I am not an expert on distributions, but I would like to know if there is a meaningful way to define the convolution
$$ sinc * P_T $$
between $ sinc(x)=\frac{sin(x)}{x} $ and $ P_T = \sum_{k=-\infty}^{\infty} \delta_{kT}$ where $T$ is an arbitrary real number and $\langle \delta_x,f \rangle = f(x)$.
Defining the convolution in a naive way
$$ (sinc * P_T)(x) = \sum_{k=-\infty}^{\infty} sinc(x-kT) $$
leads to immediate problems since the sum does not converge for most cases. Is there an more abstract route which lead to a meaningful definition?
Thank you for your help.