In my class of functional analysis we've defined two types of convolutions:
Suppose $f,g: K \to \mathbb C$ are Riemann integrable functions with compact support in $K \subseteq \mathbb R$, then the convolution $f * g$ is defined as $$ (f * g)(x) = \int_Kf(y)g(x-y)dy$$
Suppose $f,g: \mathbb T \to \mathbb C$ (with $\mathbb T$ the unit circle in $\mathbb C$ equipped with the metric defined as the shortest arc between two points in it) are Riemann integrable functions (in the sense they are integrable as periodic functions over $[-\pi, \pi]$), then their circular convolution is defined as $$ (f * g)(x) =\frac{1}{2\pi}\int_{-\pi}^{\pi}f(y)g(x-y)dy$$
We did say in class that we can identify the space of functions defined on $\mathbb T$ with the periodic functions over $[-\pi,\pi]$, but it seems to me that the two definitions are basically the same (except we normalize the integral in the circular case). So is there a difference?