The sup-interpretation method is proposed as a new tool to control memory resources of first order functional programs with pattern matching by static analysis. It has beeen introduced in order to increase the intensionality, that is the number of captured algorithms, of a previous method, the quasi-interpretations. Basically, a sup-interpretation provides an upper bound on the size of function outputs. A criterion, which can be applied to terminating as well as non-terminating programs, is developed in order to bound the stack frame size polynomially. Since this work is related to quasi-interpretation, dependency pairs and size-change principle methods, we compare these notions obtaining several results. The first result is that, given any program, we have heuristics for nding a sup-interpretation when we consider polynomials of bounded degree. Another result consists in the characterizations of the sets of functions computable in polynomial time and in polynomial space. A last result consists in applications of sup-interpretations to the dependency pair and the size-change principle methods.