On pointwise estimates of positive definite functions with given support

The following problem originated from a question due to Paul Turán. Suppose Ω is a convex body in Euclidean space R or in T, which is symmetric about the origin. Over all positive definite functions supported in Ω, and with normalized value 1 at the origin, what is the largest possible value of their integral? From this Arestov, Berdysheva and Berens arrived to pose the analogous pointwise extremal problem for intervals in R. That is, under the same conditions and normalizations, and for any particular point z ∈ Ω, the supremum of possible function values at z is to be found. 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 R and non-convex domains as well. We present another approach to the problem, giving the solution in R and for several cases in T. In fact, we elaborate on the fact that the problem is essentially one-dimensional, and investigate nonconvex open domains as well. We show that the extremal problems are equivalent to more familiar ones over trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relation of the problem for the space R to that for the torus T is given, showing that the former case is just the limiting case of the latter. Thus the hiearachy of difficulty is established, so that trigonometric polynomial extremal problems gain recognition again. MSC 2000 Subject Classification. Primary 42B10 ; Secondary 26D15, 42A82, 42A05.