scholarly journals ON ITERATED POWERS OF POSITIVE DEFINITE FUNCTIONS

2015 ◽  
Vol 92 (3) ◽  
pp. 440-443
Author(s):  
MEHRDAD KALANTAR
Keyword(s):  
Compact Group ◽  
Strong Operator Topology ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Definite Function ◽  
Completely Positive Maps ◽  
Locally Compact ◽  
Completely Positive ◽  
Positive Maps

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.

On Positive Definite Functions over a Locally Compact Group

1970 ◽  
Vol 22 (4) ◽  
pp. 892-896 ◽  
Author(s):  
J. F. Price
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Complete Characterization ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Definite Function ◽  
Locally Compact ◽  
Main Result Theorem

In this note we are concerned with several questions on positive definite functions over a Hausdorff locally compact group. The main result, Theorem A, gives some necessary and sufficient conditions for to be a positive definite function when μ is a (complex Radon) measure. In particular, is a positive definite function if and only if μ ∊ L2, and Theorem B then follows by giving a complete characterization of functions of the type , where f ∊ L2. Perhaps the most interesting aspect of these results is that they provide further examples of results over a non-abelian, non-compact group, which otherwise are simple consequences (with μ, a bounded measure in Theorem A) of the theorems of Plancherel and Bochner.Unless otherwise specified, all notation and definitions will follow [1;2]. The underlying group will always be G, a Hausdorff locally compact group with identity e, and with left Haar measure dx.

scholarly journals The Point Value Maximization Problem for Positive Definite Functions Supported in a Given Subset of a Locally Compact Group

2018 ◽  
Vol 61 (1) ◽  
pp. 179-200
Author(s):  
Sándor Krenedits ◽  
Szilárd Gy. Révész
Keyword(s):  
Extremal Problem ◽  
Positive Definite Function ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Maximization Problem ◽  
Definite Function ◽  
Locally Compact ◽  
Value Maximization ◽  
Compact Abelian Groups ◽  
Complex Exponential

AbstractThe century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$, 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.

scholarly journals On Godement's characterisation of amenability

1998 ◽  
Vol 57 (1) ◽  
pp. 153-158 ◽  
Author(s):  
Alain Valette
Keyword(s):  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Groups ◽  
Definite Function ◽  
Compact Groups ◽  
Locally Compact ◽  
Unimodular Group ◽  
Compactly Supported ◽  
Assembly Map

Motivated by a question related to the construction of the Baum-Connes analytical assembly map for locally compact groups, we refine a criterion of Godement for amenability: for a unimodular group G, our criterion says that G is amenable if and only if every compactly supported, positive-definite function has non-negative integral over G.

On various types of convergence of positive definite functions on foundation semigroups

1992 ◽  
Vol 111 (2) ◽  
pp. 325-330 ◽  
Author(s):  
M. Lashkarizadeh-Bami
Keyword(s):  
Compact Group ◽  
Pointwise Convergence ◽  
Positive Definite ◽  
Limit Problem ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Interesting Discussion ◽  
Central Limit Problem ◽  
The Relationship

As is known, on a locally compact group G, the mere assumption of pointwise convergence of a sequence (n) of continuous positive definite functions implies uniform convergence of (n) to on compact subsets of G. This result was first proved in 1947 by Raikov8 (and independently by Yoshizawa9). An interesting discussion of the relationship between such theorems and various Cramr-Lvy theorems of the 1920s and 1930s, concerning the Central Limit Problem of probability, is given by McKennon(7, p. 62).

On a question of R. Godement about the spectrum of positive, positive definite functions

1991 ◽  
Vol 110 (1) ◽  
pp. 137-142
Author(s):  
Mohammed B. Bekka
Keyword(s):  
Compact Group ◽  
Uniform Convergence ◽  
Unitary Representation ◽  
Convex Set ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Image Position ◽  
Compact Subsets

Let G be a locally compact group, and let P(G) be the convex set of all continuous, positive definite functions ø on G normalized by ø(e) = 1, where e denotes the group unit of G. For ø∈P(G) the spectrum spø of ø is defined as the set of all indecomposable ψ∈P(G) which are limits, for the topology of uniform convergence on compact subsets of G, of functions of the form(see [5], p. 43). Denoting by πø the cyclic unitary representation of G associated with ø, it is clear that sp ø consists of all ψ∈P(G) for which πψ is irreducible and weakly contained in πø (see [3], chapter 18).

scholarly journals Positive Definite Functions for the Class Lp(G)

1975 ◽  
Vol 27 (5) ◽  
pp. 1149-1156
Author(s):  
T. Husain ◽  
S. A. Warsi
Keyword(s):  
Compact Group ◽  
Relevant Information ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Topological Groups ◽  
Locally Compact

There are several notions of positive definiteness for functions on topological groups, the two of which are: Bochner type positive definite functions and integrally positive definite functions. The class P(F) of positive definite functions for the class F can be defined more generally and it is interesting to observe that a change in F produces a different class P(F) of positive definite functions. The purpose of this paper is to study the functions in P(LP(G)) which are positive definite for the class LP(G) (1 ≦ p < ∞), where G is a compact or locally compact group. The relevant information about the class P(F) can be found in [1; 2; 3 and 8].

scholarly journals Interpolation in wavelet spaces and the HRT-conjecture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Eirik Berge
Keyword(s):  
Compact Group ◽  
Wavelet Transforms ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Space Structure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Positive Type ◽  
Time Frequency ◽  
Hrt Conjecture

AbstractWe investigate the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })\subset L^{2}(G)$$ W g ( H π ) ⊂ L 2 ( G ) arising from square integrable representations $$\pi :G \rightarrow \mathcal {U}(\mathcal {H}_{\pi })$$ π : G → U ( H π ) of a locally compact group G. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong restrictions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time–frequency analysis, this problem turns out to be equivalent to the HRT-conjecture. Finally, we consider the problem of whether all the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })$$ W g ( H π ) of a locally compact group G collectively exhaust the ambient space $$L^{2}(G)$$ L 2 ( G ) . We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.

The Superstability of the Generalized D'alembert Functional Equation

2003 ◽  
Vol 10 (3) ◽  
pp. 503-508 ◽  
Author(s):  
Elhoucien Elqorachi ◽  
Mohamed Akkouchi
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Compact Group ◽  
Locally Compact Group ◽  
Complex Numbers ◽  
Locally Compact

Abstract We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

Sign in / Sign up

Export Citation Format

Share Document