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 . 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 _\mathbb{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 c}(}\textbf{  some explicit examples of computation of the convolution distance.
Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves . We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To achieve so, we explicitly compute all morphisms in D^b_{R c}( k _\mathbb{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, give a precise description of connected components of D^b_{\mathbb{R c}(}\textbf{k _\mathbb{R some explicit examples of computation of the convolution distance.
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Following Curry and Kashiwara Schapira seminal works on adapting persistence theory ideas to sheaves on a real vector space
Persistent homology has been recently studied with the tools of sheaf theory
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, we extensively study their convolution distance on the category of constructible sheavesover the real line D^b_{\mathbb{R
by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves
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between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes
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give a precise description of connected components of D^b_{\mathbb{R
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