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).