For a general vector field we exhibit two Hilbert spaces, namely the space of so called \emph{closed functions} and the space of \emph{exact functions} and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field.

The following problem has been suggested by Paul Tur\' an. Let $\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$ or in the torus $\TT^d$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega$ and normalized with the value $1$ at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\in\Omega$. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\RR^d$ and to non-convex domains as well. Here we present another approach to the problem, giving the solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for $\RR^d$ and that for $\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.