Before revision: Following Curry and Kashiwara Schapira seminal works on adapting persistence theory ideas to sheaves on a real vector space in the derived setting , we extensively study their convolution distance on the category of constructible sheavesover the real line D^b_{\mathbb{R .
After revision: Persistent homology has been recently studied with the tools of sheaf theory in the derived setting , we extensively study their convolution distance on the category of constructible sheavesover the real line D^b_{\mathbb{R .