Int. J. Anal. Appl. (2022), 20:32 A Plancherel Theorem On a Noncommutative Hypergroup Brou Kouakou Germain1,∗, Ibrahima Toure2, Kinvi Kangni2 1Université de Man, Côte d’Ivoire 2Université Félix Houphouet Boigny, Côte d’Ivoire ∗Corresponding author: germain.brou@univ-man.edu.ci, broukouakou320@yahoo.fr Abstract. Let G be a locally compact hypergroup and let K be a compact sub-hypergroup of G. (G,K) is a Gelfand pair if Mc(G//K), the algebra of measures with compact support on the double coset G//K, is commutative for the convolution. In this paper, assuming that (G,K) is a Gelfand pair, we define and study a Fourier transform on G and then establish a Plancherel theorem for the pair (G,K). 1. Introduction Hypergroups generalize locally compact groups. They appear when the Banach space of all bounded Radon measures on a locally compact space carries a convolution having all properties of a group convolution apart from the fact that the convolution of two point measures is a probability measure with compact support and not necessarily a point measure. The intention was to unify harmonic analysis on duals of compact groups, double coset spaces G//H (H a compact subgroup of a locally compact group G), and commutative convolution algebras associated with product linearization formulas of special functions. The notion of hypergroup has been sufficiently studied (see for example [2,4,6,7]). Harmonic analysis and probability theory on commutative hypergroups are well developed meanwhile where many results from group theory remain valid (see [1]). When G is a commutative hypergroup, the convolution algebra Mc(G) consisting of measures with compact support on G is commutative. The typical example of commutative hypergroup is the double coset G//K when G is a locally compact group, K is a compact subgroup of G such that (G,K) is a Gelfand pair. In [4], R. I. Jewett has shown the existence of a positive measure called Plancherel measure on the dual space Ĝ of a commutative hypergroup G. When the hypergroup G is not commutative, it is possible to involve a Received: May 24, 2022. 2010 Mathematics Subject Classification. 43A62, 43A22, 20N20. Key words and phrases. hypergroups; Gelfand pair; probability measure; Plancherel theorem; multiplicative function. https://doi.org/10.28924/2291-8639-20-2022-32 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-32 2 Int. J. Anal. Appl. (2022), 20:32 compact sub-hypergroup K of G leading to a commutative subalgebra of Mc(G). In fact, if K is a compact sub-hypergroup of a hypergroup G, the pair (G,K) is said to be a Gelfand pair if Mc(G//K) the convolution algebra of measures with compact support on G//K is commutative. The notion of Gelfand pairs for hypergroups is well-known (see [3, 8, 9]). The goal of this paper is to extend Jewett work’s by obtaining a Plancherel theorem over Gelfand pair associated with non-commutative hypergroup. In the next section, we give notations and setup useful for the remainder of this paper. In section 3, we introduce first the notion of K-multiplicative functions and obtain some of their characterizations. Thanks to these results, we establish a one to one correspondence between the space of K-multiplicative functions and the dual space of G. Then, we define a Fourier tranform on Mb(G), the algebra of bounded measures on G and on K(G), the algebra of continuous functions on G with compact support. Finally, using the fact that G//K is a commutative hypergroup, we prove that there exists a nonnegative measure (the Plancherel measure) on the dual space of G. 2. Notations and preliminaries We use the notations and setup of this section in the rest of the paper without mentioning. Let G be a locally compact space. We denote by: - C(G) (resp. M(G)) the space of continuous complex valued functions (resp. the space of Radon measures) on G, - Cb(G) (resp. Mb(G)) the space of bounded continuous functions (resp. the space of bounded Radon measures) on G, - K(G) (resp. Mc(G)) the space of continuous functions (resp. the space of Radon measures) with compact support on G, - C0(G) the space of elements in C(G) which are zero at infinity, - C(G) the space of compact sub-space of G, - δx the point measure at x ∈ G, - spt(f ) the support of the function f . Let us notice that the topology on M(G) is the cône topology [4] and the topology on C(G) is the topology of Michael [5]. Definition 2.1. G is said to be a hypergroup if the following assumptions are satisfied. (H1) There is a binary operator ∗ named convolution on Mb(G) under which Mb(G) is an associative algebra such that: i) the mapping (µ,ν) 7−→ µ∗ν is continuous from Mb(G)×Mb(G) in Mb(G). ii) ∀x,y ∈ G, δx ∗δy is a measure of probability with compact support. iii) the mapping: (x,y) 7−→ supp(δx ∗δy) is continuous from G ×G in C(G). (H2) There is a unique element e (called neutral element) in G such that δx ∗ δe = δe ∗ δx = δx,∀x ∈ G. Int. J. Anal. Appl. (2022), 20:32 3 (H3) There is an involutive homeomorphism: x 7−→ x from G in G, named involution, such that: i) (δx ∗ δy)− = δy ∗ δx,∀x,y ∈ G with µ−(f ) = µ(f−) where f−(x) = f (x),∀f ∈ C(G) and µ ∈ M(G). ii) ∀x,y,z ∈ G, z ∈ supp(δx ∗δy) if and only if x ∈ supp(δz ∗δy). The hypergroup G is commutative if δx ∗δy = δy ∗δx,∀x,y ∈ G. For x,y ∈ G, x ∗y is the support of δx ∗δy and for f ∈ C(G), f (x ∗y)≡ (δx ∗δy)(f )= ∫ G f (z)d(δx ∗δy)(z). The convolution of two measures µ,ν in Mb(G) is defined by: ∀f ∈ C(G) (µ∗ν)(f )= ∫ G ∫ G (δx ∗δy)(f )dµ(x)dν(y)= ∫ G ∫ G f (x ∗y)dµ(x)dν(y), For µ in Mb(G), µ∗ =(µ)−. So Mb(G) is a *-Banach algebra. Definition 2.2. H ⊂ G is a sub-hypergroup of G if the following conditions are satisfied. (1) H is non empty and closed in G, (2) ∀x ∈ H,x ∈ H, (3) ∀x,y ∈ H, supp(δx ∗δy)⊂ H. Let us now consider a hypergroup G provided with a left Haar measure µG and K a compact sub- hypergroup of G with a normalized Haar measure ωK. Let us put MµG(G) the space of measures in Mb(G) which are absolutely continuous with respect to µG. MµG(G) is a closed self-adjoint ideal in Mb(G). For x ∈ G, the double coset of x with respect to K is K∗{x}∗K = {k1 ∗x ∗k2;k1,k2 ∈ K}. We write simply KxK for a double coset and recall that KxK = ⋃ k1,k2∈K supp(δk1∗δx ∗δk2). All double coset form a partition of G and the quotient topology with respect to the corresponding equivalence relation equips the double cosets space G//K with a locally topology ( [1], page 53). The natural mapping pK : G −→ G//K defined by: pK(x) = KxK ,x ∈ G is an open surjective continuous mapping. A function f ∈ C(G) is said to be invariant by K or K − invariant if f (k1 ∗x ∗k2)= f (x) for all x ∈ G and for all k1,k2 ∈ K. We denote by C\(G), (resp. K\(G)) the space of continuous functions (resp. continuous functions with compact support) which are K−invariant. For f ∈ C\(G), one defines the function f̃ on G//K by f̃ (KxK)= f (x) ∀x ∈ G. f̃ is well defined and it is continuous on G//K. Conversely, for all continuous function ϕ on G//K, the function f = ϕ◦pK ∈ C\(G). One has the obvious consequence that the mapping f 7−→ f̃ sets up a topological isomorphism between the topological vector spaces C\(G) and C(G//K) (see [8,9]). So, for any f in C\(G), f = f̃ ◦pK. Otherwise, we consider the K-projection f 7−→ f \ (by identifying f \ and f̃ \) from C(G) into C(G//K) where for x ∈ G,f \(x) = ∫ K ∫ K f (k1 ∗ x ∗ k2)dωK(k1)dωK(k2). If f ∈ K(G), then f \ ∈ K(G//K). For a measure µ ∈ M(G), one defines µ \ by µ \ (f )= µ(f \) for f ∈K(G). µ is said to be K−invariant if µ \ = µ and we denote by M\ (G) the set of all those measures. Considering these properties, one 4 Int. J. Anal. Appl. (2022), 20:32 defines a hypergroup operation on G//K by: δKxK ∗δKyK(f̃ )= ∫ K f (x∗k∗y)dωK(k) (see [2, p. 12] ). This defines uniquely the convolution (KxK) ∗ (KyK) on G//K. The involution is defined by: KxK = KxK and the neutral element is K. Let us put m = ∫ G δKxKdµG(x), m is a left Haar measure on G//K. We say that (G,K) is a Gelfand pair if the convolution algebra Mc(G//K) is commutative. Mc(G//K) is topologically isomorphic to M \ c (G). Considering the convolution product on K(G), K(G) is a convolution algebra and K\(G) is a subalgebra. Thus (G,K) is a Gelfand pair if and only if K\(G) is commutative ( [3], theorem 3.2.2). 3. Plancherel theorem Let G be a locally compact hypergroup and let K be a compact sub-hypergroup of G. In this section, we assume that (G,K) is a Gelfand pair. 3.1. K-multiplicative functions. Let us put G\ b the space of continuous, bounded function φ on G such that: (i) φ is K- invariant, (ii) φ(e)=1, (ii) ∫ K φ(x ∗k ∗y)dwK(k)= φ(x)φ(y) ∀x,y ∈ G. Let Ĝ be the sub-space of G\ b containing the elements φ in G\ b such that φ(x)= φ(x) ∀x ∈ G. Ĝ is called the dual space of the hypergroup G. Remark 3.1. (1) If φ ∈ Ĝ, then φ− ∈ Ĝ. (2) Equipped with the topology of uniform convergence on compacta, Ĝ is a locally compact Hausdorff space. (3) In general, Ĝ is not a hypergroup. Definition 3.2. A complex-valued function χ on G will be called a multiplicative (resp. K- multiplicative) function if χ is continuous and not identically zero, and has the property that: χ(x ∗y)= χ(x)χ(y) (resp. ∫ K χ(x ∗k ∗y)dwK(k)= χ(x)χ(y)) ∀x,y ∈ G. A multiplicative (resp. K-multiplicative) function on Mb(G) is a continuous complex-valued function F not identically zero on M\ b (G), and has the property that: F(µ∗ν)= F(µ)F(ν) (resp. F(µ∗wK ∗ν)= F(µ)F(ν)) ∀µ,ν ∈ Mb(G). For χ ∈ Cb(G), not identically zero, let put Fχ(µ)= ∫ G χdµ for µ ∈ Mb(G). Int. J. Anal. Appl. (2022), 20:32 5 Proposition 3.3. Let F be a K-multiplicative function on Mb(G), then: i) F is multiplicative on M\ b (G). ii) F(wK)= F(δe)=1. iii) ∀µ ∈ Mb(G), F(µ\)= F(µ) iv) ∀k ∈ K, F(δk)=1. Proof. i) Just remember that µ∗wK = µ,∀µ ∈ M \ b (G). ii) Let ν ∈ M\ b (G) such that F(ν) 6=0. F(ν)= F(ν ∗wK)= F(ν)F(wK) =⇒ F(wK)=1. F(ν)= F(ν ∗wK ∗δe)= F(ν)F(δe) =⇒ F(δe)=1. iii) Let µ ∈ Mb(G). Since µ\ = wK ∗µ∗wK, we have F(µ\) = F(wK ∗µ∗wK) = F(wK ∗µ∗wK ∗wK) = F(wK ∗µ) = F(δe ∗wK ∗µ) = F(µ). iv) If k ∈ K, δ\ K = wK. Using (ii) and (iii), we have F(δk)=1. � Proposition 3.4. Let φ ∈ G\ b . i) Fφ is a bounded linear K-multiplicative function on Mb(G). ii) Fφ is not identically zero on M \ µG(G). Proof. i) That is clear that Fφ is linear and bounded. Let µ,ν ∈ Mb(G). We have Fφ(µ∗wK ∗ν)= ∫ G ∫ K ∫ G φ(x ∗k ∗y)dµ(x)dwK(k)dν(y) = ∫ G φ(x)dµ(x) ∫ G φ(x)dν(y) = Fφ(µ)Fφ(ν). Morever, Fφ(wK)= ∫ K φ(k)dwK(k)=1 6=0. ii) If µ ∈ MµG(G), then µ \ = wK ∗µ∗wK ∈ M \ µG(G). Let f ∈K(G) with spt(f )⊂ K such that∫ G f (x)duG(x)=1. f \µG ∈ M \ µG(G) and Fφ(f \µG)= Fφ(f µG) = ∫ G φ(x)f (x)duG(x) = ∫ K f (x)duG(x) =1 6=0. 6 Int. J. Anal. Appl. (2022), 20:32 � Theorem 3.5. 1) Let E be a multiplicative linear function on M\µG(G) not identically zero. There exists a unique K-multiplicative linear function F on Mb(G) such that F = E on M \ µG(G). 2) Let F be a bounded linear K-multiplicative function on Mb(G) not identically zero on M \ µG(G).There exists a unique function φ in G \ b such that F = Fφ. Proof. 1) Let ν ∈ M\µG(G) such that E(ν) 6=0 and put F(µ)= E(µ\ ∗ν) E(ν) , f or µ ∈ Mb(G) F is well defined since MµG(G) is an ideal in Mb(G). Let us first see that F is multiplicative on M\ b (G). For µ and µ′ in M\ b (G), we have F(µ∗µ′)= E(µ∗µ′ ∗ν) E(ν) = E(ν ∗µ∗µ′ ∗ν) E(ν)2 = E(ν ∗µ) E(ν) E(µ′ ∗ν) E(ν) = E(ν ∗µ∗ν) E(ν)2 F(µ′) = F(µ)F(µ′). Moreover F(wK)= E(wK ∗ν) E(ν) = E(ν) E(ν) =1. So for µ and µ′ in Mb(G), we have F(µ∗wK ∗µ′)= F(wK ∗ (wK ∗µ∗wK)∗ (wK ∗µ′ ∗wK)∗wK) = F((wK ∗µ∗wK)∗ (wK ∗µ′ ∗wK)) = F(µ\ ∗µ′\) = F(µ)F(µ′). The uniqueness stems from proposition 3.3. 2) Let F be a bounded linear K-multiplicative function on Mb(G). Let ν ∈ M \ µG(G) such that F(ν) 6=0. If µ1,µ2 ∈ Mb(G) then |F(µ1)−F(µ2)|= ∣∣∣F(µ\1)−F(µ\2)∣∣∣ = ∣∣∣F(µ\1 ∗ν)−F(µ\2 ∗ν)∣∣∣ |F(ν)| = ∣∣F((µ1 ∗ν −µ2 ∗ν)\)∣∣ F(ν) ≤ ‖F‖ F(ν) ‖µ1 ∗ν −µ2 ∗ν‖ . Int. J. Anal. Appl. (2022), 20:32 7 Thus F is positive-continuous by ( [4], Theorem 5.6B). By ( [4], Theorem 2.2D) there exists a bounded continuous function h on G such that F(µ)= ∫ G h(x)dµ(x). So φ = h. � 3.2. Fourier transform on Mb(G). Definition 3.6. Let µ ∈ Mb(G), the Fourier transform of µ is the map µ̂ : Ĝ −→ C defined by: µ̂(φ)= ∫ G φ(x)dµ(x). Proposition 3.7. i) For µ ∈ Mb(G), µ̂ ∈ Cb(Ĝ). ii) For µ ∈ Mb(G), µ̂ = µ̂ \ . iii) For µ ∈ MµG(G), µ̂ ∈ C0(Ĝ). iv) If µ ∈ M\ b (G) and ν ∈ Mb(G), then µ̂∗ν = µ̂ν̂. Proof. i) We can see that, µ̂(φ)= µ(φ) ∀φ ∈ Ĝ. ii) For φ ∈ Ĝ, we have µ̂(φ)= Fφ−(µ). So µ̂ \ (φ)= Fφ−(µ \ )= Fφ−(µ)= µ̂(φ). iii) This comes from theorem 3.5 and ( [4], theorem 6.3G) iv) Let φ belongs to Ĝ, we have µ̂∗ν(φ)= ∫ G φ−(x)dµ∗ν(x) = ∫ G ∫ G φ−(x ∗y)dµ(x)dν(y) = ∫ G [∫ G ( ∫ K ∫ K φ−(k1 ∗x ∗k2 ∗y)dωK(k1)dωK(k2))dµ(x) ] dν(y) = ∫ G [∫ G ( ∫ K ( ∫ K φ−((k1 ∗x)∗k2 ∗y)dωK(k2))dωK(k1))dµ(x) ] dν(y) = ∫ G φ−(y) [∫ G ( ∫ K φ−(k1 ∗x)dωK(k1))dµ(x) ] dν(y) = ∫ G φ−(y) [∫ G (φ−(x)dµ(x) ] dν(y) = ∫ G φ−(x)dµ(x) ∫ G φ−(y)dν(y) = µ̂(φ)ν̂(φ). � Remark 3.8. By the definition, the mapping µ 7−→ µ̂ from Mb(G) to Cb(Ĝ) is continuous. 3.3. Fourier transform on G. Definition 3.9. Let f ∈ K\(G), the Fourier transform of f is the map f̂ : Ĝ −→ C defined by: f̂ (φ)= ∫ G φ(x)f (x)duG(x) 8 Int. J. Anal. Appl. (2022), 20:32 Proposition 3.10. i) For f ∈K(G), f̂ \ = f̂ µG ∈ C0(Ĝ). ii) If f ∈K\(G) and g ∈K(G), then ̂f ∗g = f̂ ĝ\. Proof. i) For any f in K(G), we have f̂ \(φ)= ∫ G φ−(x)( ∫ K ∫ K f (k1 ∗x ∗k2)dωK(k1)dωK(k2))duG(x) = ∫ G f (x)( ∫ K ∫ K φ−(k1 ∗x ∗k2)dωK(k1)dωK(k2))duG(x) = ∫ G φ(x)f (x)duG(x)= f̂ µG(φ) ∀φ ∈ Ĝ Since f µG ∈ MµG(G), then f̂ µG ∈ C0(Ĝ). ii) Let f ∈K\(G) and g ∈K(G). For φ ∈ Ĝ, we have ̂f ∗g(φ)= ∫ G φ−(x)f ∗g(x)dµG(x) = ∫ G φ−(x)( ∫ G f (x ∗y)g(y)dµG(y))dµG(x) = ∫ G g(y)( ∫ G φ−(x ∗y)f (x)dµG(x))dµG(y) = ∫ G g(y) ∫ K ∫ K ∫ G φ−(k1 ∗x ∗k2 ∗y)f (x)dµG(x)dωK(k1)dωK(k2)dµG(y) = ∫ G g(y)φ−(y)dµG(y) ∫ G f (x) ∫ K φ−(k1 ∗x)dωK(k1)dµG(x) = ∫ G φ−(y)g(y)dµG(y) ∫ G φ−(x)f (x)dωK(k1)dµG(x) = f̂ (φ)ĝ(φ). � We therefore extend the spherical Fourier transform to all K(G) with f̂ = f̂ \ for any f ∈K(G) and to L1(G,µG) and L 2(G,µG). We have the following result. Theorem 3.11. There exists a unique nonnegative measure π on Ĝ such that∫ G |f (x)|2dµG(x)= ∫ Ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ) for all f in L1(G,µG)∩L2(G,µG). The space { f̂ : f ∈K(G) } is dense in L2(Ĝ,π). Proof. Considering the space Ĝ//K defined by [4], φ̃ ∈ Ĝ//K if and only if φ = φ̃◦pK ∈ Ĝ. Let ϕ̃ belongs to Cb(Ĝ//K). Let us consider ϕ : Ĝ −→C defined by: ϕ(φ)= ϕ̃(φ̃). Int. J. Anal. Appl. (2022), 20:32 9 ϕ ∈ Cb(Ĝ) and the mapping Cb(Ĝ//K) −→ Cb(Ĝ) ϕ̃ 7−→ ϕ is a linear bijection, specificaly ϕ ∈ K(Ĝ) ⇐⇒ ϕ̃ ∈ K(Ĝ//K). By ( [4], theorem. 7.3I), there exist a unique nonnegative measure π̃ on Ĝ//K such that ∫ G//K ∣∣∣f̃ (KxK)∣∣∣2dm(KxK) =∫ Ĝ//K ∣∣∣∣̂̃f (φ̃) ∣∣∣∣2dπ̃(φ̃) for f̃ ∈ L1(G//K,m)∩ L2(G//K,m). Let us consider the mapping π defined by π(ϕ) = π̃(ϕ̃) for ϕ ∈ K(Ĝ).π is a measure on Ĝ. Since π̃ is nonnegative, then π is nonnegative. Otherwise, note that ˜̂ f = ̂̃ f for f ∈ K\(G). Indeed since f ∈ K\(G) then f̃ ∈ K(G//K) and f̂ ∈ Cb(Ĝ). So ̂̃ f and ˜̂ f belong to Cb(Ĝ//K). For φ̃ ∈ Ĝ//K, we have ̂̃ f (φ̃)= ∫ G//K φ̃(KxK)f̃ (KxK)dm(KxK) = ∫ G//K φ̃−(KxK)f̃ (KxK)dm(KxK) = ∫ G φ−(x)f (x)duG(x) = f̂ (φ)= ˜̂ f (φ̃) Let f ∈K\(G). We have ∫ Ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ)= ∫ Ĝ//K ∣∣∣∣˜̂f (φ̃) ∣∣∣∣2dπ̃(φ̃) = ∫ Ĝ//K ∣∣∣∣̂̃f (φ̃) ∣∣∣∣2dπ̃(φ̃) = ∫ G//K ∣∣∣f̃ (KxK)∣∣∣2dm(KxK) = ∫ G |f (x)|2dµG(x). As f̂ = f̂ \ ∀f ∈K(G) and G unimodular, we deduce that ∫ Ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ)= ∫G |f (x)|2dµG(x) ∀f ∈ K(G). By the continuity of the Fourier transform and by application of the dominated convergence theorem, we conclude that ∫ G |f (x)|2dµG(x) = ∫ Ĝ ∣∣∣f̂ (φ)∣∣∣2dπ(φ) for any f belongs to L1(G,µG)∩ L2(G,µG). Let π ′ a nonnegative measure on Ĝ such that ∫ G |f (x)|2dµG(x)= ∫ Ĝ ∣∣∣f̂ (φ)∣∣∣2dπ′(φ) for all f in L1(G,µG)∩L2(G,µG). As above but in reverse order π′ defines a nonnegative measure π̃′ on Ĝ//K such that ∫ G//K ∣∣∣f̃ (KxK)∣∣∣2dm(KxK) = ∫ Ĝ//K ∣∣∣∣̂̃f (φ̃) ∣∣∣∣2dπ̃(φ̃) for f̃ ∈ L1(G//K,m)∩ L2(G//K,m). That is π̃′ = π̃ seen the uniqueness of π̃, so π = π′. Let us put F(K(G)) = 10 Int. J. Anal. Appl. (2022), 20:32{ f̂ ; f ∈K(G } . Let ϕ ∈K(Ĝ) such that 〈 f̂ ,ϕ 〉 = ∫ Ĝ f̂ (φ)ϕ(φ)dπ(φ)=0 ∀f ∈K\(G). We have〈 f̂ ,ϕ 〉 =0 ∀f ∈K\(G) =⇒ ∫ Ĝ f̂ (φ)ϕ(φ)dπ(φ)=0 ∀f ∈K\(G) =⇒ ∫ Ĝ ˜̂ f (φ̃)ϕ̃(φ̃)dπ̃(φ̃)=0 ∀f ∈K\(G) =⇒ 〈˜̂ f , ϕ̃ 〉 =0 ∀f ∈K(G) =⇒ 〈̂̃ f , ϕ̃ 〉 =0 ∀f̃ ∈K(G//K) =⇒ ϕ̃ =0 since F(K(G//K)) is dense in L2(Ĝ//K,π̃) =⇒ ϕ =0. So (F(K(G)))⊥ ∩K(Ĝ) = {0}. Since K(Ĝ) is dense in L2(Ĝ,π), then (F(K(G)))⊥ = {0} and F(K(G)) is dense in L2(Ĝ,π). � Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. References [1] W.R. Bloom, H. Heyer, Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter Studies in Mathe- matics 20, Walter de Gruyter, Berlin, 1995. [2] C.F. Dunkl, The Measure Algebra of a Locally Compact Hypergroup, Trans. Amer. Math. Soc. 179 (1973), 331–348. https://doi.org/10.1090/s0002-9947-1973-0320635-2. [3] B.K. Germain, K. Kinvi, On Gelfand Pair Over Hypergroups, Far East J. Math. 132 (2021), 63–76. https://doi. org/10.17654/ms132010063. [4] R.I. 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Hungar. 160 (2019), 101–108. https://doi.org/10. 1007/s10474-019-00946-1. https://doi.org/10.1090/s0002-9947-1973-0320635-2 https://doi.org/10.17654/ms132010063 https://doi.org/10.17654/ms132010063 https://doi.org/10.1016/0001-8708(75)90002-x https://doi.org/10.1016/0001-8708(75)90002-x https://doi.org/10.1090/S0002-9939-1961-0123197-5 https://doi.org/10.1090/s0002-9947-1978-0493161-2 https://doi.org/10.1090/s0002-9947-1978-0493161-2 https://doi.org/10.1090/s0002-9947-1978-0463806-1 https://doi.org/10.1090/s0002-9947-1978-0463806-1 https://doi.org/10.1007/s10474-020-01068-9 https://doi.org/10.1007/s10474-020-01068-9 https://doi.org/10.1007/s10474-019-00946-1 https://doi.org/10.1007/s10474-019-00946-1 1. Introduction 2. Notations and preliminaries 3. Plancherel theorem 3.1. K-multiplicative functions. 3.2. Fourier transform on Mb(G) 3.3. Fourier transform on G References