Nonstandard Proofs of Herglotz, Bochner and Bochner-Minlos Theorems

We describe a unified approach to Herglotz, Bochner and BochnerMinlos theorems using a combination of Daniell integral and nonstandard analysis. The proofs suggest a natural extension of the last two theorems to the case when the characteristic function is not continuous. This extension is proven and is demonstrated to be the best one possible. The goal of this paper is to show how the classic theorems of Herglotz, Bochner and Bochner-Minlos are in fact simple consequences of the trivial discrete case, i.e. the obvious analogue of these theorems for functions on Zn. It turns out that Herglotz follows immediately (Theorem 1), while Bochner and Bochner-Minlos require a little more work for topological reasons (Theorems 2 and 4). We shall use a combination of Daniell integral and nonstandard analysis to demonstrate the claims above1. We shall also show that Bochner and Bochner-Minlos theorems can be extended to the case when the characteristic function is not continuous (in the Herglotz case, since the domain has discrete topology, no function is discontinuous), and shall prove that this extension is the best one can hope for (Theorem 3). Let us start with the simplest case, the discrete one: Lemma: Let {μ̂(k)}N ′ k=−N ′ be a positive-definite collection of complex numbers2, such that μ̂(0) = 1, then there is a positive collection of real numbers {μ(z)}N ′ z=−N ′ such that μ̂(k) = 1 N N ′ ∑ z=−N ′ e−2πi kz N μ(z), (1) 2013 Mathematics Subject Classification 28E05, 42A82.