A note on the representation of positive polynomials with structured sparsity

We consider real polynomials in finitely many variables. Let the variables consist of finitely many blocks that are allowed to overlap in a certain way. Let the solution set of a finite system of polynomial inequalities be given where each inequality involves only variables of one block. We investigate polynomials that are positive on such a set and sparse in the sense that each monomial involves only variables of one block. In particular, we derive a short and direct proof for Lasserre's theorem of the existence of sums of squares certificates respecting the block structure. The motivation for the results can be found in the literature and stems from numerical methods using semidefinite programming to simulate or control discrete-time behaviour of systems.