Strictly and Strongly Positive Definite Functions on Locally Compact Hypergroups

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
László Székelyhidi ◽  
Seyyed Mohammad Tabatabaie ◽  
Kedumetse Vati
Keyword(s):  
Positive Definite ◽  
Positive Definite Functions ◽  
Locally Compact

Bochner's Theorem and the Hausdorff Moment Theorem on foundation Topological Semigroups

1985 ◽  
Vol 37 (5) ◽  
pp. 785-809 ◽  
Author(s):  
M. Lashkarizadeh Bami
Keyword(s):  
Harmonic Analysis ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Commutative Semigroups ◽  
Bochner’S Theorem ◽  
Topological Semigroups ◽  
Bochner's Theorem ◽  
Extensive Class

One of the most basic theorems in harmonic analysis on locally compact commutative groups is Bochner's theorem (see [16, p. 19]). This theorem characterizes the positive definite functions. In 1971, R. Lindhal and P. H. Maserick proved a version of Bochner's theorem for discrete commutative semigroups with identity and with an involution * (see [13]). Later, in 1980, C. Berg and P. H. Maserick in [6] generalized this theorem for exponentially bounded positive definite functions on discrete commutative semigroups with identity and with an involution *. In this work we develop these results, and also the Hausdorff moment theorem, for an extensive class of topological semigroups, the so-called “foundation topological semigroups”. We shall give examples to show that these theorems do not extend in the obvious way to general locally compact topological semigroups.

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).

scholarly journals On Positive Definiteness Over Locally Compact Quantum Groups

2016 ◽  
Vol 68 (5) ◽  
pp. 1067-1095 ◽  
Author(s):  
Volker Runde ◽  
Ami Viselter
Keyword(s):  
Quantum Groups ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Separation Property ◽  
Locally Compact ◽  
Square Roots ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Quantum Framework

AbstractThe notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on “square roots” of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

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].

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