $L^1*L^2$ definition guarantees absolute convergence almost everywhere. Values at exceptional points can be assigned arbitrarily without changing the resulting $L^2$ function.
• Treating an $L^2$ function as necessarily integrable. On an infinite-measure space, $L^2$ does not in general sit inside $L^1$. The convolution is guaranteed to be in $L^2$, not necessarily in $L^1$.
• Forgetting the measure and domain. The formula and norm estimate here use Lebesgue measure on $\mathbb{R}^n$. On a locally compact group, the analogous definition uses Haar measure, with conventions adjusted to the group operation and choice of left or right Haar measure.