This may be a silly question and I am certainly missing something obvious, but I can't figure it out at the moment. Following Rudin's RCA, Theorem 6.19, the space $M(\mathbb{R})$ of regular Borel complex measures on $\mathbb{R}$ is isomorphic to the space of bounded linear functionals on $C_0(\mathbb{R})$. Moreover by Convolution of regular measures is regular, I know that convolution is regular, and I also know that convolution of finite measures is finite. So convolution is actually a product in $M(\mathbb{R})$.
Now, the question: is it possible to describe the functional representing the convolution of two measures by using only the functionals of the two measures? Or, in other words, is it possible to define convolution directly in $C_0(\mathbb{R})^*$, without making any reference to measure theory or Riesz representation theorem?
I think the answer is yes, but I can't find it anywhere and I'm too dumb or tired to find it by myself.