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This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group S , i.e., the unit 3-sphere in the quaternion algebra. Secondly, we reveal a new phenomenon: The compact convex set of such functions is not a Bauer simplex except when G is of exponent ≤ 2 . In contrast, its complex counterpart is always a Bauer simplex, as is well known. We also present an integral representation for such functions as an application and some other minor results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a Delsarte-type extremal problem. Denote by P (G) the set of positive definite continuous functions on a locally compact abelian group G. We consider the function class, which was originally introduced by Gorbachev, G (W , Q) G = f ∈ P (G) ∩ L 1 (G) : f (0) = 1 , supp f + ⊆ W , supp f ^ ⊆ Q where W ⊆ G is closed and of finite Haar measure and Q ⊆ G ^ is compact. We also consider the related Delsarte-type problem of finding the extremal quantity D (W , Q) G = sup ∫ G f (g) d λ G (g) : f ∈ G (W , Q) G. The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem D (W , Q) G . The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where G = R d . So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész. [ABSTRACT FROM AUTHOR]

The century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$T(t)=a0+∑k=1nakcos(2πkt)+bksin(2πkt)⩾0, normalized by a0=1, where the quantity to be maximized is the coefficient a1 of cos (2π t). Carathéodory and Fejér found that for any given degree n, the maximum is 2 cos(π/n+2). In the complex exponential form, the coefficient sequence (ck) ⊂ ℂ will be supported in [−n, n] and normalized by c0=1. Reformulating, non-negativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c: ℤ → ℂ, supported in [−n, n]. Boas and Kac, Arestov, Berdysheva and Berens, Kolountzakis and Révész and, recently, Krenedits and Révész investigated the problem in increasing generality, reaching analogous results for all locally compact abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups. [ABSTRACT FROM PUBLISHER]

Suppose that f is the characteristic function of a probability measure on the real line $$\mathbb R$$ . In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part $$\mathfrak {I}f$$ ? In other words, is it true that for any characteristic function f there exists a characteristic function g such that $$\mathfrak {I}f\equiv \mathfrak {I}g$$ but $$ f\not \equiv g$$ ? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts. [ABSTRACT FROM AUTHOR]

In his seminal paper, Schoenberg (Duke Math J 9:96-108, 1942) characterized the class $$\mathcal P(\mathbb {S}^d)$$ of continuous functions $$f:[-1,1] \rightarrow \mathbb {R}$$ such that $$f(\cos \theta (\xi ,\eta ))$$ is positive definite on the product space $$\mathbb {S}^d \times \mathbb {S}^d$$ , with $$\mathbb {S}^d$$ being the unit sphere of $$\mathbb {R}^{d+1}$$ and $$\theta (\xi ,\eta )$$ being the great circle distance between $$\xi ,\eta \in \mathbb {S}^d$$ . In the present paper, we consider the product space $$\mathbb {S}^d \times G$$ , for G a locally compact group, and define the class $$\mathcal P(\mathbb {S}^d, G)$$ of continuous functions $$f:[-1,1]\times G \rightarrow \mathbb {C}$$ such that $$f(\cos \theta (\xi ,\eta ), u^{-1}v)$$ is positive definite on $$\mathbb {S}^d \times \mathbb {S}^d \times G \times G$$ . This offers a natural extension of Schoenberg's theorem. Schoenberg's second theorem corresponding to the Hilbert sphere $$\mathbb {S}^\infty $$ is also extended to this context. The case $$G=\mathbb {R}$$ is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of planet Earth. [ABSTRACT FROM AUTHOR]

Continuing research begun in [15] and [16] , we investigate the problem of when a closed subgroup H of a nilpotent locally compact group G has the property that every continuous positive definite function on H extends to some such function on G . The main result is an identification of all subgroups of the Heisenberg groups which share this property. [ABSTRACT FROM AUTHOR]

We prove that if ${\it\rho}$ is an irreducible positive definite function in the Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\Vert {\it\rho}\Vert _{B(G)}=1$, then the iterated powers $({\it\rho}^{n})$ as a sequence of unital completely positive maps on the group $C^{\ast }$-algebra converge to zero in the strong operator topology. [ABSTRACT FROM PUBLISHER]

Let $G={\mathcal{A}ut(\mathcal{T})}$ be the group of all automorphisms of a homogeneous tree $\mathcal{T}$ of degree q + 1 ≥ 3 and ( X, m) a compact metrizable measure space with a probability measure m. We assume that μ has no atoms. The group $\mathcal{G}={\mathcal{A}ut(\mathcal{T})}^X=G^X$ of bounded measurable currents is the completion of the group of step functions $f:X\to{\mathcal{A}ut(\mathcal{T})}$ with respect to a suitable metric. Continuos functions form a dense subgroup of $\mathcal{G}$. Following the ideas of I.M. Gelfand, M.I. Graev and A.M. Vershik we shall construct an irreducible family of representations of $\mathcal{G}$. The existence of such representations depends deeply from the nonvanisching of the first cohomology group $H^1({\mathcal{A}ut(\mathcal{T})},\pi)$ for a suitable infinite dimensional π. [ABSTRACT FROM AUTHOR]

The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries. [ABSTRACT FROM AUTHOR]