On a Generalization of O'connor's and Gajda's Functional Equations

Math-Recherche et Application, Vol.7, (2005), pp. 49–70

On a generalization of O’Connor’s and Gajda’s functional equations

Belaid Bouikhalene and Samir Kabbaj

Department of Mathematics, University of Ibn Tofail, Faculty of Sciences, BP : 133, K´enitra - Morocco. E-mail: [email protected], [email protected]

AMS classification : 39B32. 39B42. 22D10. 22D12. 22D15.

Abstract In this paper, we study functional equations of the form Z n X −1 X −1 E(K) f(xkϕ(y)k )dωK (k) = |Φ| ai(x )ai(y), x, y ∈ G, ϕ∈Φ K i=1 where G is a locally compact group, K is a compact subgroup of G, ωK is the normalized Haar measure of K, Φ is a ﬁnite group of K- invariant morphisms of G and f, a1, ..., an : G −→ C are continuous complex-valued functions such that f and a satisfy the Kannappan type condition (∗) Z Z −1 −1 f(zkxk hyh )dωK (k)dωK (h) = K K Z Z −1 −1 f(zkyk hxh )dωK (k)dωK (h), K K for all x, y, z ∈ G. Our results extend the ones obtained for O’Connor’s and Gajda’s functional equations.

1. Introduction Let G be a locally compact group. Let K be a compact subgroup of G and let ωK be the normalized Haar measure of K. A mapping

c Math-Rech. & Appl. 50 Belaid Bouikhalene and Samir Kabbaj

ϕ : G → G is a morphism of G if ϕ is a homeomorphism of G onto itself which is either a group-homomorphism, i.e. (ϕ(xy) = ϕ(x)ϕ(y), x, y ∈ G), or a group-antihomomorphism, i.e. (ϕ(xy) = ϕ(y)ϕ(x), x, y ∈ G). We denote by Mor(G) the group of morphism of G and Φ a ﬁnite subgroup of Mor(G) of a K-invariant morphisms of G (i.e. ϕ(K) ⊂ K, for all ϕ ∈ Φ). The number of elements of a ﬁnite group Φ will be designated by |Φ|. Cb(G) (resp. C(G)) designates the Banach space of bounded continuous ( resp. continuous) complex valued functions on G. The Banach space of all complex measurable and essentially bounded functions on G is denoted by L∞(G). By L1(G) designates the Banach algebra of all integrable K functions on G and L1 (G) the subalgebra of the functions in L1(G) −1 K that are K-invariant (i.e. f(kxk ) = f(x), x ∈ G, k ∈ K). If L1 (G) is commutative, we say that (G, K) is a Gelfand pair ([7],[11],[13], [16]).

A non-zero function f ∈ C(G) is said a K-spherical function if it satisﬁes the functional equation ([4], [7], [10], [11], [13], [16], [19], [20], [21]) Z −1 f(xkyk )dωK (k) = f(x)f(y), x, y ∈ G. K In this paper, we consider the functional equation Z n X −1 X −1 f(xkϕ(y)k )dωK (k) = |Φ| ai(x )ai(y), x, y ∈ G. (1) ϕ∈Φ K i=1 When Φ = {I} (resp. Φ = {I, σ}), where σ is a continuous involution on G such that σ(K) ⊂ K), (1.1) is reduced to the equation Z n −1 X −1 f(xkyk )dωK (k) = ai(x )ai(y), x, y ∈ G, (2) K i=1 (resp. Z Z −1 −1 f(xkyk )dωK (k) + f(xkσ(y)k )dωK (k) = (3) K K n X −1 2 ai(x )ai(y), x, y ∈ G.) i=1 on a generalization of o’connor’s and gajda’s equations 51

In a recent paper [23], Stetkær studied the equation (1.2) in the case where K is a compact subgroup of Aut(G) such that (G, K) is a Gelfand pair.

If G is a locally compact abelian group K = {e} and Φ = {I} (resp. Φ = {I, −I}), the equation (1.2) (resp. (1.3)) is reduced to the O’Connor’s functional equation [18]

n X F (x − y) = bi(x)bi(y), x, y ∈ G, (4) i=1

(resp. Gajda’s functional equation [14]

n X F (x + y) + F (x − y) = 2 bi(x)bi(y), x, y ∈ G), (5) i=1 where F (x) = f(−x), x ∈ G and bi = ai, i ∈ {1, ..., n}.

When n = 1, the equation (1.1) becomes Z X −1 −1 f(xkϕ(y)k )dωK (k) = a1(x )a1(y), x, y ∈ G, (6) ϕ∈Φ K which encompass several functional equations of type d’Alembert and Wilson ([9],[12], [21], [25]).

Furthermore the equation (1.1) may be considered as a common generalization of a many functional equations of generalized O’Connor’s and Gajda’s functional equations like

Z n X X −1 f(xkϕ(y))dωK (k) = |Φ| ai(x )ai(y), x, y ∈ G, (7) ϕ∈Φ K i=1

Z n X −1 f(xky)dωK (k) = ai(x )ai(y), x, y ∈ G, (8) K i=1 52 Belaid Bouikhalene and Samir Kabbaj

Z Z n X −1 f(xky)dωK (k) + f(xkσ(y))dωK (k) = 2 ai(x )ai(y), K K i=1 x, y ∈ G, (9)

Z n X −1 Σϕ∈Φ f(xkϕ(y))χ(k)dωK (k) = |Φ| ai(x )ai(y), x, y ∈ G, K i=1 (10) Z n X −1 f(xky)χ(k)dωK (k) = ai(x )ai(y), x, y ∈ G, (11) K i=1

Z Z f(xky)χ(k)dωK (k) + f(xkσ(y))χ(k)dωK (k) = (12) K K n X −1 2 ai(x )ai(y), x, y ∈ G, i=1 where χ is a unitary character of K. It is also a generalization of the functional equations

n X X −1 f(xϕ(y)) = |Φ| ai(x )ai(y), x, y ∈ G, (13) ϕ∈Φ i=1

Z n X −1 X −1 f(xtϕ(y) )dt = |Φ| ai(x )ai(y), x, y ∈ G, (14) ϕ∈Φ G i=1 Z n −1 X −1 f(xtyt )dt = ai(x )ai(y), x, y ∈ G, (15) G i=1 Z Z n −1 −1 X −1 f(xtyt )dt + f(xtσ(y)t )dt = 2 ai(x )ai(y), x, y ∈ G, G K i=1 (16) where G is a compact group and σ is a continuous involution on G.

The aim of this work is to study functional equation (1.1). This paper is organized as follows. In the ﬁrst section after this introduction, we on a generalization of o’connor’s and gajda’s equations 53 study the functional equation (1.6). In the second section, with the assumption that the set of functions {a1, ..., an} is linearly independent and f, a satisfy the Kannappan type condition (∗) ([8], [12]), we deal with the equation (1.1). In section 3, we assume that Φ is a compact subgroup of Aut(G) and (G, K) is a Gelfand pair, we determine the continuous bounded solutions of (1.6) and (1.1). In the last section, we give some applications.

P R −1 2. On the functional equation ϕ∈Φ K f(xkϕ(y)k )dωK (k) = −1 a1(x )a1(y) In what follows, we study general properties of the solutions of (1.6). Let G be a locally compact group, K be a compact subgroup of G and let Φ be a ﬁnite group of K-invariant morphisms of G.

PROPOSITION 2.1 ([8],[9]). For an arbitrary fixed τ ∈ Φ, and f ∈ C(G), we have i) the mapping ϕ −→ ϕ ◦ τ is a bijection of Φ. R −1 ii) Σϕ∈Φ K f(xkϕ(τ(y))k )dωK (k) = R −1 Σϕ∈Φ K f(xkϕ(y)k )dωK (k), x, y ∈ G, . iii) If f is K-invariant and satisfies the condition (∗), then for all z, y, x ∈ G, we have Z Z −1 −1 f(ykxk )dωK (k) = f(xkyk )dωK (k), K K 4i) If f satisfies the condition (∗), then we have Z Z −1 −1 f(zhϕ(ykxk )h )dωK (h)dωK (k) = K K Z Z −1 −1 f(zhϕ(xkyk )h )dωK (h)dωK (k), K K Z Z −1 −1 f(xkϕ(hy)k )dωK (k) = f(xkϕ(yh)k )dωK (k), K K for all z, y, x ∈ G. 5i) If (G, K) is a Gelfand pair. Then the condition (∗) holds. 54 Belaid Bouikhalene and Samir Kabbaj

In the next lemma, we consider the functional equation Z X −1 f(xkϕ(y)k )dωK (k) = |Φ|f(x)g(y), x, y ∈ G. (17) ϕ∈Φ K Lemma 2.1. Let G be a locally compact group and let K be a compact subgroup of G. Let Φ be a ﬁnite subgroup of the group morphisms of G such that K is Φ-invariant. Let f, g ∈ C(G) be a solution of (2.1) such that f 6= 0 and satisﬁes the condition (∗). Then g is a solution of the functional equation Z X −1 g(xkϕ(y)k )dωK (k) = |Φ|g(x)g(y), x, y ∈ G. (18) ϕ∈Φ K

Proof. Let x0 ∈ G such that f(x0) 6= 0. By using proposition 2.1 one has for all x, y ∈ G Z −1 |Φ|f(x0) .Σϕ∈Φ g(xkϕ(y)k )dωK (k) K Z −1 = Σϕ∈Φ |Φ|f(x0)g(xkϕ(y)k )dωK (k) K Z −1 −1 = Σϕ∈Φ Στ∈Φf(x0hτ(xkϕ(y)k )h )dωK (h)dωK (k) K Z −1 −1 = Σϕ∈Φ Στ∈Φf(x0hτ(x)kτ(ϕ(y))k h )dωK (h)dωK (k) K Z −1 −1 = Σϕ∈Φ Σψ∈Φf(x0hτ(x)kψ(y)k h )dωK (h)dωK (k) K Z −1 −1 = Σϕ∈Φ Σψ∈Φf(x0hτ(x)h kψ(y)k )dωK (h)dωK (k) K Z −1 = Στ∈Φ |Φ|f(x0hτ(x)h )dωK (h) K 2 = |Φ| f(x0)g(x)g(y). i.e. Z X −1 g(xkϕ(y)k )dωK (k) = |Φ|g(x)g(y), x, y ∈ G. ϕ∈Φ K on a generalization of o’connor’s and gajda’s equations 55

Theorem 2.1. Let f, a1 ∈ Cb(G) \{0} be a solution of the functional equation (2.1) such that f satisﬁes the condition (∗). Then f, a1, satisfy the equations Z X −1 a1(e) f(xkϕ(y)k )dωK (k) = |Φ|f(x)a1(y), x, y ∈ G. (19) ϕ∈Φ K and Z X −1 a1(e) a1(xkϕ(y)k )dωK (k) = |Φ|a1(x)a1(y), x, y ∈ G. (20) ϕ∈Φ K

Proof. By taking y = e in (1.6), we get

−1 f(x) = a1(x )a1(e), x ∈ G.

−1 f(x) Since f 6= 0, then a1(e) 6= 0 and it follows that a1(x ) = , x ∈ G. a1(e) So Z X −1 f(xkϕ(y)k )dωK (k) = |Φ|f(x)g(y), x, y ∈ G, ϕ∈Φ K where g = a . By using lemma 2.1, we get the remainder. a1(e)

Corollary 2.1. Let f, a1 ∈ C(G) \{0} be a solution of the functional equation Z Z −1 −1 −1 f(xkyk )dωK (k) + f(xkσ(y) k )dωK (k) = K K −1 2a1(x )a1(y), x, y ∈ G, (21) where σ is a continuous involution on G. Then there exists a K- spherical function ω such that

|a(e)|2 f(x) = (ω(x−1) + ω(σ(x−1))), x ∈ G. 2 a(e) a (x) = (ω(x) + ω(σ(x))), x ∈ G. 1 2 56 Belaid Bouikhalene and Samir Kabbaj

Proof. In view of theorem 2.1, one has Z Z −1 −1 −1 a1(e) f(xkyk )dωK (k) + a1(e) f(xkσ(y) k )dωK (k) = K K 2a1(x)a1(y), x, y ∈ G, (22)

By using [12], we conclude.

Corollary 2.2. [21] Let f, a1 ∈ C(G) \{0} be a solution of the functional equation Z −1 −1 f(xkyk )dωK (k) = a1(x )a1(y), x, y ∈ G. (23) K Then there exists a K-spherical function ω such that

f(x) = |a(e)|2ω(x−1), x ∈ G.

a1(x) = a(e)ω(x), x ∈ G.

3. On the functional equation E(K) We will use the following notations in the rest of the paper. n The usual inner product on C which is linear in the ﬁrst component, is denoted < ., . >. We let At, resp. A∗ denote the transpose, resp. the adjoint, of the matrix A. The algebra of n × n complex matrices is −1 ∗ denoted M(n, C). For all A ∈ M(n, C), we put A˜ = A(x ) .

The following proposition produces a necessary condition for (1.1) to have a solution.

Proposition 3.1 . If {f, a1, ..., an} is a solution of (1.1) such that f, a satisfy the condition (∗). Then for all x, y, z ∈ G, we have n Z X X −1 ( aˇi(xkϕ(y)k )dωK (k))ai(z) = i=1 ϕ∈Φ K on a generalization of o’connor’s and gajda’s equations 57

n Z X X −1 aˇi(x)( ai(ykϕ(z)k )dωK (k)). (24) i=1 ϕ∈Φ K

Proof. By easy computations.

In the remainder of this paper we will assume that the set of functions {a1, ..., an} is linearly independent. In the case where n = 1, it means that a1 6= 0.

Theorem 3.1 . Let {a1, ..., an} be a solution of (3.1) such that a1, ..., an satisfy the condition (∗). Then there exists a matrix function M ∈ M(n, C) such that i) Z −1 Σϕ∈Φ M(xkϕ(y)k )dωK (k) = |Φ|M(x)M(y), x, y ∈ G. (25) K ii) M(kxk−1) = M(x), x ∈ G, k ∈ K. iii) M ◦ ϕ = M for all ϕ ∈ Φ. 4i) M˜ = M. 5i) M(x)M(y) = M(y)M(x) for all x, y ∈ G. 6i) M(x) = diag(ω (x)) , where ω is a solution of the functional i 16i6n i equation (2.2).

Proof. Since the set of functions {a1, ..., an} is linearly independent, 0 0 0 0 then by [1] there exist x1, ..., xn, y1, ..., yn ∈ G such that the matrix ˇ n n {aj(xi}i,j=1, {ai(xj}i,j=1 are invertible. From (3.1), one has for all x1, ..., xn, z ∈ G Z X −1 { a˘j(xikϕ(z)k )dωK (k)}{ai(yj)} = ϕ∈Φ K Z X −1 {a˘j(xi)}{ ai(zkϕ(yj)k )dωK (k))}. (26) ϕ∈Φ K Let M(z) be the common value Z 0 −1 X 0 −1 M(z) = {a˘j(xi )} { a˘j(xi kϕ(z)k )dωK (k))} ϕ∈Φ K 58 Belaid Bouikhalene and Samir Kabbaj

Z X 0 −1 −1 = { ai(zkϕ(yj )k )dωK (k)}{aj(yj)} . ϕ∈Φ K

So one has Z X −1 { a˘j(xikϕ(z)k )dωK (k)} = {a˘j(xi)}M(z) (27) ϕ∈Φ K and Z X −1 { ai(zkϕ(xj)k )dωK (k)} = M(z){ai(xj)} (28) ϕ∈Φ K i) For all x, y ∈ G, we have Z X −1 M(xkϕ(y)k )dωK (k) ϕ∈Φ K Z Z 0 −1 X X ˘0 −1 −1 = {a˘j(xi )} { aj (xikτ(xhϕ(y)h )k ) τ∈Φ K ϕ∈Φ K

dωK (k)dωK (h)} Z Z ˘0 −1 X X −1 −1 = {aj (xi)} { a˘j(xihτ(x)h kψ(y)k )) ψ∈Φ K ϕ∈Φ K

dωK (k)dωK (h)} Z Z ˘0 −1 X X ˘0 −1 = {aj (xi)} { aj (xikϕ(x)k ))dωK (k)}M(y) ϕ∈Φ K τ∈Φ K

˘0 −1 ˘0 = {aj (xi)} {aj (xi)}M(x)M(y) = M(x)M(y).

Since a satisfies the condition (∗) it follows that M(x)M(y) = M(y)M(x), for all x, y ∈ G. Then we have (5i). (ii) and (iii) by easy computations. n 4i) Let a(x) = (a1(x), ..., an(x)) ∈ C . Then a satisfies the condition (∗). Since the set of functions {a1, ..., an} is linearly independent then there exist points x1, ..xn ∈ G such that the vectors a(x1), ..a(xn) form n ˇ a basis of C . In the next we view a(x) as a row vector and a(x) as a on a generalization of o’connor’s and gajda’s equations 59 column vector. Let us write (3.3) and (3.4) out with indices, we find that Z X −1 a˘(xkϕ(z)k )dωK (k) = a˘(x)M(z) (29) ϕ∈Φ K and Z X −1 a(zkϕ(x)k )dωK (k) = M(x)a(z). (30) ϕ∈Φ K By taking the adjoint we get for all x, y ∈ G Z ∗ ∗ X −1 ∗ M(y) aˇ(x) = a˘(xkϕ(y)k ) dωK (k), ϕ∈Φ K so Z X −1 −1 −1 ∗ M˜ (x)a(y) = a˘(x kϕ(y )k ) dωK (k) ϕ∈Φ K Z X −1 = a(xkϕ(y)k )dωK (k) ϕ∈Φ K Z X −1 = a(ykτ(x)k )dωK (k) τ∈Φ K = M(x)a(y).

n Since span{a(x), x ∈ G} = C , it follows that M˜ = M. 6i) Since M(x)M(y) = M(y)M(x) for all x, y ∈ G and M˜ = M. Then {M(x), x ∈ G} is a commuting set of normal matrices. By the spectral theorem, the matrices M(x), x ∈ G are diagonalized. So there exists a unitary matrix P ∈ M(n, ) such that M(x) = P −1diag(ω (x)) P , C i 16i6n x ∈ G. Since M satisﬁes the equation (3.2) and M˜ = M it follows that for all i ∈ {1, ..., n},ω ˜i = ωi and Z X −1 ωi(xkϕ(y)k )dωK (k) = |Φ|ωi(x)ωi(y), x, y ∈ G. ϕ∈Φ K

Theorem 3.2 . Let {f, a1, ..., an} be a solution of (1.1) such that f, a satisfy the condition (∗). Then there exist functions ω1, ..., ωn solutions 60 Belaid Bouikhalene and Samir Kabbaj

of (2.2) for which ωˇi = ωi for i = 1, ..., n, a unitary matrix P ∈ M(n, C) n and ξ ∈ C such that f(x) =< ξ, diag(ω (x)) ξ >, x ∈ G, i 16i6n a(x) = P diag(ω (x)) ξ, x ∈ G. i 16i6n

n Proof. Let a : G −→ C given by the formula a(x) = (a1(x), ..., an(x)). Since Z X −1 a(xkϕ(y)k )dωK (k)) = M(x)a(y), x, y ∈ G. ϕ∈Φ K

By taking y = e one has

a(x) = M(x)a(e) = P (diag(ω (x))) P −1a(e). i 16i6n and

f(x) = < a(e), a(x−1) > = < a(e), P diag(ω (x)) P −1a(e) > i 16i6n = < P −1a(e), diag(ω (x)) P −1a(e) >, i 16i6n by setting ξ = P −1a(e) we end the proof of theorem.

Corollary 3.1. Let {f, a1, ..., an} be a solution of the functional equation (1.2). Then there exist K-spherical functions ω1, ..., ωn for which ωˇi = ωi for i = 1, ..., n, a unitary matrix P ∈ M(n, C) and n ξ ∈ C such that f(x) =< ξ, diag(ω (x)) ξ >, x ∈ G, i 16i6n a(x) = P diag(ω (x)) ξ, x ∈ G. i 16i6n

Remark 3.1 If (G, K) is a Gelfand pair then the condition (∗) holds ([8], [9], [12]). consequently we obtain the theorem 6.1 in [23]. on a generalization of o’connor’s and gajda’s equations 61

Corollary 3.2. Let {f, a1, ..., an} be a solution of the functional equation (1.3) such that f, a satisfy the condition (∗). Then there exist K-spherical functions ω1, ..., ωn for which ωˇi = ωi for i = 1, ..., n, a n unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 f(x) =< ξ, diag( (ω (x) + ω (σ(x))) ξ >, x ∈ G, 2 i i 16i6n 1 a(x) = P diag( (ω (x) + ω (σ(x))) ξ, x ∈ G. 2 i i 16i6n In the next corollary we assume that K ⊂ Z(G) (the center of G.)

Corollary 3.3. Let {f, a1, ..., an} be a solution of the functional equation (1.13) such that f, a satisfy the Kannappan condition f(zxy) = f(zyx), for all x, y, z ∈ G [17]. Then there exist ω1, ..., ωn for which ωˇi = ωi, solutions of functional equation X ωi(xϕ(y)) = |Φ|ωi(x)ωi(y), x, y ∈ G. ϕ∈Φ

n for i = 1, ..., n, a unitary matrix P ∈ M(n, C) and ξ ∈ C such that

f(x) =< ξ, diag(ω (x)) ξ >, x ∈ G, i 16i6n a(x) = P diag(ω (x)) ξ, x ∈ G. i 16i6n

Remark 3.2 If G is an abelian group, K = {e}, Φ = {I} (resp. Φ = {I, −I}), F (x) = f(−x) and bi = ai, i = 1, ..., n, we get the solutions of O’Connor’s [18] (resp. Gajda’s [14]) functional equation without imposing Gajda’s assumption.

In the next corollary, we suppose that K = G is a compact group.

Corollary 3.4. Let G be a compact group. Let {f, a1, ..., an} ∈ L∞(G) such that f, a are central, be a solution of the functional equation 62 Belaid Bouikhalene and Samir Kabbaj

(1.14). Then there exist continuous unitary representation π1, ..., πn of n G, a unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 X f(x) =< ξ, diag( tr(π (ϕ(x)))) ξ >, x ∈ G, i 16i6n dπ |Φ| i ϕ∈Φ 1 X a(x) = P diag( tr(π (ϕ(x)))) ξ, x ∈ G, i 16i6n dπ |Φ| i ϕ∈Φ where dπi is the dimension of πi.

Proof. Since f is central then f satisﬁes the condition (∗) [8]. By using [15] theorem 2.1 we get the remainder.

4. the equation E(K) on Gelfand pairs In this section we determine continuous bounded solutions of the equation E(K) in the case where Φ is a ﬁnite subgroup of Aut(G) and (G, K) is a Gelfand pair.

Theorem 4.1 ([3],[9]). Let G be a locally compact group and let K be a compact subgroup of G such that (G, K) is a Gelfand pair. Let Φ be a ﬁnite subgroup of the group Aut(G) such that K is Φ-invariant. Let f ∈ Cb(G) \{0}. Then f is a solution of the equation (2.2) if and only if there exists a bounded K-spherical function ω such that 1 X f(x) = ω(ϕ(x)), x ∈ G. (31) |Φ| ϕ∈Φ

Theorem 4.2. Let G be a locally compact group and let K be a compact subgroup of G such that (G, K) is a Gelfand pair. Let Φ be a ﬁnite subgroup of the group Aut(G) such that K is Φ-invariant. Let {f, a1, ..., an} ∈ Cb(G) be a solution of (1.1). Then there exist bonded K-spherical functions ω1, ..., ωn for which ωˇi = ωi for i = 1, ..., n, a n unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 X f(x) =< ξ, diag( ω(ϕ(x))) ξ >, x ∈ G, |Φ| 16i6n ϕ∈Φ on a generalization of o’connor’s and gajda’s equations 63

1 X a(x) = P diag( ω(ϕ(x))) ξ, x ∈ G. |Φ| 16i6n ϕ∈Φ

Proof. Since (G, K) is a Gelfand pair then the condition (∗) holds. By using theorems 4.1 and 3.2 we get the remainder.

In the next corollary we suppose that f(kxh) = χ(k)f(x)χ(h), for all k, h ∈ K and x ∈ G

Corollary 4.1. Let G be a locally compact group and let K be a compact subgroup of G such that (G, K) is a Gelfand pair. Let Φ be a ﬁnite subgroup of the group Aut(G) such that K is Φ-invariant. Let {f, a1, ..., an} ∈ Cb(G) be a solution of (1.10). Then there exist bonded K-spherical functions ω1, ..., ωn for which ωˇi = ωi for i = 1, ..., n, a n unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 X f(x) =< ξ, diag( ω (ϕ(x))) ξ >, x ∈ G, |Φ| i 16i6n ϕ∈Φ 1 X a(x) = P diag( ω (ϕ(x))) ξ, x ∈ G. |Φ| i 16i6n ϕ∈Φ

In the next corollary we suppose that f is bi-K-invariant i.e. f(kxh) = f(x), for all k, h ∈ K and x ∈ G

Corollary 4.2. Let G be a locally compact group and let K be a compact subgroup of G such that (G, K) is a Gelfand pair. Let Φ be a ﬁnite subgroup of the group Aut(G) such that K is Φ-invariant. Let {f, a1, ..., an}Cb(G) be a solution of (1.7). Then there exist bonded K-spherical functions ω1, ..., ωn for which ωˇi = ωi for i = 1, ..., n, a n unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 X f(x) =< ξ, diag( ω (ϕ(x))) ξ >, x ∈ G, |Φ| i 16i6n ϕ∈Φ 1 X a(x) = P diag( ω (ϕ(x))) ξ, x ∈ G. |Φ| i 16i6n ϕ∈Φ 64 Belaid Bouikhalene and Samir Kabbaj

In the next corollary we suppose that G = K

Corollary 4.3. Let G be an abelian group. Let Φ be a ﬁnite subgroup of the group Aut(G). Let {f, a1, ..., an} ∈ C(G) be a solution of (1.13). Then there exist continuous homomorphisms m1, ..., mn of n G, a unitary matrix P ∈ M(n, C) and ξ ∈ C such that

1 X f(x) =< ξ, diag( m (ϕ(x))) ξ >, x ∈ G, |Φ| i 16i6n ϕ∈Φ

1 X a(x) = P diag( m (ϕ(x))) ξ, x ∈ G. |Φ| i 16i6n ϕ∈Φ

5. Applications As a consequence of theorem 3.2 we have the following results If f(kxh) = χ(k)f(x)χ(h), for all k, h ∈ K and x ∈ G then we obtain

Corollary 5.1. Let {f, a1, ..., an} ∈ C(G), such that f, a satisfy the condition (∗) Z Z f(zkxhy)χ(k)χ(h)dωK (k)dωK (h) = K K Z Z f(zkyhx)χ(k)χ(h)dωK (k)dωK (h), K K be a solution of the functional equation Z Z f(xky)χ(k)dωK (k) + f(xkσ(y))χ(k)dωK (k) = K K n X −1 2 ai(x )ai(y), x, y ∈ G, (32) i=1 on a generalization of o’connor’s and gajda’s equations 65 where σ ∈ Aut(G) such that σ ◦ σ = σ. Then there exists K-spherical n functions ω1, ..., ωn, a unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 f(x) =< ξ, diag( (ω (x) + ω (σ(x))) ξ >, x ∈ G, 2 i i 16i6n 1 a(x) = P diag( (ω (x) + ω (σ(x))) ξ, x ∈ G. 2 i i 16i6n

Corollary 5.2. Let {f, a1, ..., an} ∈ C(G) be a solution of the functional equation

Z n X −1 f(xky)χ(k)dωK (k) = ai(x )ai(y), x, y ∈ G, (33) K i=1 where σ ∈ Aut(G) such that σ ◦ σ = σ. Then there exists K-spherical n functions ω1, ..., ωn, a unitary matrix P ∈ M(n, C) and ξ ∈ C such that f(x) =< ξ, diag(ω (x)) ξ >, x ∈ G, i 16i6n a(x) = P (diag(ω (x)) ξ, x ∈ G. i 16i6n

If f is K-biinvariant i.e f(kxh) = f(x) for all x ∈ G and k ∈ K, we get

Corollary 5.3. Let {f, a1, ..., an} ∈ C(G) be a solution of the functional equation

Z n X −1 f(xky)dωK (k) = ai(x )ai(y). (34) K i=1

Then there exists K-spherical functions ω1, ..., ωn, a unitary matrix n P ∈ M(n, C) and ξ ∈ C such that

f(x) =< ξ, diag(ω (x)) ξ >, x ∈ G, i 16i6n 66 Belaid Bouikhalene and Samir Kabbaj

a(x) = P diag(ω (x)) ξ, x ∈ G. i 16i6n

Corollary 5.4. Let {f, a1, ..., an} ∈ C(G) such that f, a satisfy the condition (∗) Z Z Z Z f(zkxhy)dωK (k)dωK (h) = f(zkyhx)dωK (k)dωK (h), K K K K be a solution of the functional equation Z Z n X −1 f(xky)dωK (k)+ f(xkσ(y))dωK (k) = 2 ai(x )ai(y), x, y ∈ G, K K i=1 (35) where σ ∈ Aut(G) such that σ ◦ σ = σ. Then there exists K-spherical n functions ω1, ..., ωn, a unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 f(x) =< ξ, diag( (ω (x) + ω (σ(x))) ξ >, x ∈ G, 2 i i 16i6n 1 a(x) = P diag( (ω (x) + ω (σ(x))) ξ, x ∈ G. 2 i i 16i6n

If G = K is a compact group then we have

Corollary 5.5. Let {f, a1, ..., an} ∈ L∞(G) be a solution of the functional equation Z n −1 X −1 f(xtyt )dt = ai(x )ai(y). x, y ∈ G, (36) K i=1

Then there exists continuous unitary representation π1, ..., πn of G, a n unitary matrix P ∈ M(n, C) and ξ ∈ C such that 1 f(x) =< ξ, diag( tr(π (x)) ξ >, x ∈ G, i 16i6n dπi on a generalization of o’connor’s and gajda’s equations 67

a(x) = P (diag(tr(π (x)) ξ, x ∈ G. i 16i6n

Corollary 5.6. Let {f, a1, ..., an} ∈ L∞(G), such that f, a are central, be a solution of the functional equation Z Z n −1 −1 X −1 f(xtyt )dt + f(xtσ(y)t )dt = 2 ai(x )ai(y), x, y ∈ G, K K i=1 (37) where σ ∈ Aut(G) such that σ ◦ σ = σ. Then there exists continuous unitary representation π1, ..., πn, a unitary matrix P ∈ M(n, C) and n ξ ∈ C such that 1 f(x) =< ξ, diag( (tr(π (x)) + tr(π (σ(x))) ξ >, x ∈ G, i i 16i6n 2dπi 1 a(x) = P diag( (tr(π (x)) + tr(π (σ(x))) ξ, x ∈ G. i i 16i6n 2dπi

Corollary 5.7. Let G be an abelian group and let {f, a1, ..., an} ∈ C(G) be a solution of the functional equation n X −1 f(x + y) + f(x + σ(y)) = 2 ai(x )ai(y), x, y ∈ G, (38) i=1 where σ is an involution of G. Then there exist continuous homomor- ? phisms m1, ..., mn : G −→ C , a unitary matrix P ∈ M(n, C) and n ξ ∈ C such that 1 f(x) =< ξ, diag( (m (x) + m (σ(x))) ξ >, x ∈ G, 2 i i 16i6n 1 a(x) = P diag( (m (x) + m (σ(x))) ξ, x ∈ G. 2 i i 16i6n

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