  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.
