THE SOLUTION OF THE FUNCTIONAL EQUATION OF D ’ ALEMBERT ’ S TYPE FOR COMMUTATIVE GROUPS

A functional equation of the form l(x+Y) + 2(x-Y) [. ei(x) 8i(y), 1 where functions i,2,i,8i, i l,...,n are defined on a co:,utatlve group, is solved. We also obtain conditions for the solutions of this equation to be matrix elements of a finite dimensional representation of the group.


INTRODUCTION.
Consider the functional equation l(x+Y) + 2(x-Y) al(X)l (y) + + (x)8 (Y) n n (i.i) where l,#2,ei, Si, i I, n are functions given on a commutative group , tak- ing values in a field of characteristic zero.
The functional equations (1.2) and (1.3) follow from the equation (x+y) a l(x) B l(y) + + a (x)B (x) m m (1.5) which also was an object of detailed study.Clearly (1.5) implies that the space obtained by taking finite linear combinations of translates of is of finite di- mension, and this of course means that is a matrix element of a finite dimension- al representation of the group .It is known (cf Engert [4], Laird [8], Stone [17]) that, for a locally compact group, every finite dimensional (or only closed in the space of all continuous functions) translation invariant subspsce consists of exponential polynomials.In other terms #. must have the form n (x) L Pi(x) gi(x), i=l where gi are multiplicative homomorphisms of into and Pi are polynomials in different additive homomorphisms of Q into .Actually, the solutions of the functional equation (1.5) are known to be of such a form in a more general sit- uation when Q is a groupoid or a semigroup and is a commutative ring (see Aczel [2], McKiernan [I0], [ii]).However, as we shall see, not every exponential poly- nomial is a solution of (1.5) in the nonlocally compact case.
In Section 3 we obtain the general form of the solutions f and g of equa- tions (1.2) and (1.3).These solutions are expressed as linear combinations of matrix elements of inequivalent finite dimensional representations of the group and also of terms involving homomorphlsms of into a vector space n over the field and homomorphisms of into additive matrix group over .Section 2 con- tains some preliminary results about polynomials on Abelian groups.The discussion of the main result is given in Section 4, where the equations (1.2) and (1.3) are proved to have all solutions being exponential polynomials.We also prove that while every solution of (1.2) is a solution of (1.5), there are solutions of (1.3) which are not matrix elements of any finite dimensional representation of , i.e.
which do not satisfy (1.5), and give sufficient conditions for a solution of (1.3) to have such form.
Let De a finite dimensional vector space over the field.
(In this paper, will be a vector space of all nn matrices over the field , or the vector n space of dimension n on. .If is an -valued function defined on the Abe- lian group q, then L(x), x E is the translation operator, L(x)(') ('+x).
Thus L is a regular representation of which acts in the linear space spanned by the translates of the function .The function is called a polynomial if, for some n, (L(x)-l)n+l(y) E 0 for all x, y E .The smallest number n for which this identity holds is called the degree of the polynomial.
Thus a polynomial of degree one satisfies the identity (x+y) + (x-y) 2(x).
The following elementary results [9] will be used in Section 3.
If is a homogenous polynomial of degree n, then for all integer j (jx) jn(x), x q.
If is a polynomial of degree n, then (x) (Lx)-l)n(y) does not de- depend on y and is an homogenous polynomial of degree n in x.
If @ is a polynomial of degree n, then (L(Xl)-l)...(L(xj)-l(x) is a poly- nomial in x of degree n-j.

4
If is a polynomial of degree n, then (x) n (x) + + 0 (x), where j (x) is a homogenous polynomial of degree j, j=O n.One has i n(X) -. (L(x)-l)n(") and for j=n-i O, j+l(.)).
5 .If 9 is a homogenous polynomial of degree n, then x) X(x,... ,x) where X(X I ,x n) is a symmetric function of Xl,...,x and for fixed n x 2 Xn,X(-,x 2 Xn) 6Horn If is a polynomial of even degree and 2 then in all formulas above L (x)-I can be replaced by L (x/2)-L (-x/2).
If k 6 n and k E n, where *n is the dual space, then <h,k> will always denote the value of the linear functional k on the element h.With this conven- tion, equation (1.2), for instance, can be rewritten as where h(x).E m, and k(y) m.Also, A t will denote the transpose of a linear transformation A.
A structure theorem for the solutions of the functional equations (1.2) and (1. hj,kj, j I ,m if, and only if, there exist nonnegatlve integers m I mR,m I +

R. r
We do not prove the next Theorem 2 since its proof is analogous to that of Theorem i. Theorem 2. Under assumptions of Theorem i a function g is a solution of the equation (1.3) with linearly independent function ui,v i i=l,...,p if, and only if, there exist nonnegative integers pl,...,pR,Pl+...+pR p, such that g 'mr> Here C(x) ,S(x),Fr(X),Qr and Qr have the same meaning as in Theorem i with mr re- placed by pr, Hom(, l),br,g r mr,gr_br 2Qrar,ar r, r= 2 R, cE C t

Pl
The vectors (x)a I, x span and the vectors C(X)al+ S(x)(x), x , span Pl; the spacesPr and *Pr f=2 ,R are spanned by the vectors (x)g r + Fr(-X)b and by the vectors [F$(x)+F$(-x)]a correspondingly.The ma- F r r r trix functions H(x,x) and F (x) r=2 R are defined uniquely up to equivalence.r We break up the proof of Theorem i into three lemmas.
Lemma i. Assume that the functional equation fl(x+Y) fl(x-Y) <w(x) ,k(y)> has a symmetric solution fl' fl (-x) fl (x)" Then the vector function w has the form, w(x) Bk(x), where B is an invertible linear operator from *m to m, Bt=B.Also k(x) Tt(x), where T is an invertible linear operator from *m to *m and there exist nonnegative integers m I m R such that *m *ml ... mr and the projections of onto *mr satisfy the following functional equa- r tion, (x+y) + (x-y) 2B r(y)r(x).
r r Here B (y) is an upper triangular matrix of dimension m with the same diagonal r r elements b(r) (y)' b(r)(y) # b(S)(y), r # s, such that QrBr(Y) B tr(y)Qr with some t invertible operators Qr' Qr Qr' r=l R, and Therefore there exists an invertible linear operator B from m to m such that B t B and for all x Q Now let V denote the linear space over spanned by the translates f(-+x), x E Q of the function f which satisfies (1.2).Then the regular representation L(x): L(x)g g(-+), gV acts in V, and the functional equation (1. "" Here f denotes the {cyclic Ivector of V corresponding to the function f(.), and h I h are vectors from V which correspond to the functions hi(.) If V_ denotes the subspace of V spanned by the vectors [L(y) L(-y)]f, y Q, then it follows from (3.3) that V has dimension m.
variant under all operators L(x) + L(-x), x Q.
Also, as is easy to see, V is in- Then, [kj (x+y) + kj (-x+y) ]hj, j=l j J j=l so that k(x+y) + k(-x+y) 2At(x)k(y). (3.4) It is evident that all matrices A(x) commute.Therefore (see Suprunenko and   *m Tyshkevich [18] p. 16) the whole space can be represented as a direct sum of invariant subspaces W r, with respect to all A(x), for r=l R. The irreducible parts of A(x) IW are equivalent while for r # s the irreducible parts of A(x) IW r r and A(x) IW s are not equivalent.Since the field is algebraically closed, Shur's lemma shows that all irreducible parts of A(x) IWr, r=l R, are one-dimensional operators.Thus all matrices A(t) have the form A(x) T-IBt(x)T, where B(x) is a quasi-diagonal matrix with blocks BI(X),...,BR(X) on the principal, diagonal, and B r(x) is an upper triangular matrix of dimension mr dim Wr, r=l'''''R with the same diagonal elements b(r QIO"" Qr where Qr is of dimension mr, and QrBr(X) Btr(x)Qr' r=l,...,R.Also, if k(y) Tt(y) then (x+y) + (x-y) 2B(y)(x).
Let (y) 1(y)O... OR(y) with rQ*mr' r=l,.." ,R be the partition of (y) into direct sum corresponding to that of the matrix B(x).Then for r=l R (x+y) + (x-y) 2B (y)r(X).
It is easy to deduce from the definition of A(x) that the matrices A(x) satis- fy D'Alembert's functional equation A(x+y) + A(x-y) 2A(x)A(y).
It follows from (3.6) that where X r is a multiplicative homomorphism of into , Xr(X+y) Xr(X)r(y).
Xr is not identically one, then All solutions of this D'Alembert's functional equation are known to be of the form (cf. Kannappan [7]) [Xr(X) + Xr(-X)]/2 where Xr is a multiplicative homorphism of q into Xr(X+y) Xr(X)Xr(Y).
If r is not identically one there exists x 0 E such that Xr(2XO) # 1 and the matrix B2(x0 )-Ir [Br(2Xo)-I]/2 is nonsingular.Moreover, one can find a nonsing- ular lower triangular matrix G such that G 2 B2(xo)-I.Indeed Clearly Gr commutes with all matrices Br(X) and QrGr GrQr .t It is easy to check (cf.[5]) that G (x+y) G r(x)Gr(y) r and QrGr (x) Gtr(x)Qr" Evidently Gr(X) and Gs(x) are inequivalent for r # s and It is also clear that G (x) is a lower triangular matrix with all diagonal r elements (and hence eigenvalues) equal to Xr(X).
It follows from (3.5) so that (x+y) + r(Y-X) 2B (X)r (y) r r r (x-y) B r (y) r (x)-Br (x) r (Y)" Using again (35) we see that 2B (x)[r(X+y) + (x-y)] 2B (y)r(2X)  We prove now that every element z K has this form.r r If there exists x 0 Kr such that Xr(XO) # i we put z (z+x0)-x 0. (3.8) Clearly z + xov Kr and x0/2 Kr.If for all X E Kr one has r(X) i, then we show that z x+y with x, y K Indeed in this case it suffices to take r y z/2.Thus (3.8) holds for all z K We prove now that (3.8) is valid for all r z. 6 .Let z 6 Kr, x Kr, then x + z Kr and x-z Kr.Therefore (z+x) + (z-x)   [G (x+z)-G (-x-z) + Gr(-Z-x)-Gr(X-Z)] r r r r r 2Br(X) [Gr(Z)-Gr(-Z) ]r" From this relation and (3.5) it follows that (3.8) holds if there exists x K such that the matrix B (x) is nonsingular.The latter condition is met if r r 2x K If 2x 6 K for all x then because of the condition 2 it r r follows x Kr for all x.Thus Xr(X) i for all x contrary to our assumption.
Thus (3.8) is true for all z q and Lemma 2 is proven.
Lemma 3. Assume that Xl(X) i and let q m I. Then 1(x) is a polynomial of degree 2q-l, which has the form q-i Mk(x,x).
polynomials are defined by the formulas i N2q-2(x) (2q-2)! [L(x/2)-L(-x/2)]2q-2N(')' We prove at first that These j=k where the coefficients djk can be found in the following way.If D is the lower -I triangular matrix formed by djk k < j, then D P where the elements Pjk of P have the form We prove now that Therefore [e(y/2)-e(-y/2) 2 [2N(x) and [e(y/2)-L(-y/2) ]2N2k(X) Note that there exists a function M(x,y) on with values in q such that 2N2(x)=M(x,x) and it possesses the properties from the condition of Lemma 3.
Now we return to the equation (3.9) which can be rewritten in the following forln [L(y/2)-L(-y/2)]E l(x) 2N(y)E l(x).
Since l(X) is a polynomial of degree one, 4 1 is an additive homomorphism.
We used here the formula which follows from the properties of M(x,x).
The formula (3.1) follows with H(x,x) Mt(x,x) and F (x) Gt(x).r r We prove now that every function f of the form (3.1) satisfies the equation The last identity follows from the formula 2k (ki)()2k-i (2p_1 i:k-i odd 2j+k-i=2p-i which is easily obtained by comparison of coefficients of a2Pb 2k-2p  Thus (1.2) holds and the statements of the Theorem 1 about vectors St(x)9(x), C(x)f I + S(X)Ql.(x ),Fr(X)Cr-Fr(-X)d r and [Ft(x)-Ftr(-X)]r ,r xE r=2,...,R follow from the assumed linear independence of functions hj ,kj, j=l m.
The uniqueness up to equivalence of the matrices in formula (3.1) is a corollary of the uniqueness of the decomposition of the space m into direct sum of subspaces invariant with respect to commuting matrices A(x) from (3.4).Theorem i is proved.
REMARK i.If q is a topological group and f (or g) is assumed to be a contin- uous (or only a measurable) function, then the condition 2 Q of the Theorem 1 (or 2) can be replaced by the following one: the subgroup 2 is dense in q In- cidentally, this condition means that the dual group does not have elements of order two.
REMARK 2. Theorems i and 2 are true if the field is not algebraically closed.
In this case all homomorphisms from into corresponding vector spaces over should be replaced by homomorphisms from into vector spaces over a finite extension of the field .Of course if is the field of reals, this extension coincides with the field of complex numbers.
For instance, any solution of the classical D'Alembert's equation (1.4) has the form [X(X) +X(-X)]/2 where X is a multiplicative homomorphism into a simple ex- tension of the initial field .
REMARK 3. The general form of a solution of (i.I) easily follows from Theorems i and 2. Namely, if m and p denote the maximal number of linearly independent functions among j (x), 8j (Y)-Sj (-Y) and among j (x), 8j (Y)+j (-Y), j=l n, re- spectively, then l(X) [f(x)+g(x)]/2, 2(x) [f(x)-g(x)]72 where the forms of f(x) and g(x) are given in Theorems i and 2.
It follows from the proof of Theorem 1 that every solution f of (1.2) has the fo rm f(x) <L(x) f,A>.(4.1) Here L is a cyclic representation of the group Q in the space V with a cyclic vec- tor f, and the space V spanned by the vectors [L(x)-L(-x)]f, x E Q has dimension m.The element A of the dual space V is a cyclic vector for the contragradient L t representation L L (x) (-x). (Indeed we define A in the following way: <h, A> h(0) for all h from V. Then <h, L (x)A> h(-x) and the vectors , L (x)A, x. 6 Q must span the whole space V .)Clearly the representation L under these conditions is defined uniquely up to equivalence.A natural question where ij (x) are elements of the matrix H(x,x)T(X)Qland 9i(y) and j (y) are coor- dinates of the functions #(y) and Tt(y)(y).
The proof of Theorem 4 under condition (ii) follows from the following for- mua valid for any solution of (1. (1.2) and (1.3) are considered.The (x) 0 for all x belonging to a compact sub- group of Q. Therefore the first term in the formula (3.1) vanishes if is a compact group.
If the group does not contain nontrivial compact groups, then any matrix homomorphism F(x) has the form F(x) exp{H(x)} where H E Horn (,). (cf. [5p. 393] for one dimensional result.)In this case, the power series for, say, [F(x)+F(-x)]/2 bears some resemblance to the function C(x) and explains the struc- ture of the latter.
Thus if G is a compact commutative group, it follows from Theorem i, that every solution of (1.2) has the form f(x) [ [ckXk(x)+dkXk(-x) ], k=l where Xk(X+y) Xk(X)Xk(Y), k=l m are different multiplicative homomorphlsms of q.The same is true for equation (1.3).
As another application of Theorems i and 2 notice that every solution of (1.2) or (i. 3) is an exponential polynomial. (However, as we noticed, not every exponen- tial polynomial can be a solution.)Indeed, the proof of Theorem 1 shows that <S(x)f I, (x)> and <T(X)Qlg(X),9(x)> are polynomials in components of (x) of degree 2m and Fr(X) gr(X) (I+C r(x)) where gr(X) is a multiplicative homomorphism, I is m r the identity matrix, and the matrix C (*) is a nilpotent one, C (x) 0. Thus r r <Fr(X)fr 'r >= r(x)<l + Cr(x)fr,r > gr(X)Pr (x) where Pr(X) is a polynomial of degree m In the case = R m one can indicate con- ditlons on the coefficients of these polynomials (See [16].).
)-g(x-y) 2<H(x,y)S(X)Qla I (Y)al>+2<S(y)(y) ,C t (x)al> R + <[Fr(x)gr-Fr(-X)b r [Ftr(Y)-Ft(-y)]a > r r r=2This identity implies, that the dimension of the subspace V is less or equal to R is a topological group and continuous solutions of equations 3) is obtained in this Section.
defined uniquely up to equivalence.Theorem 4. Every solution g of the euation (1.3) has the form (4.1) with a finite dimensional representation L under one of the two following conditions: is