in this paper we propose an unified construction of lyapunov functions based on lur'e type functions that allows us to go from persidskii's to pure quadratic functions. such lyapunov functions allows the establishment of the absolute stability of nonlinear systems with nonlinearities belonging to given sectors. the sectors can be finite or infinite. an stability criterion is formulated in terms of linear matrix inequalities (lmi). the parameters that define the sectors are present in the given lmi and are made available for optimization, leading to several optimization problems that are able to determine the available robustness levels for several sector configurations.