An extremal problem for positive definite functions

Schoenberg’s Problem on Positive Definite Functions

An Extremal Problem for Characteristic Functions

Suppose E is a subset of the unit circle T and H∞ ⊂ L∞ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znH∞. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of...

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...

Some Extremal Problems for Positive Definite Matrices and Operators

Let C be a real-valued function defined on the set 9& of all positive definite complex hermitian or real symmetric matrices according as F = C (the complex field) or F = R (the real field). Suppose A, B E 9&. We study the optimization problems of (1) finding max C(X) subject to A X, B X positive semidefinite, (2) finding minG(X) subject to X A, X B positive semidefinite. For a general class of ...