Hairer's reconstruction theorem without regularity structures

This survey is devoted to Martin Hairer's Reconstruction Theorem, which is one of the cornerstones of his theory of Regularity Structures [6]. Our aim is to give a new self-contained and elementary proof of this theorem, together with some applications, including a characterization, based on a single arbitrary test function, of negative Hölder spaces. We present the Reconstruction Theorem as a general result in the theory of distributions that can be understood without any knowledge of Regularity Structures themselves, which we do not even need to define.