5 N ov 2 01 4 A renormalisation group method . II . Approximation by local polynomials

This paper is the second in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. The method is set within a normed algebra N of functionals of the fields. In this paper, we develop a general method—localisation—to approximate an element of N by a local polynomial in the fields. From the point of view of the renormalisation group, the construction of the local polynomial corresponding to F ∈ N amounts to the extraction of the relevant and marginal parts of F . We prove estimates relating F and its corresponding local polynomial, in terms of the Tφ semi-norm introduced in part I of the series.