Abstract : The reconstruction theorem, a cornerstone of Martin Hairer's theory of regularity structures,appears in this article as the unique extension of the explicitly given reconstruction operatoron the set of smooth models due its inherent Lipschitz properties. This new proof is a directconsequence of constructions of mollification procedures on spaces of models and modelled distributions: more precisely, for an abstract model Z of a given regularity structure, a mollifiedmodel is constructed, and additionally, any modelled distribution f can be approximated byelements of a universal subspace of modelled distribution spaces. These considerations yield inparticular a non-standard approximation results for rough path theory. All results are formulatedin a generic (p, q) Besov setting. There are also implications on learning solution maps from amachine learning perspective.Joint work with Harprit Singh.

Keywords : universal approximation, reconstruction theorem, regularity structures