On Cauchy type functional equations

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PII. S0161171204304254 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON CAUCHY-TYPE FUNCTIONAL EQUATIONS

ELQORACHI ELHOUCIEN and MOHAMED AKKOUCHI

Received 24 April 2003

LetGbe a Hausdorﬀ topological locally compact group. LetM(G)denote the Banach algebra of all complex and bounded measures onG. For all integersn≥1 and allµ∈M(G), we consider the functional equationsGf (xty)dµ(t)=

n

i=1gi(x)hi(y),x, y∈G, where the functionsf,{gi},{hi}:G→Cto be determined are bounded and continuous functions on G. We show how the solutions of these equations are closely related to the solutions of the µ-spherical matrix functions. WhenGis a compact group andµis a Gelfand measure, we give the set of continuous solutions of these equations.

2000 Mathematics Subject Classiﬁcation: 39B32, 39B42.

1. Introduction. LetGbe a locally compact Hausdorﬀ group, that is, a locally com-pact group which satisﬁes the following separation axiom: every pair of distinct points inGhave disjoint neighborhoods. Letµbe a complex bounded measure onG. We con-sider the functional equation

G

f (xty)dµ(t)= n

i=1

gi(x)hi(y), x, y∈G. (1.1)

In the particular case whereµ =δe (the Dirac complex measure concentrated at the

identity element ofG) (i.e.,f , δe =

Gf (t)dδe(t)=f (e), for allf:G→C), (1.1) reduces

to Levi-Civita equation

f (xy)=

n

i=1

gi(x)hi(y), x, y∈G, (1.2)

a special case of which is Cauchy’s functional equationf (xy)= f (x)f (y), for all x, y∈G. This explains the choice of the title of this note.

Solutions of general equations like (1.2) were studied by many authors. The trigono-metric addition and subtraction formulas and their relations have been studied by Wil-son [19], Vietoris [17], and Vincze [18].

In their work, Chung et al. [7] found the solutions of the equation

f (xy)=f (x)g(y)+g(x)f (y)+h(x)h(y), x, y∈G, (1.3)

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O’Connor [11] studied a solution of the equation

fxy−1= n

i=1

ai(x)ai(y), x, y∈G, (1.4)

on a locally compact abelian group.

Poulsen and Stetkær [12] introduced and solved the following functional equations:

fxσ (y)=f (x)f (y)+g(x)g(y), x, y∈G,

fxσ (y)=f (x)g(y)+g(x)f (y), x, y∈G, (1.5)

whereσ is a homomorphism ofGsuch thatσ (σ (x))=x, for allx∈G.

For other references and more information about (1.2), we can see the monographs by Aczél [1], by Aczél and Dhombres [2], and by Székelyhidi [16].

LetKbe a compact subgroup of the group Aut(G)of all mappings ofGontoGthat are simultaneously automorphisms and homeomorphisms. Letdkbe the normalized Haar measure onK, that is, the normalized nonnegative measure onKwhich is invariant by translations ofK(see [10]) and consider

Kf

xk(y)dk= n

i=1

gi(x)hi(y), x, y∈G. (1.6)

Equation (1.6) were considered by Stetkær in several works (see, e.g., [13,14,15]) and it was solved in the particular case whenG is compact and commutative (see [13]). Furthermore, on an abelian groupG, the bounded solutions of equations like

Kf

xk(y)dk=f (x)f (y), x, y∈G, (1.7)

were discussed by Chojnacki [6] and Badora [5].

Consider the group ˜G=G×sK, the semidirect product ofGandK, where the topology

is the product topology and the group operation is given by

g1, k1g2, k2=g1k1g2, k1k2, (1.8)

Ks= {e}×Kis a closed compact subgroup of ˜G. So the functional equation (1.6) onG

is closely related to the functional equation

Ks

f (xky)dk= n

i=1

gi(x)hi(y) (1.9)

on ˜G, which is a particular case of (1.1).

Whenn=2, there are two interesting cases of (1.1):

G

f (xty)dµ(t)=f (x)g(y)+f (y)g(x), x, y∈G, (1.10)

G

f (xty)dµ(t)=f (x)f (y)+g(y)g(x), x, y∈G, (1.11)

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The aim of this note is to study (1.1). Our discussion is organized as follows. In

Section 2, we make a general setup and recall some deﬁnitions. InSection 3, we establish some general properties about the solutions of (1.1). InSection 4, we suppose thatG is compact and µ is a Gelfand measure (see [3]). In this case, when{g1, . . . , gn} and {h1, . . . , hn} are two sets of linearly independent functions andf is µ-invariant, we

completely solve (1.1). The solutions are described inTheorem 4.2. As a particular case, we obtain the result given by Stetkær whenGis compact and commutative (see [13]).

In Sections 4.3 and 4.4, we solve (1.10) and (1.11) without any assumption of µ-invariance nor of independence of the unknown functionsfandg. We notice that the solutions of (1.1), (1.10), and (1.11) are expressed in terms ofµ-spherical function of the compact groupGcharacterized in [3]. The approach adopted here is based on a general process of diagonalization of matricialµ-spherical functions.

The results obtained in this note may be viewed as some generalizations of important works studied in the literature (see [1,2,5,6,7,11,12,13,14,15,19]). Our work uniﬁes many of the results presented in these references.

2. Setup and notations. Throughout this note, G will be a Hausdorﬀ topological locally compact group. LetM(G)denote the Banach algebra of complex bounded mea-sures; it is the dual ofC0(G), the Banach space of continuous functions vanishing at

inﬁnity. For allµ, ν∈M(G), we recall that the convolution productµ∗νis the measure given by

f , µ∗ν=

G

Gf (ks)dµ(k)dν(s), (2.1)

and the involution is deﬁned inM(G)byµ∗=_{µ, where}ˇ_¯

f ,µ¯ =_{f , µ}¯ _, _{f ,}_µ_ˇ₌_{f , µ}ˇ _, _{with ˇ}_{f (x)}₌_f_x−1_, _(2.2)

for allx∈G.

LetC(G)(resp.,Cb(G)) designate the Banach space of continuous (resp., continuous

and bounded) complex valued functions. Forf∈Cb(G), we deﬁne

fµ(x)=

G

Gf (kxt)dµ(t)dµ(k), x∈G, (2.3)

and we say thatf isµ-invariant iffµ(x)=f (x), for allx∈G.

For everyn∈N,M(n,n)(C)(resp.,GLn(C)) will be the algebra of all complexn×n

matrices (resp., invertible matrices). IfA∈M(n,n)(C), we letAtdenote the transpose of

the matrixA.

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Remark2.1. IfKis a compact subgroup ofGand µ=dkis the normalized Haar measure ofK, thendkis a Gelfand measure onGif and only if(G, K)is a Gelfand pair (see [8,9]).

Definition2.2. A nonzero functionΦ∈Cb(G)is aµ-spherical function if it satisﬁes

the functional equation_GΦ(xty)dµ(t)=Φ(x)Φ(y), for allx, y∈G.

A functionΦ∈M(n,n)(C)(n≥2), is aµ-spherical matrix function ifΦ(e)=InandΦ

satisﬁes the functional equationGΦ(xty)dµ(t)=Φ(x)Φ(y), for allx, y∈G.

3. Cauchy-type functional equations. In this section, we study the general prop-erties of the functional equation (1.1) whereµ∈M(G) and the unknown functions f , gi, hi∈Cb(G), for alli∈ {1,2, . . . , n}.

The following useful lemma produces a necessary condition for (1.1) to have a solu-tion.

Lemma3.1. If(f ,{gi},{hi})are solutions of (1.1), then

n

i=1 G

gi(xty)dµ(t)

hi(z)= n

i=1 G

hi(ytz)dµ(t)

gi(x), (3.1)

for allx, y, z∈G.

Proof. Using Fubini’s theorem, (1.1) shows that

n

i=1 G

gi(xty)dµ(t)

hi(z)=

G n

i=1

gi(xty)hi(z)

dµ(t)

=

G G

f (xtysz)dµ(s)

dµ(t)

=

G G

f (xtysz)dµ(t)

dµ(s)

=

G n

i=1

gi(x)hi(ysz)

dµ(s)

= n

i=1 G

hi(ytz)dµ(t)

gi(x),

(3.2)

which proves equality (3.1).

Assumptions. For the remainder of the note, we will make the following assump-tions:

(H1) µ∈M(G)andµ∗µ=µ;

(H2) the two sets of functions{gi},{hi}are linearly independent.

We recall from [2] that assumption (H2) implies that there exist{ai}i∈{1,2,...,n}∈G×

G×···×Gand{bi}i∈{1,2,...,n}∈G×G×···×Gsuch that the matrices{gj(ai)},{hi(bj)}

are invertible.

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Theorem3.2. Letµ∈M(G)such thatµ∗µ=µ. Let(f ,{gi},{hi})be a solution of (1.1). Then

(i) for allz∈G, the following identity holds inM(n,n)(C):

G

gj

xitz

dµ(t)hi

yj

=gj

xi

G

hi

ztyj

dµ(t)

; (3.3)

(ii) letΦ(z)forz∈Gbe then×nmatrix function defined by

Φ(z)=gjai−1

Ggj

aitzdµ(t)

=

G

hi

ztbj

dµ(t)hi

bj −1

,

(3.4)

thenΦ is a µ-spherical matrix function. Furthermore, there exist the following identities:

Ggj

xitz

dµ(t)

=gj

xi

Φ(z), (3.5)

G

hiztxjdµ(t)

=Φ(z)hixj, ∀z∈G; (3.6)

(iii) let

g1, . . . , gn

=g1, . . . , gn

A−1, h1, . . . , hn

=h1, . . . , hn

At, (3.7)

whereAis an invertible matrix ofM(n,n)(C). Then(g1, . . . , gn)and(h1, . . . , hn) are linearly independent sets. There exists

n

i=1

gi(x)hi(y)= n

i=1

gi(x)hi(y), (3.8)

for allx, y∈G, and theµ-spherical matrix functionΦ _{corresponding to}_{_g i} and{h_i}defined in (ii) above is given by

Φ_(z)₌_A_Φ_(z)A−1_; _(3.9)

(iv) assume thatΦ_(z)_{has diagonal form, that is,}

Φ_(z)₌_Diag_Φ i(z)

i∈{1,...,n}, (3.10)

thenΦ

1, . . . ,Φnareµ-spherical functions with complex values and

G

g_i(xty)dµ(t)=g_i(x)Φ i(y),

G

h_i(xty)dµ(t)=Φ

i(x)hi(y),

(3.11)

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Proof. (i) FromLemma 3.1, we can easily derive (3.3). (ii) For allx1, . . . , xn,z∈G,

gj

xi

Φ(z)=gj

xi

Ghi

ztbj

dµ(t)hi

bj −1

. (3.12)

By using (3.3), we get

gj

xi

Φ(z)=

G

gj

xitz

dµ(t)hi bj hi bj −1 = G

gjxitzdµ(t)

,

(3.13)

which proves (3.5).

From the assumed form ofΦ, we compute

GΦ

(xsy)dµ(s)=gj ai −1 G G gj

aitxsy

dµ(t)dµ(s)

=gj

ai −1

Ggj

aitx

dµ(t)

Φ(y)

=gj ai −1 gj ai

Φ(x)Φ(y)=Φ(x)Φ(y).

(3.14)

Using (1.1) and assumption (H1), we obtain

n

i=1

G

gi(xs)dµ(s)hi(y)= n

i=1

gi(x)hi(y), ∀x, y∈G. (3.15)

In view of assumption (H2), we deduce that

Ggi(xs)dµ(s)=gi(x), for allx∈Gand Φ(e)=In. Therefore, according toDeﬁnition 2.2,Φis aµ-spherical matrix function.

Now, by applying (3.3) and (3.4), we get

Φ(z)hi

xj

=gj

ai −1

Ggj

aitz

dµ(t)hi

xj

=gjai−1gjai G

hjztxjdµ(t)

=

Ghi ztxj dµ(t) , (3.16)

which proves (3.6). Thus (i) and (ii) are proved.

To prove (iii), we write(g₁, . . . , g_n)=(g1, . . . , gn)A−1and(h1, . . . , hn)=(h1, . . . , hn)At,

whereAis an invertiblen×nmatrix. Then by a simple computation we get (3.8). As mentioned above, the matrices{gj(ai)},{hi(bj)}are invertible, then{gj(ai)}, {h_i(bj)}are also invertible, and from (ii), the matrix µ-spherical function Φ

corre-sponding to{g_i}and{h_i}is deﬁned by

Φ(z)=gj ai −1 Gg j

aitz

dµ(t)

=Agj ai −1 G gj

aitz

dµ(t)

A−1=AΦ(z)A−1.

(3.17)

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In the next section, we study (1.1) in the case of compact groups.

4. Cauchy-type functional equations on compact groups

4.1. The aim of this section is to treat (1.1) in the case whereGis compact (Haus-dorﬀ) not necessarily commutative. We suppose only thatGis endowed with a Gelfand measure µ (see [3]). For such a measure, we let Σµ denote the set of allµ-spherical functions onG; it is the Gelfand spectrum of the commutative Banach algebraLµ1(G)=

µ∗L1(G, dx)∗µ. We recall (see [3]) that in compact groups, aµ-spherical functionω

is also a positive deﬁnite function and in particular, ˘ω(x)=ω(x), for allx∈G.

4.2. Solutions of (1.1) for compact groups. We start by the following result concern-ingµ-spherical matricial function.

Theorem 4.1. LetΦ : G→M(n,n)(C)be a continuous µ-spherical matrix function, that is,

GΦ(xty)dµ(t)=Φ(x)Φ(y), x, y∈G, Φ(e)=In.

(4.1)

Then there existsω1, . . . , ωn∈Σµand an invertible matrixA∈M(n,n)(C)such that

AΦA−1=Diagωii∈{1,...,n}. (4.2)

Proof. Let ˆGdenote the set of irreducible characters ofG(see [10]) and consider the mapping from ˆGtoM(n,n)(C)deﬁned by

ˆ

Φχπ

=

GΦ

(x)χπ(x)dx. (4.3)

From (4.1), we have_GΦ(xt)dµ(t)=Φ(x). Consequently, we get

GΦ

(x)χπ(x)dx=

G

GΦ

(xt)χπ(x)dµ(t)dx

=

GΦ

(x)

G

χπxt−1dµ(t)

dx

=

GΦ(x) Gχπ

tx−1dµ(t)

=

GΦ

(x)ωπ

x−1dx

=

GΦ

(x)ωπ(x)dx,

(4.4)

where

ωπ(x)=

Gχπ(tx)dµ(t). (4.5)

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For ally∈G, we have

Φ(y)Φˆχπ

=

GΦ(y)Φ(x)ωπ(x)dx =

G

GΦ(ytx)ωπ(x)dµ(t)dx.

(4.6)

In view ofGΦ(kx)dµ(k)=Φ(x), it follows that

Φ(y)_Φˆχπ

=

G

GΦ

(kytx)ωπ(x)dµ(k)dµ(t)dx

=

G

GΦ

(x)ωπ

t−1_y−1_k−1_x_{dµ(k)dµ(t)dx}

=

G

GΦ

(x)ωπx−1kytdµ(k)dµ(t)dx

=

GΦ(x)ωπ(x)ωπ(y)dx

=ωπ(y)

GΦ(x)ωπ(x)dx=ωπ(y)

ˆ

Φχπ

.

(4.7)

By the Peter-Weyl theorem (see, e.g., [10]), we know that the irreducible characters of the compact groupsGform a basis of the Banach spaceL2_{(G, dx). Therefore, the Span} {ˆ_Φ(χπ)ς, χπ∈G, ςˆ ∈Cn} =Cn, and it follows that there existsχπ1, χπ2, . . . , χπn ∈Gˆ

andς1, ς2, . . . , ςn∈Cn such that{Φˆ(χπi)ςi, i=1, . . . , n}is a basis ofC

n_{. By combining}

this result with (4.7), we obtain the desired conclusion.

By using Theorems4.1and3.2, we get the following result.

Theorem4.2. Let(f ,{gi},{hi})be a solution of (1.1) such thatfisµ-invariant. Then there existsA∈GLn(C), and for eachi=1, . . . , n, there exist positive definiteµ-spherical functionsωiandαi, βi∈Csuch that

g1, . . . , gn=α1ω1, . . . , αnωnA,

h1, . . . , hn=β1ω1, . . . , βnωnAt−1,

(4.8)

f= n

i=1

αiβiωi. (4.9)

Conversely, formulas (4.8) and (4.9) define a solution of (1.1).

To solve the functional equation (1.1) in compact groups, we note the following gen-eral result that describes its solutions in case ofn=1.

Proposition4.3. Letµ∈M(G). Letf , g, h∈C(G)\{0}be a solution of the functional equation

G

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Then there exists aµ-spherical functionφonGsuch that

G

g(xty)dµ(t)=g(x)φ(y), x, y∈G,

G

h(xty)dµ(t)=φ(x)h(y), x, y∈G,

(4.11)

for allx, y∈G.

Furthermore, ifµ∗µ=µandf isµ-invariant, then there existα, β∈C∗_{such that}

g=αφ, h=βφ, f=αβφ. (4.12)

Proof. Leta, b∈Gsuch thatg(a)≠0 andh(b)≠0. In view of (4.10) andLemma 3.1, we get

h(b)

G

g(atx)dµ(t)=g(a)

G

h(xtb)dµ(t), (4.13)

and

φ(x)=_h(b)1

Gh(xtb)dµ(t)=

1 g(a)

Gg(atx)dµ(t) (4.14)

is aµ-spherical function.

Now according toTheorem 3.2, we have the rest of the proof.

Proof ofTheorem4.2. ByTheorem 3.2, there exists aµ-sphericaln×nmatrix functionΦ(z),z∈Gsuch that (3.5) and (3.6) hold. SinceGis compact, then by using

Theorem 4.1, there existω1, . . . , ωn∈ΣµandA∈GLn(C)such that

AΦA−1=     

ω1 ··· 0

..

. . .. ...

0 ··· ωn

   

. (4.15)

Now, if we put(g1, . . . , gn)=(g1, . . . , gn)A−1and(h1, . . . , hn)=(h1, . . . , hn)At, it follows

immediately from Theorem 3.2(ii) that the functions hi and gi are solutions of the

system of functional equations

G

g_i(xty)dµ(t)=g_i(x)ωi(y),

G

h_i(xty)dµ(t)=ωi(x)hi(y),

(4.16)

for allx, y∈Gand for alli=1, . . . , n.

In view ofProposition 4.3, there existsαi, βi∈C∗such that

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which implies that

g1, . . . , gn=α1ω1, . . . , αnωnA,

h1, . . . , hn=β1ω1, . . . , βnωnAt−1,

(4.18)

andf=n

i=1αiβiωi. This ends the proof of the theorem.

The subject of the following subsections is to treat some particular cases with n=2. More precisely, we are interested in solving (1.10) and (1.11). We will see that the assumptions of independence orµ-invariance required inTheorem 4.2are not needed here.

4.3. Solutions of (1.10). By using Theorems4.2and3.2, and without assuming the µ-invariance off, we get the following result.

Theorem4.4. The complete list of functionsf , g∈C(G)\{0}satisfying the functional equation (1.10) consists of the following two cases, whereΦ1,Φ2,(Φ1≠Φ2),Φare positive definiteµ-spherical functions andα, β∈C\{0}:

(i) f=(1/2α)Φ,g=Φ/2;

(ii) f=β(Φ1−Φ2)/2,g=(Φ1+Φ2)/2.

To solve this equation we need to prove the following lemmas.

Lemma4.5. Iff∈C(G)is a solution of the functional equation

G

f (xty)dµ(t)=f (x)ω(y)+f (y)ω(x), x, y∈G, (4.19)

in whichωis aµ-spherical function such thatf andωare linearly independent, then

f (x)=0, for allx∈G.

Proof. Letf satisfy (4.19). By a small computation, we show thatf isµ-invariant. Multiplying (4.19) byω(x)and integrating the result overG, we get

G

Gf (xty)ω(x)dxdµ(t)=f (y)

G

ω(x)2dx+ω(y)

Gf (x)ω(x)dx. (4.20)

SinceG|ω(x)|2dx≠0 (see [3]), and

G

Gf (xty)ω(x)dx dµ(t)

=

G

f (xtyk)ω(x)dx dµ(t)dµ(k),

G

f (x)ωxk−1_y−1_t−1_{dx dµ(t)dµ(k)}

=

G

f (x)ωtykx−1dxdµ(t)dµ(k)

=ω(y)

G

f (x)ω(x)dx,

(4.21)

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Lemma4.6. Iff , g∈C(G)withf≠0constitute a solution of the functional equation (1.10), then there existsα∈C\{0}such that

Gg(xty)dµ(t)=g(x)g(y)+αf (x)f (y), x, y∈G. (4.22)

Proof. Equation (1.10) shows that

f (x)

Gg(ytz)dµ(t)−g(y)g(z)

=f (z)

Gg(xty)dµ(t)−g(y)g(x)

, (4.23)

for allx, y, z∈G.

Leta∈G such thatf (a)≠0, Φ(x, y)=Gg(xty)dµ(t)−g(x)g(y), then we get

f (x)Φ(y, a)= f (a)Φ(x, y) and Φ(x, y)= f (x)Ψ(y), where Ψ(y)= Φ(y, a)/f (a). Consequently,f (x)f (y)Ψ(a)=f (y)f (a)Ψ(x), from which we see that there exists α∈Csuch thatα=ω(a)/f (a)andGg(xty)dµ(t)=g(y)g(x)+αf (x)f (y), for all

x, y∈G.

Now, byLemma 4.5,α≠0. This provesLemma 4.6.

Proof ofTheorem4.4. Iff,gare linearly independent, by usingLemma 4.6, the matrixµ-spherical function deﬁned inTheorem 3.2is

Φ(z)=

g(z) αf (z) f (z) g(z)

, z∈G. (4.24)

Sinceα≠0, then we can diagonalizeΦ(z)as follows:

Φ(z)=

g(z)+βf (z) 0 0 g(z)−βf (z)

, (4.25)

whereβ2₌_{α. This implies that}_g(z)₊_{βf (z)}₌_Φ1_and_g(z)₋_{βf (z)}₌_Φ2_are_µ-spherical

functions. Consequently, we obtain case (ii). The rest of the proof is obvious.

4.4. Solutions of (1.11)

Theorem4.7. The complete list of functionsf , g∈C(G)\{0}satisfying the functional equation (1.11) consists of the following two cases, whereΦ1,Φ2,(Φ1≠Φ2),Φare positive definiteµ-spherical functions andα, β∈C\{0,±i}:

(i) f=(1/(1+α2₎₎_Φ_,_g₌_α_Φ_/(1₊_α2₎

(ii) f=(βΦ1+β−1Φ2)/(β+β−1),g=(Φ1−Φ2)/(β+β−1).

To solve (1.11) we need the following lemma.

Lemma4.8. Iff , g∈C(G)constitute a solution of the functional equation (1.11), then there existsα∈Csuch that

G

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Proof ofLemma4.8. Letf,gsatisfy (1.11). A small computation shows that

g(x)

G

g(ytz)dµ(t)−f (y)g(z)−g(y)f (z)

=g(z)

Gg(xty)dµ(t)−g(x)f (y)−f (x)g(y)

.

(4.27)

The caseg=0 is trivial. Now we assume that g≠0. Leta∈G such that g(a)≠0 and Φ(x, y)=Gg(xty)dµ(t)−f (x)g(y)−g(x)f (y), then we get g(x)Φ(y, a)=

g(a)Φ(x, y)andΦ(x, y)=g(x)Ψ(y), whereΨ(y)=Φ(y, a)/g(a). Consequently, for allx, y∈G, we haveg(a)g(y)Ψ(x)=g(x)g(y)Ψ(a), which proves thatg(a)Ψ(x)= g(x)Ψ(a) and Ψ(x)=αg(x), where α= Ψ(a)/g(a), from which we getΦ(x, y)= αg(x)g(y). This completes the proof ofLemma 4.8.

Proof ofTheorem4.7. Iff,gare linearly independent, then by usingLemma 4.8, the matrixµ-spherical function deﬁned inTheorem 3.2is

f (z) g(z) g(z) f (z)+αg(z)

, (4.28)

and fromLemma 4.5,α∈C\{0,±i}, then we may diagonalizeΦ(z)as follows:

Φ(z)=      

f (z)+

α 2+

α2

4 +1

g(z) 0

0 f (z)+

α 2−

α2

4 +1

g(z)     

. (4.29)

Now if we takeβ=(α/2+α2_/4₊_{1), by a small computation, we produce the solution}

formulas of (1.11).

References

[1] J. Aczél,Vorlesungen über Funktionalgleichungen und ihre Anwendungen, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, vol. 25, Birkhäuser Verlag, Basel, 1961 (German).

[2] J. Aczél and J. Dhombres,Functional Equations in Several Variables, Encyclopedia of Math-ematics and Its Applications, vol. 31, Cambridge University Press, Cambridge, 1989. [3] M. Akkouchi and A. Bakali,Une généralisation des paires de Guelfand[A generalization of

Gelfand pairs], Boll. Un. Mat. Ital.6(1992), no. 4, 795–822 (French).

[4] ,Fonctions sphériques associées à une mesure bornée, Hermitienne et idempotente

[Spherical functions associated with a bounded idempotent Hermitian measure], Com-positio Math.90(1994), no. 3, 245–266 (French).

[5] R. Badora,On a joint generalization of Cauchy’s and d’Alembert’s functional equations, Aequationes Math.43(1992), no. 1, 72–89.

[6] W. Chojnacki,On some functional equation generalizing Cauchy’s and d’Alembert’s func-tional equations, Colloq. Math.55(1988), no. 1, 169–178.

[7] J. K. Chung, Pl. Kannappan, and C. T. Ng,A generalization of the cosine-sine functional equation on groups, Linear Algebra Appl.66(1985), 259–277.

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[9] J. Faraut,Analyse harmonique sur les espaces Riemanniens symétriques de rang un, Cours CIMPA., Université de Nancy I, 1980.

[10] M. A. Na˘ımark,Normed Algebras, Wolters-Noordhoﬀ Publishing, Groningen, 1972. [11] T. A. O’Connor,A solution of the functional equationϕ(x−y)=n

1aj(x)aj(y)on a

locally compact abelian group, J. Math. Anal. Appl.60(1977), no. 1, 120–122. [12] T. A. Poulsen and H. Stetkær,On the trigonometric subtraction and addition formulas,

Aequationes Math.59(2000), no. 1-2, 84–92.

[13] H. Stetkær,Functional equations and spherical functions, Aarhus University Preprint Series no. 18, 1994.

[14] ,Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), no. 1-2, 144–172.

[15] ,Functional equations on abelian groups with involution. II, Aequationes Math.55 (1998), no. 3, 227–240.

[16] L. Székelyhidi, Convolution Type Functional Equations on Topological Abelian Groups, World Scientiﬁc Publishing, New Jersey, 1991.

[17] L. Vietoris,Zur Kennzeichnung des Sinus und verwandter Funktionen durch Funktionalgle-ichungen, J. reine angew. Math.186(1944), 1–15 (German).

[18] E. Vincze,Eine allgemeinere methode in der theorie der funktionalgleichungen. II, Publ. Math. Debrecen9(1962), 314–323 (German).

[19] W. H. Wilson,On certain related functional equations, Bull. Amer. Math. Soc.26(1919), 300–312.

Elqorachi Elhoucien: Department of Mathematics, Faculty of Sciences, University of Ibnou Zohr, Agadir 80000, Morocco

E-mail address:[email protected]

Mohamed Akkouchi: Department of Mathematics, Faculty of Sciences, Semlalia, University of Cadi Ayyad, Marrakech 40000, Morocco