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1. CJM 2008 (vol 60 pp. 685)

Savu, Anamaria
 Closed and Exact Functions in the Context of Ginzburg--Landau Models 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. Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetriesCategories:42B05, 81Q50, 42A16

2. CJM 2006 (vol 58 pp. 401)

Kolountzakis, Mihail N.; Révész, Szilárd Gy.
 On Pointwise Estimates of Positive Definite Functions With Given Support 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. Keywords:Fourier transform, positive definite functions and measures, TurÃ¡n's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problemsCategories:42B10, 26D15, 42A82, 42A05
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