My objective function that is to be minimized is as follows:
$$\|y-Ax\|_2^2 + \alpha\|Lx\|_1$$
where $L$ is the gradient operator.
Now this problem seems convex because the first term is quadratic and hence convex and the second is a norm so it has to be convex (the triangular inequality), but is it also strictly convex? and why?