On functions of positive type on commutative monoids

1977 ◽  
Vol 156 (2) ◽  
pp. 115-121 ◽  
Author(s):  
Walter Schempp
Keyword(s):  
Commutative Monoids ◽  
Positive Type ◽  
Functions Of Positive Type

Olshanski spherical pairs of semigroups type

2019 ◽  
Vol 22 (03) ◽  
pp. 1950021
Author(s):  
Mohamed Bouali
Keyword(s):  
Integral Representation ◽  
Unitary Group ◽  
Inductive Limit ◽  
Complete Classification ◽  
Spherical Functions ◽  
Hermitian Matrices ◽  
Positive Type ◽  
Negative Type ◽  
Functions Of Positive Type

Let [Formula: see text] be the infinite semigroup, inductive limit of the increasing sequence of the semigroups [Formula: see text], where [Formula: see text] is the unitary group of matrices and [Formula: see text] is the semigroup of positive hermitian matrices. The main purpose of this work is twofold. First, we give a complete classification of spherical functions defined on [Formula: see text], by following a general approach introduced by Olshanski and Vershik.10 Second, we prove an integral representation for functions of positive-type analog to the Bochner–Godement theorem, and a Lévy–Khinchin formula for functions of negative type defined on [Formula: see text].

scholarly journals Linear transformations and functions of positive type

1921 ◽  
Vol 99 (696) ◽  
pp. 19-38
Keyword(s):  
Symmetric Function ◽  
Orthogonal System ◽  
Linear Transformations ◽  
Positive Type ◽  
First Case ◽  
Functions Of Positive Type

In a communication received by the Society on April 12 last I gave in outline a proof that a symmetrisable function k ( s, t ), which belongs to one of certain classes, admits an expansion in terms of its bi-orthogonal fundamental functions. Two classes were explicitly mentioned, viz., (i) the functions defined by k ( s, t ) = α( s ) ϒ ( s, t ), and (ii) those defined by k ( s, t ) = ∫ a b α( s, x ) ϒ ( x, t ) dx . It was assumed in each case that ϒ( s, t ) was of positive type in the square a ≤ s ≤ b , a ≤ t ≤ b ; in the first case, that α( s ) was a function defined in the interval ( a, b ); and, in the second case, that α ( s, t ) was a symmetric function defined in the square a ≤ s ≤ b , a ≤ t ≤ b . When k ( s, t ) belongs to either of the classes mentioned and certain conditions of continuity are satisfied it was shown that k ( s, t ) = Σ n=1 ⌽ n ( s ) Ψ n ( t )/λ n + Σ n=1 ξ n ( s ) η n ( t )/ v n , where (i) ⌽ 1 ( s ), ⌽ 2 ( s ), ... ⌽ n ( s ), ... Ψ 1 ( s ), Ψ 2 ( s ), ... Ψ n ( s ), ... is a bi-orthogonal system of fundamental functions for the interval ( a, b ), and ⌽ n ( s ), Ψ n ( t ) satisfy the homogeneous equations ⌽ n ( s ) = λ n ∫ a b k ( s, t ) ⌽ n ( t ) dt , Ψ n ( t ) = λ n ∫ a b Ψ n ( s ) k ( s, t ) ds ;}

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