scholarly journals Infinitely divisible representations and positive definite functions on a compact group

1970 ◽  
Vol 16 (2) ◽  
pp. 148-156 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Compact Group ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Infinitely Divisible

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].

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.

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
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