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Abstract

Moments of probability measures on a hypergroup can be obtained from so-called (generalized) moment functions of a given order. The aim of this paper is to characterize generalized moment functions on a non-commutative affine group. We consider a locally compact group G and its compact subgroup K. First we recall the notion of the double coset space G / / K of a locally compact group G and introduce a hypergroup structure on it. We present the connection between K-spherical functions on G and exponentials on the double coset hypergroup G / / K. The definition of the generalized moment functions and their connection to the spherical functions is discussed. We study an important class of double coset hypergroups: specyfing K as a compact subgroup of the group of inverible linear transformations on a finitely dimensional linear space V we consider the affine group \({\mathrm {Aff\,}}K\). Using the fact that in the finitely dimensional case \(({\mathrm {Aff\,}}K,K)\) is a Gelfand pair we give a description of exponentials on the double coset hypergroup \({\mathrm {Aff\,}}K//K\) in terms of K-spherical functions. Moreover, we give a general description of generalized moment functions on \({\mathrm {Aff\,}}K\) and specific examples for \(K=SO(n)\), and on the so-called \(ax+b\)-group.
Results Math (2019) 74:5
c
2018 The Author(s)
1422-6383/19/010001-13
published online November 17, 2018
https://doi.org/10.1007/s00025-018-0926-2 Results in Mathematics
Moment Functions on Affine Groups
˙
Zywilla Fechner and L´aszl´oSz´ekelyhidi
Abstract. Moments of probability measures on a hypergroup can be ob-
tained from so-called (generalized) moment functions of a given order.
The aim of this paper is to characterize generalized moment functions on
a non-commutative affine group. We consider a locally compact group G
and its compact subgroup K. First we recall the notion of the double coset
space G//K of a locally compact group Gand introduce a hypergroup
structure on it. We present the connection between K-spherical functions
on Gand exponentials on the double coset hypergroup G//K. The def-
inition of the generalized moment functions and their connection to the
spherical functions is discussed. We study an important class of double
coset hypergroups: specyfing Kas a compact subgroup of the group of
inverible linear transformations on a finitely dimensional linear space V
we consider the affine group Aff K. Using the fact that in the finitely
dimensional case (Aff K, K) is a Gelfand pair we give a description of
exponentials on the double coset hypergroup Aff K//K in terms of K-
spherical functions. Moreover, we give a general description of generalized
moment functions on Aff Kand specific examples for K=SO(n), and
on the so-called ax +b-group.
Mathematics Subject Classification. 20N20, 43A62, 39B99.
Keywords. Hypergroup, generalized moment function, affine group,
spherical functions.
1. Introduction
In this paper we shall consider the generalized moment functions on some type
of hypergroups called affine groups (see [3] for details). By a hypergroup we
mean a locally compact hypergroup. The identity element of the hypergroup
Kis denoted by o.Forxin Kthe symbol δxdenotes the point mass with
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5Page 2 of 13 ˙
Z. Fechner and L. Sz´ekelyhidi Results Math
support {x}. In the sequel Cdenotes the set of complex numbers. The classical
monograph on hypergroups is [3]. A comprehensive discussion on the subject
is in [14] and reference therein.
Let Kbe a hypergroup. We begin with recalling the notion of exponen-
tial, additive and m-sine functions on hypergroups which are strongly con-
nected with the generalized moment functions. The non-identically zero con-
tinuous function m:KCis called an exponential on Kif msatisfies
m(xy)=m(x)m(y)foreachx, y in K. The continuous function a:KC
is called additive function if it satisfies a(xy)=a(x)+a(y)foreachx, y
in K. The description of exponentials and additive functions on some types
of commutative hypergroups can be found in [14]. The continuous function
f:KCis called an m-sine function if it satisfies
f(xy)=f(x)m(y)+f(y)m(x) (1)
for each x, y in K. The function fis called a sine function if fis an m-
sine function for some exponential m. Clearly, every sine function fsatisfies
f(o) = 0. For a given exponential mall m-sine functions form a linear space.
Obviously, m1 is an exponential on any hypergroup, and 1-sine functions
are exactly the additive functions. The description of sine functions on some
types of hypergroups can be found in [5].
Let Gbe a locally compact group with identity eand Ka compact
subgroup with the normalized Haar measure ω.AsKis unimodular ωis left
and right invariant, and it is also inversion invariant. For each xin Gwe define
the double coset of xas the set
KxK ={kxl :k, l K}.
Let
L:= G//K ={KxK :xG}
be the set of all double cosets. We introduce a hypergroup structure on the
set Lof all double cosets: the topology of Lis the factor topology, which is
locally compact. The identity is the coset K=KeK itself and the involution
is defined by
(KxK)=Kx1K
for xin G. Finally, the convolution of δKxK and δKyK is defined by
δKxK δKyK =K
δKxkyK (k),
where xand yare in G. It is known that this gives a hypergroup structure
on L(see [3], p. 12.), which is non-commutative, in general. If Kis a normal
subgroup, then Lis isomorphic to the hypergroup arising from the factor group
G/K.
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In [8] representations of double coset hypergroups are investigated. Such
representations can be canonically obtained from those of the group in ques-
tion. Nevertheless, not every representation arises in this way. However, our
results show that on some affine groups, basic representing functions, like ex-
ponentials, additive functions and quadratic functions are closely related to
the corresponding functions on the original group.
We note that continuous functions on Lcan be identified with those
continuous functions on Gwhich are K-invariant: f(x)=f(kxl)foreachxin
Gand k, l in K. Hence, for a continuous function f:LCthe simplified – and
somewhat loose – notation f(x) can be used for the function value f(KxK).
Using this convention we can write for each continuous function f:LC
and for each x, y in G:
f(xy)=K
f(xky)(k).
The following theorem exhibits a close connection between exponen-
tials on double coset hypergroups and spherical functions on locally compact
groups. Following the terminology of [3](seealso[4]) we recall the concept of
spherical functions. The continuous bounded K-invariant function f:GC
is called a K-spherical function if f(e)=1and
K
f(xky)(k)=f(x)f(y) (2)
holds for each x, y in G.Ageneralized K-spherical function on Gis the same
as above without the boundedness hypothesis. For the sake of simplicity in this
paper we use the term spherical function for continuous functions satisfying
(2) without the boundedness assumption. The following theorem, which is an
immediate consequence of the previous considerations gives the link between
spherical functions and exponentials of double coset hypergroups.
Theorem 1. Let Gbe a loca l ly compact group, and KGa compac t subgroup
of G. Then a nonzero continuous complex valued function mis a K-spherical
function on Gif and only if it is an exponential on the double coset hypergroup
G//K. In particular, K-spherical functions on Gcan be identified with the
multiplicative functions on G//K.
Here a multiplicative function means an exponential map m:G//K C
such that m(e) = 1. See [3, Def 1.4.32]
It is obvious, that if Gis a locally compact Abelian group, and Kis a
compact subgroup of G, then K-spherical functions are exactly the exponen-
tials of the (locally compact) Abelian group G/K .
Given the locally compact group Gand the compact subgroup Kthe
space Mc(G) of compactly supported regular complex Borel measures on G
is identified with the topological dual of the locally convex topological vector
space C(G) of all continuous complex valued functions on Gwhen equipped
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5Page 4 of 13 ˙
Z. Fechner and L. Sz´ekelyhidi Results Math
with the topology of uniform convergence on compact sets. The pairing be-
tween C(G)andMc(G) will be denoted by ·,·, hence
μ, f =G
fdμ
holds for each μin Mc(G)andfin C(G). All K-invariant functions in C(G)
form a closed subspace, which is identified with C(G//K), the locally convex
topological vector space of all continuous complex valued functions on the
double coset space G//K. The topological dual of C(G//K) is identified with
the space Mc(G//K) of all K-invariant measures on G– the measure μin
Mc(G) is called K-invariant if
μ, f =GKK
f(kxl)(k)(l)(x)
holds for each fin C(G). Using the hypergroup structure on L=G//K we see
easily that Mc(G//K) is a complex unital topological algebra with unit δe,
the point mass with support {e}. We say that (G, K)isaGelfand pair if this
algebra is commutative (see e.g. [4,16]). It turns out that (G, K) is a Gelfand
pair if and only if the double coset hypergroup Lis commutative ([16]).
A sequence (gn)1nNof continuous K-invariant functions on Gis a
generalized K-moment function sequence if they satisfy
K
gn(xky)(k)=
n
j=0 n
jgj(x)gnj(y) (3)
for n=0,1,...,N and x, y in G. In fact, this is exactly the case if (gn)1nN
is a generalized moment function sequence on the double coset hypergroup
G//K. In particular, g0is a K-spherical function on G, or what is the same,
g0is an exponential on G//K. In the case K={e}it is clear that this concept
coincides with the concept of a generalized moment function sequence.Inthe
case g01 the first order moment functions are exactly the additive functions,
which, for general K, can be called K-additive functions. In the generalized
moment function sequence (gn)1nNwith N1 the function g1is a g0-sine
function, according to the terminology introduced in [7](seealso[5]), which, in
the general case can be called K-sine functions associated with the K-spherical
function g0. Generalized moment functions and generalized moment function
sequences have been described on different types of commutative hypergroups
(see e.g. [1114]). Nevertheless, non-commutative hypergroups have not yet
been considered in this context.
2. Affine Groups
An important type of double coset hypergroups arises form the concept of affine
groups. Let Vbe an n-dimensional vector space over the field Kand let GL(V)
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denote the general linear group of V, the invertible linear transformations on
V. For each subgroup Hof GL(V)weformthesemidirect product
Aff H=HV
in the following way: we equip the set H×Vwith the following multiplication:
(x, u)·(y, v)=(x·y, x ·v+u)
for each x, y in Hand u, v in V. Here x·vis the image of vunder the linear
mapping x. Then Aff (H) is a group with identity (id, 0), where id is the
identity operator and 0 is the zero of the vector space V.Theinverseof(x, u)
is
(x, u)1=(x1,x1·u)
for each xin Hand uin V. We note that, in general, Aff His non-commutative,
even if His commutative. In any case V– as a commutative group – is iso-
morphic to the normal subgroup of Aff Hconsisting of all elements of the form
(id, u) with uis in Vthe isomorphism provided by the mapping u→ (id, u).
Indeed, we have
(x, u)·(id, v)·(x, u)1=(id, x ·v),
which proves that the image of V, which we identify with V, is normal. The set
of all elements of the form (x, 0) with xin His a subgroup of Aff Hisomorphic
to H, and it will be identified with H. Affine groups play an important role in
geometry and physics. For instance, the Poincar´egroupAff O(1,3) is the affine
group of the Lorentz group O(1,3): O(1,3)R1,3, where O(1,3) is the isometry
group of the real vector space R1,3=RR3equipped with the quadratic form
v, w=
p
t=1
vtwt
p+q
t=p+1
vtwt,
where v=(v1,...,v
p,v
p+1,...,v
p+q)andw=(w1,...,w
p,w
p+1,...,w
p+q).
For more about affine groups and geometry see e.g. [2,10].
In the case V=Knwe have GL(V)=GL(K,n), which is a matrix
group. With the usual Euclidean topology Vand GL(V) are locally compact
topological groups. If His a subgroup of GL(V) a function f:VCis called
H-invariant if f(h·v)=f(v) holds for each hin Hand vin V. Suppose that
His a closed subgroup of GL(V)andKis a compact subgroup in H. Then
K– identified with the set of all elements of the form (x, 0) with xin K
is a compact subgroup in HV.LetLdenote the double coset hypergroup
Aff H//K.AswehaveseenaboveK-invariant continuous functions on Aff H
can be identified with the continuous functions on L– we use the notation
f(x, u)forfK(x, u)K. Using this notation K-invariance means f(kx, lu)=
f(x, u)foreachk, l in K,xin Hand uin V.Ifdenotes convolution in L
then we have
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Z. Fechner and L. Sz´ekelyhidi Results Math
f(x, u)(y, v)=K
f(xky, xk v +u)(k),
for each x, y in Hand u, v in V, where ωis the normalized Haar measure on
K.
A basic observation about affine groups is the following theorem.
Theorem 2. Let Vbe a finite dimensional vector space and Ka compact sub-
group of GL(V).Then(Aff K, K )is a Gelfand pair.
For the sake of completeness we present here the proof of this statement
which is based on the following theorem of Gelfand (see e.g. [4,17]).
Theorem 3. (Gelfand) Let Gbe a locally compact group and Kacompact
subgroup of G. Suppose that there exists a continuous involutive automorphism
θ:GGsuch that θ(x)is in Kx1Kholds for each xin G.Then(G, K )is
a Gelfand pair.
Proof (Proof of Theorem 2). We define θ:AKAff Kas
θ(k, u)=(k, u)
for each kin Kand uin V. Then clearly θis a continuous involutive auto-
morphism of Aff K. On the other hand, we can write
(k, u)=(k, 0) ·(k1,k1u)·(k, 0),
as it is easy to verify.
Our purpose is to describe generalized moment functions on double coset
hypergroups. We shall consider the case of double coset hypergroups of affine
groups. Recall that a continuous K-invariant function on Vsatisfying
K
ϕ(k·u+v)(k)=ϕ(u)ϕ(v) (4)
for each u, v in Vis called K-spherical function on V.
The next theorem is a generalization of [6, Theorem 2.1].
Theorem 4. Let (Aff H, K)be a Gelfand pair. Then a continuous K-invariant
function m:AHCis an exponential on Aff Hif and only if it has the
form
m(x, u)=e(x)ϕ(u)
for each xin Hand uin V,wheree:HCand ϕ:VCare continuous
K-invariant functions satisfying
K
e(xky)ϕ(xkv +u)(k)=e(x)e(y)ϕ(u)ϕ(v) (5)
for each x, y in Hand u, v in V. In particular, eis a K-spherical function
on H:
K
e(xky)(k)=e(x)e(y)for each x, y H,
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and ϕ:VCis a K-spherical functions on V.
Proof. Observe that
(x, u)(y, v)=K
δ(x,u)δ(k,0) δ(y,v)(k)=K
δ(xky,xkv +u)(k)
for each x, y in Hand u, v in V, hence for every continuous K-invariant func-
tion f:AHCit follows
K
f(xky, xk v +u)(k)=f(x, u)(y, v)
=f(y, v)(x, u)=K
f(ykx, yku +v)(k) (6)
whenever x, y is in Hand u, v is in V. In particular, putting y=id and u=0
in (6)wehave
K
f(x, xkv)(k)=f(x, v) (7)
for each K-invariant continuous function f:LCand xin H,vin V.
Now assume that mis an exponential on L. Then it satisfies
K
m(xky, xk v +u)(k)=m(x, u)·m(y, v) (8)
whenever x, y are in Hand u, v are in V.Foreachkin Kwe have
m(x, 0)m(id, u)=m(x, 0) ·m(id, ku)=m(x, xku),
and, by integrating
m(x, 0)m(id, u)=K
m(x, xku)(k)=m(x, u),(9)
by (7). Denoting e(x)=m(x, 0) and putting u=v=0in(8)wehave
K
e(xky)(k)=e(x)e(y)
as it was stated. Similarly, denoting ϕ(u)=m(id, u) and putting x=y=id
in (8) we conclude
K
ϕ(kv +u)(k)=ϕ(u)ϕ(v).
Finally, substitution into (9) and (8) gives (5).
Conversely, if mhas the desired form then (5) implies that mis expo-
nential.
Lemma 1. Let Vbe a finite dimensional K-vector space and Kacompact
subgroup of GL(V). Then a continuous function f:AKCis K-invariant
if and only if it has the form f(k, u)=ϕ(u)for each kin Kand uin V,where
ϕ:VCis a continuous K-invariant function, i.e. ϕ(u)=ϕ(k·u)holds for
each kin Kand uin V.
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Z. Fechner and L. Sz´ekelyhidi Results Math
Functions ϕwith this property are called K-radial functions on V.
Proof. By the K-invariance of fwe have for each (k, u)inAKand lin K
f(k, u)=f[(l1,0) ·(id, l ·u)·(lk, 0)] = f(id, l ·u),
as (l1,0) and (lk, 0) are in K.
Corollary 1. Let Vbe a finite dimensional K-vector space and Kacompact
subgroup of GL(V). Then a continuous K-invariant function m:AKC
is an exponential on Aff K//K if and only if it has the form
m(k, u)=ϕ(u)
for each kin Kand uin V,whereϕ:VCis a K-spherical function.
As an example we consider the case V=Rnand K=SO(n), the special
orthogonal group on Rn. It is known (see e.g. [17]) that the SO(n)-spherical
functions on Rnare exactly the radial eigenfunctions of the Laplacian in Rn,
hence they can be described in terms of the Bessel functions Jλin the form
ϕλ(u)=Jλ(u)
for each uin Rnand complex number λ, where
Jλ(r)=Γn
2
k=0
λk
kk+n
2r
22k
.
It follows that the function (k, u,λ)→ Φ(k, u, λ) defined by
Φ(k, u, λ)=Jλ(u)
represents an exponential family for the double coset hypergroup
L=ASO(n)//SO(n).
In particular, for n=3wehave
Φ(k, u, λ)=sinh λu
λu
whenever uis in R3and λis in C, where in the case λu= 0 we take the
limit, which gives the spherical function identically 1.
Suppose that Vis a finite dimensional K-vector space, His a closed
subgroup of GL(V)andKis a compact subgroup of Hwith the normalized
Haar measure ω. Specializing the concept of generalized K-moment function
sequence to the case G=HVwith the compact subgroup {(k, 0) : k
K}
=Kwe can rewrite the system of Eq. (3) in the following way:
K
gn(xky, xk ·v+u)(k)=
n
j=0 n
jgj(x, u)gnj(y, v) (10)
which holds for n=0,1,...,N and for each (x, u),(y, v)inG. Here we sup-
pose that the functions gn:GCare continuous and K-invariant for
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Vol. 74 (2019) Moment Functions on Affine Groups Page 9 of 13 5
n=0,1,...,N. In terms of the double coset hypergroup L=G//K this
is exactly the system of Eq. (3).
In the case H=Kwe have the following theorem.
Theorem 5. Let Vbe a finite dimensional K-vector space and let Kbe a com-
pact subgroup of GL(V). Then a sequence (gn)0nNof K-invariant continu-
ous complex functions on Aff Kis a generalized moment function sequence if
and only if it has the form
gn(x, u)=ϕn(u)(n=0,1...,N) (11)
for each (x, u)in Aff K,whereϕn:VCis continuous and K-invariant,
further we have
K
ϕn(k·u+v)(k)=
n
j=0 n
jϕj(u)ϕnj(v) (12)
for n=0,1...,N and for each u, v in V.
Proof. By Lemma 1,wehavethatgn(x, u)=ϕn(u) holds for each (x, u)in
Aff K, where ϕn:VCis continuous and K-invariant (n=0,1,...,N).
Substitution into (10) gives
K
ϕn(xk ·v+u)(k)=
n
j=0 n
jϕj(u)ϕnj(v) (13)
for each xin Kand u, v in V. By the translation invariance of ωand inter-
changing uand v, we obtain the system (12).
The converse statement follows by direct calculation.
In general, it is not easy to describe all solutions of the functional equation
system (12). In the next section we shall give an example where this is possible.
Now we are going to use the form of K-spherical functions for a certain compact
subgroup Kof GL(Rn). More about spherical functions on Euclidean spaces
from the viewpoint of Riemannian symmetric spaces can be found in [18]and
reference therein. Using results from [18] one can seen that similar examples
can be performed using any compact subgroup of O(n) which is transitive on
the spheres around the origin. We restrict ourselves to the case of V=Rn
and K=SO(n). We have seen above that the K-spherical functions, that is
the generalized K-moment functions can be expressed in terms of the Bessel
functions in the form
ϕ0(u, λ)=Jλ(u)
for each uin Rn, where λis a complex number. On the other hand, by Theorem
3.4 and Corollary 2 in [15], it follows that ϕnis a linear combination of the
derivatives of λ→ ϕ0(u, λ)uptothen-th order. In other words, we have
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5Page 10 of 13 ˙
Z. Fechner and L. Sz´ekelyhidi Results Math
Theorem 6. Let Nbe a natural number and let ϕl:RnC(l=0,1,...,N)be
a generalized SO(n)-moment function sequence on the double coset hypergroup
Aff SO(n)//SO(n). Then for each uin Rnwe have
ϕ0(u)=Jλ(u),
and
ϕl(u)=
l
j=1
cl,j
dj
jJλ(u),
where λ, cl,j are complex numbers for l=1,2...,N,j =1,2,...,l.
The particular value of the coefficients cl,j can be determined by substi-
tution into (13).
3. An Example
Now we are going to give an explicit example of solutions of (12). We consider
the following group G: it is the multiplicative group of matrices of the form
xu
01
where x, u are complex numbers with x= 0. All these matrices form a subgroup
of GL(2,C) which can be identified with a subset of C2and it is a locally
compact topological group Gwhen equipped with the topology inherited from
C2.Aswehave
xu
01
·yv
01
=xy xv +u
01
,
where x, y are nonzero complex numbers and u, v are in C, we can describe
the group operation on the set G={(x, u): x, u C,x =0}in the following
way:
(x, u)·(y, v)=(xy, xv +u)
where (x, u) and (y, v)areinG.LetKdenote the set of all matrices of the
form
z0
01
where zis a complex number with |z|= 1. Then Kis topologically isomorphic
to the complex unit circle group with multiplication. Clearly, Kis a com-
pact subgroup of G. Finally, Gis topologically isomorphic to the affine group
Aff (K)=KC. For more about this group see e.g. [9, p. 201] or [5].
The normalized Haar measure on Kis given by
K
ϕ(u, 0) du =1
2π2π
0
ϕ(eit,0) dt
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Vol. 74 (2019) Moment Functions on Affine Groups Page 11 of 13 5
for each continuous function ϕ:KC. It is easy to check that Kis not a
normal subgroup, hence the hypergroup structure on the double coset space
G//K is not induced by a group operation. The function f:GCis K-
invariant if and only if it satisfies the compatibility condition
f(x, u)=f(eitx, eis u)
for each (x, u)inGand t, s in R.
Therefore, for each nin Nby K-invariance of ϕnwe obtain
1
2π2π
0
ϕn(xeiky)dk =ϕn(xy)
for all x, y in C\{0}and
ϕ(xeit)=ϕ(x)
for all xin C\{0}and tR.NowEq.(12) becomes a binomial type equation
ϕn(xy)=
n
j=0 n
jϕj(x)ϕnj(y),(14)
for x, y in C\{0}, where ϕn:C\{0}→Cis continuous and K-invariant. By
K-invariance we obtain that ϕkfor k=0,...,n depends only on |x|,thatis
ϕ(x)=ϕ(|x|)foreachx=0inC. There exist u, v in Rsuch that |x|=euand
|y|=ev.Letψkfor k=0,...,n be given by ϕkexp : RCfor k=0,...,n.
Then
ψn(u+v)=
n
j=0 n
jψj(v)ψnj(v),(15)
for uand vin R. Equation (15) is well known and it has been considered e.g.
in [1]. Using the results from [1] we infere that there exist complex constants
α1,...,α
nsuch that
ψn(u)=n!
j1+2j2+···+njn=n
n
m=1 αnu
m!jm
/jm!
for uin R. Therefore
ϕn(x)=n!
j1+2j2+···+njn=n
n
m=1 αnln(|x|)
m!jm
/jm!
for x=0inC.
Acknowledgements
The study was funded by Hungarian National Foundation for Scientific Re-
search (OTKA) with Grant No. K111651.
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5Page 12 of 13 ˙
Z. Fechner and L. Sz´ekelyhidi Results Math
Open Access. This article is distributed under the terms of the Creative Com-
mons Attribution 4.0 International License (http://creativecommons.org/licenses/
by/4.0/), which permits unrestricted use, distribution, and reproduction in any
medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes
were made.
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Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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˙
Zywilla Fechner
Institute of Mathematics
Lodz University of Technology
ul. W´olcza´nska 215
90-924 od´z
Poland
e-mail: zfechner@gmail.com
aszl´oSz´ekelyhidi
University of Debrecen
Debrecen
Hungary
e-mail: lszekelyhidi@gmail.com
Received: January 17, 2018.
Accepted: November 8, 2018.
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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... [3] and [8]. More on moment functions can be found e.g. in [2,4,5,6,10,11,12] and references therein. As already mentioned groups are special hypergroups and in this paper we are going to focus on a generalized moment problem on Abelian groups. ...
... In addition, this paper of J. Aczél has provided motivation for the present research. Namely, it is shown there that if pG,`q is a grupoid and R is a commutative ring, then functions ϕ n : G Ñ R satisfying (1) for each n in N are of the form (2) ϕ n ptq " n! ÿ j 1`2 j 2`¨¨¨`n j n "n n ź k"1 ...
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The main purpose of this work is to prove characterization theorems for generalized moment functions on groups. According one of the main results these are exponential polynomials that can be described with the aid of complete (exponential) Bell polynomials. These characterizations will be immediate consequences of our main result about the characterization of generalized moment functions of higher rank.
... [3] and [8]. More on moment functions can be found e.g. in [2,[4][5][6][10][11][12] and references therein. As already mentioned groups are special hypergroups and in this paper we are going to focus on a generalized moment problem on Abelian groups. ...
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