TURÁN'S EXTREMAL PROBLEM FOR POSITIVE DEFINITE FUNCTIONS ON GROUPS

Schoenberg’s Problem on Positive Definite Functions

Extensions of Positive Definite Functions on Amenable Groups

Let S be a subset of a amenable group G such that e ∈ S and S = S. The main result of the paper states that if the Cayley graph of G with respect to S has a certain combinatorial property, then every positive definite operator-valued function on S can be extended to a positive definite function on G. Several known extension results are obtained as a corollary. New applications are also presented.

Extensions of Positive Definite Functions on Free Groups

An analogue of Krein’s extension theorem is proved for operatorvalued positive definite functions on free groups. The proof gives also the parametrization of all extensions by means of a generalized type of Szegö parameters. One singles out a distinguished completion, called central, which is related to quasi-multiplicative positive definite functions. An application is given to factorization o...

On the Extremal Rays of the Cone of Positive, Positive Definite Functions

The aim of this paper is to investigate the cone of non-negative, radial, positivedefinite functions in the set of continuous functions on R. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, t...

On a Problem of Turán about Positive Definite Functions

We study the following question posed by Turán. Suppose Ω is a convex body in Euclidean space Rd which is symmetric in Ω and with value 1 at the origin; which one has the largest integral? It is probably the case that the extremal function is the indicator of the half-body convolved with itself and properly scaled, but this has been proved only for a small class of domains so far. We add to thi...