Moment Functions on Affine Groups

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 $${\mathrm {Aff\,}}K$$AffK. Using the fact that in the finitely dimensional case $$({\mathrm {Aff\,}}K,K)$$(AffK,K) is a Gelfand pair we give a description of exponentials on the double coset hypergroup $${\mathrm {Aff\,}}K//K$$AffK//K in terms of K-spherical functions. Moreover, we give a general description of generalized moment functions on $${\mathrm {Aff\,}}K$$AffK and specific examples for $$K=SO(n)$$K=SO(n), and on the so-called $$ax+b$$ax+b-group.


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 K is denoted by o. For x in K the symbol δ x denotes the point mass with support {x}. In the sequel C denotes 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 K be a hypergroup. We begin with recalling the notion of exponential, additive and m-sine functions on hypergroups which are strongly connected with the generalized moment functions. The non-identically zero continuous function m : K → C is called an exponential on K if m satisfies m(x * y) = m(x)m(y) for each x, y in K. The continuous function a : K → C is called additive function if it satisfies a(x * y) = a(x) + a(y) for each x, 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 : K → C is called an m-sine function if it satisfies for each x, y in K. The function f is called a sine function if f is an msine function for some exponential m. Clearly, every sine function f satisfies f (o) = 0. For a given exponential m all m-sine functions form a linear space. Obviously, m ≡ 1 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 G be a locally compact group with identity e and K a compact subgroup with the normalized Haar measure ω. As K is unimodular ω is left and right invariant, and it is also inversion invariant. For each x in G we define the double coset of x as the set be the set of all double cosets. We introduce a hypergroup structure on the set L of all double cosets: the topology of L is the factor topology, which is locally compact. The identity is the coset K = KeK itself and the involution is defined by Finally, the convolution of δ KxK and δ KyK is defined by where x and y are in G. It is known that this gives a hypergroup structure on L (see [3], p. 12.), which is non-commutative, in general. If K is a normal subgroup, then L is isomorphic to the hypergroup arising from the factor group G/K. In [8] representations of double coset hypergroups are investigated. Such representations can be canonically obtained from those of the group in question. Nevertheless, not every representation arises in this way. However, our results show that on some affine groups, basic representing functions, like exponentials, additive functions and quadratic functions are closely related to the corresponding functions on the original group.
We note that continuous functions on L can be identified with those continuous functions on G which are K-invariant: f (x) = f (kxl) for each x in G and k, l in K. Hence, for a continuous function f : L → C the 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 : L → C and for each x, y in G: The following theorem exhibits a close connection between exponentials on double coset hypergroups and spherical functions on locally compact groups. Following the terminology of [3] (see also [4]) we recall the concept of spherical functions. The continuous bounded holds for each x, y in G. A generalized K-spherical function on G is 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. Here a multiplicative function means an exponential map m : It is obvious, that if G is a locally compact Abelian group, and K is a compact subgroup of G, then K-spherical functions are exactly the exponentials of the (locally compact) Abelian group G/K.
Given the locally compact group G and the compact subgroup K the space M c (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 G when equipped with the topology of uniform convergence on compact sets. The pairing between C(G) and M c (G) will be denoted by ·, · , hence 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 holds for each f in C(G). Using the hypergroup structure on L = G//K we see easily that M c (G//K) is a complex unital topological algebra with unit δ e , the point mass with support {e}. We say that (G, K) is a Gelfand 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 L is commutative ( for n = 0, 1, . . . , N and x, y in G. In fact, this is exactly the case if (g n ) 1≤n≤N is a generalized moment function sequence on the double coset hypergroup G//K. In particular, g 0 is a K-spherical function on G, or what is the same, g 0 is 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. In the case g 0 ≡ 1 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 (g n ) 1≤n≤N with N ≥ 1 the function g 1 is a g 0 -sine function, according to the terminology introduced in [7] (see also [5]), which, in the general case can be called K-sine functions associated with the K-spherical function g 0 . Generalized moment functions and generalized moment function sequences have been described on different types of commutative hypergroups (see e.g. [11][12][13][14]). Nevertheless, non-commutative hypergroups have not yet been considered in this context.

Affine Groups
An important type of double coset hypergroups arises form the concept of affine groups. Let V be an n-dimensional vector space over the field K and let GL(V ) for each x, y in H and u, v in V . Here x · v is the image of v under 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 . The inverse of (x, u) is for each x in H and u in V . We note that, in general, Aff H is non-commutative, even if H is commutative. In any case V -as a commutative group -is isomorphic to the normal subgroup of Aff H consisting of all elements of the form (id, u) with u is in V the isomorphism provided by the mapping u → (id, u).
In the case V = K n we have GL(V ) = GL(K, n), which is a matrix group. With the usual Euclidean topology V and GL(V ) are locally compact topological groups. If H is a subgroup of GL(V ) a function f : Suppose that H is a closed subgroup of GL(V ) and K is a compact subgroup in H. Then K -identified with the set of all elements of the form (x, 0) with x in Kis a compact subgroup in H V . Let L denote the double coset hypergroup Aff H//K. As we have seen above K-invariant continuous functions on Aff H can be identified with the continuous functions on L -we use the notation f (x, u) for f K(x, u)K . Using this notation K-invariance means f (kx, lu) = f (x, u) for each k, l in K, x in H and u in V . If * denotes convolution in L then we have for each x, y in H and u, v in V , where ω is the normalized Haar measure on K.
A basic observation about affine groups is the following theorem.

Theorem 2. Let V be a finite dimensional vector space and K a compact subgroup 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]). holds for each x in G. Then (G, K) is a Gelfand pair.

Theorem 3. (Gelfand) Let G be a locally compact group and K a compact subgroup of G. Suppose that there exists a continuous involutive automorphism
Proof (Proof of Theorem 2). We define θ : Aff K → Aff K as for each k in K and u in V . Then clearly θ is a continuous involutive automorphism of Aff K. On the other hand, we can write 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 V satisfying The next theorem is a generalization of [6, Theorem 2.1]. Proof. Observe that for each x, y in H and u, v in V , hence for every continuous K-invariant func- whenever x, y is in H and u, v is in V . In particular, putting y = id and u = 0 in (6) we have for each K-invariant continuous function f : L → C and x in H, v in V . Now assume that m is an exponential on L. Then it satisfies as it was stated. Similarly, denoting ϕ(u) = m(id, u) and putting x = y = id in (8) we conclude Finally, substitution into (9) and (8) gives (5). Conversely, if m has the desired form then (5) implies that m is exponential.  for each k in K and u in V , where ϕ : V → C is a K-spherical function.
As an example we consider the case V = R n and K = SO(n), the special orthogonal group on R n . It is known (see e.g. [17]) that the SO(n)-spherical functions on R n are exactly the radial eigenfunctions of the Laplacian in R n , hence they can be described in terms of the Bessel functions J λ in the form for each u in R n and complex number λ, where represents an exponential family for the double coset hypergroup L = Aff SO(n)//SO(n).
In particular, for n = 3 we have Φ(k, u, λ) = sinh λ u λ u whenever u is in R 3 and λ is in C, where in the case λ u = 0 we take the limit, which gives the spherical function identically 1. Suppose that V is a finite dimensional K-vector space, H is a closed subgroup of GL(V ) and K is a compact subgroup of H with the normalized Haar measure ω. Specializing the concept of generalized K-moment function sequence to the case G = H V with the compact subgroup {(k, 0) : k ∈ K} ∼ = K we can rewrite the system of Eq. (3) in the following way: for each (x, u) in Aff K, where ϕ n : V → C is continuous and K-invariant, further we have for n = 0, 1 . . . , N and for each u, v in V .
Proof. By Lemma 1, we have that g n (x, u) = ϕ n (u) holds for each (x, u) in Aff K, where ϕ n : V → C is continuous and K-invariant (n = 0, 1, . . . , N). Substitution into (10) gives for each x in K and u, v in V . By the translation invariance of ω and interchanging u and 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 K of GL(R n ). 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 = R n 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 ϕ 0 (u, λ) = J λ ( u ) for each u in R n , where λ is a complex number. On the other hand, by Theorem 3.4 and Corollary 2 in [15], it follows that ϕ n is a linear combination of the derivatives of λ → ϕ 0 (u, λ) up to the n-th order. In other words, we have Open Access. This article is distributed under the terms of the Creative Commons 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.