A derived isometry theorem for constructible sheaves on R
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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 sheaves over the real line D^b_R c(k_R). We prove an isometry theorem in this derived setting, expressing the convolution distance as a matching distance, mirroring the situation of persistence with one parameter. We also explicitly compute all morphisms in D^b_R c(k_R), which enables us to compute distances between indecomposable objects. Then we adapt Bjerkevik's stability proof to this derived setting. As a byproduct of our isometry theorem, we prove that the convolution distance is closed, and also give some explicit examples of computation of the convolution distance.
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