I nternat. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982) 315-335
315
THE SOLUTION OF THE FUNCTIONAL EQUATION
OF D’ALEMBERT’S TYPE FOR COMMUTATIVE GROUPS
A.L. RUKHIN
Department of Statistics, Purdue University
West Lafayette, Indiana 47907 U.S.A.
(Received June 2, 1980 and in revised form December Ii, 1980)
ABSTRACT.
A functional equation of the form
where functions
solved.
i,2,i,8i,
i
l(x+Y)
+
2(x-Y)
[.
1
ei(x) 8 i(y),
l,...,n are defined on a co:,utatlve group, is
We also obtain conditions for the solutions of this equation to be matrix
elements of a finite dimensional representation of the group.
D’Alembe’s (cosine) functonal equaon, finite
dimensional representations,
groups, ids of characterstic zero.
KEY WORDS AND PHRASES.
commve
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
43A99
i.
Primary
3..9B50, Secondary
20K99,
INTRODUCTION.
Consider the functional equation
+
l(x+Y)
where
l,#2,ei, Si,
i
ing values in a field
Clearly, if f(x)
2(x-Y)
I,
al(X)l (y)
+
+
n
(x)8 (Y)
n
n are functions given on a commutative group
,
(i.i)
tak-
of characteristic zero.
l(X)-2(x)’
g(x)
l(X)
+
.
2(x),
then
m
f (x+y)-f (x-y)
i:l
i(x)
[8 i(y)-8 i(-y)]
and
7,
i=l
,
hj(x)kj(y)
(1.2)
p
n
g(x+y) + g(x-y)
j =I
ei(x)
[8 i(y)+8 i(-y)]
i=l
u
i(x)vi(y),
(1.3)
316
A.L. RIFKHIN
where the functions
1
j
m and
i
ui,vl,
i
Therefore it suffices to consider the case when
pendent.
(I.i).
hj,kj,
Note that linear independence of h. and
implies
uj
v.(y),1
vi(-y)
i
% 2
k. (-y)
3
3
m and
,p are linearly indeor
% =-%
-kj (y),
in
j
i,
p.
i
The equation (i.i) can be viewed as a generalization of D’Alembert’s (cosine)
functional equation
#(x+y)
+ #(x-y)
(1.4)
2 (x)#(y),
which has been much studied (cf Aczel [i, p. 176], Corovei [3], Hosszu [6],
Kannappan [7], O’Connor [12], Rejto [14]).
tions (Rukhln
It also arises in statistical applica-
[15]).
The functional equations (1.2) and (1.3) follow from the equation
(x+y)
a
l(x) B l(y)
+
+
a
m
(x)B (x)
(1.5)
m
Clearly (1.5) implies that the space
which also was an object of detailed study.
is of finite di-
obtained by taking finite linear combinations of translates of
.
mension, and this of course means that
al representation of the group
[17]) that, for
is a matrix element of a finite dimension-
It is known (cf Engert [4], Laird [8], Stone
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)
where
gi
L Pi(x) gi(x),
i=l
.
are multiplicative homomorphisms of
different additive homomorphisms of
Q
into
into
and
Pi
are polynomials in
Actually, the solutions of the
functional equation (1.5) are known to be of such a form in a more general situation when
Q
is a groupoid or a semigroup and
[2], McKiernan [I0], [ii]).
is a commutative ring (see Aczel
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 equations (1.2) and (1.3).
These solutions are expressed as linear combinations of
matrix elements of inequivalent finite dimensional representations of the group
FUNCTIONAL EQUATION FOR CUMMUTATIVE GROUPS
and also of terms involving homomorphlsms of
and homomorphisms of
field
317
.
n over the
into a vector space
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.
2.
POLYNOMIALS OVER COMMUTATIVE GROUPS.
Let
will be a vector space
n
of dimension n on.
space
lian group
q,
.
of all nn matrices over the
then L(x), x E
is the translation operator,
the translates of the function
(L(x)-l)n+l(y)
E
field
,
or the vector
is an -valued function defined on the Abe-
If
. .
Thus L is a regular representation of
some n,
(In this paper,
De a finite dimensional vector space over the field.
L(x)(’)
(’+x).
which acts in the linear space spanned by
is called a polynomial if, for
The function
The smallest number n for which
0 for all x, y E
this identity holds is called the degree of the polynomial.
Thus a polynomial of degree one satisfies the identity
(x+y) + (x-y)
If
2
Horn
2(x).
=
this condition implies that
(q,);
i.e., X(x+y)
(x) + (y)
(x)
(x) + c, where
for all x, y
c
,
q.
A polynomial @ is said to be homogenous, if
(e (x) -l)
n@ (.
nl@(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)
If
jn(x),
x
is a polynomial of degree n, then
q.
(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-
A.L. RUKHIN
318
nomial in x of degree n-j.
4
is a polynomial of degree n, then
If
n (x) +
(x)
where
(x),
1
jCx)
x
j.---CLCx)-l)
n) is a
Xn,X(-,x2
,x
2
If
One has
-. (L(x)-l)n(")
i
j
n (.1
(f(-)
j+l(.)).
If 9 is a homogenous polynomial of degree n, then
X(X I
n.
O,
and for j=n-i
.
0
(x) is a homogenous polynomial of degree j, j=O
j
n(X)
5
+
symmetric function of
Xn)
Xl,...,xn
x)
X(x,... ,x) where
and for fixed
6Horn
is a polynomial of even degree and
then in all formulas
2
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
f(x+y)
where h(x). E
m,
and k(y)
f(x-y)
m.
<h(x),k(y)>,
(2.1)
t
Also, A will denote the transpose of a linear
transformation A.
3.
THE .AIN RESULT.
A structure theorem for the solutions of the functional equations (1.2) and
(1.3) is obtained in this Section.
Theorem I.
Assume that
is a commutative group such that
function f taking values in an algebraically closed field
2 ]
A
].
of characteristic
zero is a solution of the equation (1.2) with linearly independent functions
I
hj,kj,
j
m
mR,mI +
I
,m if, and only if, there exist nonnegatlve integers
+ mR
m such that
<S(X)fl,X)>
f(x)
+
<T(X)QlX)>
R
+
Here
Hom(,ml),S(x)
r
__[
<F (x)f
ml-i
r
r
r
>
+ <F r (-x)d r’
k-- Hk (x,x)/(2k+l)!,T(x)
r>] +
ml-i
k=
k
c.
(3.1)
H (x,x)/(2k+2)!,where
319
FUNCTIONAL EQUATION FOR CIMMUTATIVE GROUPS
m
for each y E
m
F
H
I >_ i,
Q H(-,y) EHom
2(x,y)
(Qml),
H(x,y)
I
H(y,x), H
(x,y)
0 if
H t(x,y)x) for all x,y;
H(x,x)H(y,y),Ht(x,x) (y)
r=2,...,R are pairwise inequivalent matrix representations of the groupQ of de-
r
gree
mr where all
*mr
vertible linear operators from
R, f r + d
r
r=2
ml
C(x)
Hk(x,x)/(2k)!;
equal and not identically one;
mr,
r=2,...,R;
r=2
2Qr r,
S(X)QlX), x q span
ml-1
*ml,
to
QiHt(x,x),Fr(X)Qr QrFrt(X),
H(x,X)Ql
mr
fr,dr
C(x) f I +
Fr are
eigenvalues of
R,
spondingly.
[Ft(x)-Ft(-X)r
]r’
r
the spaces
xE
c
and the vectors S
k=0
the vectors
fl
mr
t
mr
.
are in-
*mr
Also the vectors
(x)x) span
mr
and
and by the vectors
Qr
r=2
R are spanned by
Fr(x)fr-Fr(-X)d r x
,
corre-
The representation (3.1) is unique up to equivalence for matrices H(x,x)
and F (x), r=2
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
there exist nonnegative integers
g(x)
ui,v i
pl,...,pR,Pl+...+pR
<C(X)Qlgl,al >
+
i=l,...,p if, and only if,
p, such that
<S(x)(x),a
R
+
<F (x)g
r
r= 2
,S(x),Fr(X),Qr
by pr, Hom(,
a >
r
r
+ <F r (-x)b r ’mr>
Here C(x)
and
placed
l),br,g r mr,gr_br
t
The vectors C (x)a I,
x
spacesPr
and
Pl;
the
F (x)g
r
r
+
Fr(-X)b r
trix functions
Qr
span
*Pr
Pl
f=2
and the
mr re-
R, cE
2Qrar,ar r, r= 2
span
vectors C(X)al+ S(x)(x), x
,
,R are spanned by the vectors
and by the vectors
H(x,x) and F r (x) r=2
[F$(x)+F$(-x)]a r
correspondingly.
fl(x-Y)
The ma-
R are defined uniquely up to equivalence.
Assume that the functional equation
fl(x+Y)
c
have the same meaning as in Theorem i with
We break up the proof of Theorem i into three lemmas.
Lemma i.
+
<w(x) ,k(y)>
320
A.L. RUKHIN
fl’ fl (-x) fl (x)"
has a symmetric solution
Then the vector function w has the
m,
Bt=B. Also k(x) Tt(x), where T is an invertible linear operator from *m to
mr
*m and there exist nonnegative integers mI mR such that *m *ml
form, w(x)
Bk(x), where B is an invertible linear operator from
*m
to
...
the projections
and
r
of
*mr satisfy the following functional equa-
onto
tion,
r
(x+y) +
r
2B r(y)r(x).
(x-y)
Here B (y) is an upper triangular matrix of dimension m with the same diagonal
r
r
elements
b(r) (y)’ b(r)(y)
invertible operators
#
b(S)(y),
t
Qr’ Qr
Br(X+y)
Proof of Lemma i.
+
Since
r # s, such that
Qr’ r=l
Br(x-y).
fl
QrBr(Y)
R, and
2B
r(x)Br(y)
<w(y),k(x)>.
Therefore there exists an invertible linear operator B from
B
B and for all
x
L(x):
L(x)g
to
m such
that
(3.2)
Bk(x).
spanned by the translates f(-+x),
Now let V denote the linear space over
Q of the function f
m
Q
w(x)
xE
tr(y)Qr with some
is symmetric
<w(x),k(y)>
t
B
which satisfies (1.2).
Then the regular representation
g(-+), gV acts in V, and the functional equation (1.2 means
that
m
[L(y)
L(-y)]f
k
j=l
j
(y)h
(3 3)
""
Here f denotes the {cyclic Ivector of V corresponding to the function f(.), and
h
h
I
If
V_
m
are vectors from V which correspond to the functions
denotes the subspace of V spanned by the vectors [L(y)
it follows from (3.3) that V
has dimension m.
variant under all operators L(x)
+ L(-x),
x
+ L(-x)]/2
m
j=l
k (y)A(x)h
j
J
j=l
y Q,
Q.
Then,
m
m
L(-y)]f,
Also, as is easy to see, V
Let A(x) denote the restriction of the operator [L(x)
2
h (.).
hi(.)
[kj (x+y)
+
kj (-x+y) ]hj,
on V
then
is in-
321
FUNCTIONAL EQUATION FOR CUMMUTATIVE GROUPS
so that
2At(x)k(y).
k(x+y) + k(-x+y)
Therefore (see Suprunenko and
It is evident that all matrices A(x) commute.
Tyshkevich [18] p. 16) the whole space
*m
can be represented as a direct sum of
invariant subspaces Wr, with respect to all
parts of A(x)
and A(x)
IWs
IWr
A(x), for r=l
#
while for r
are equivalent
lemma shows that all irreducible parts of A(x)
operators.
B
r(x)
IWr,
b(r) (x),b (r) (x)
and all matrices B (x), r=l
r
TBT
Let Q
Shur’s lemma Q
QrBr(X)
t.
Then
Qt
QIO"" Qr
Btr(x)Qr’
,R
T-IBt(x)T,
mr
dim
(s)
(x), r # s.
# b
Qr
where
is of dimension
Tt(y)
Also, if k(y)
1(y)O... OR(y)
with
mr,
(x+y) +
r=l’’’’’R
Clearly m
with the
ml+...+mR
r
(x-y)
B t(x)Q.
Because of
and
then
2B(y)(x).
rQ*mr’
r=l,.." ,R be the partition of (y)
into direct sum corresponding to that of the matrix B(x).
r
Wr,
commute.
(x+y) + (x-y)
Let (y)
where B(x) is
on the principal, diagonal, and
Q, Q is invertible and QB(x)
r=l,...,R.
IWr
R, are one-dimensional
r=l
BI(X),...,BR(X)
is an upper triangular matrix of dimension
same diagonal elements
The irreducible
is algebraically closed, Shur’s
Thus all matrices A(t) have the form A(x)
a quasi-diagonal matrix with blocks
R.
s the irreducible parts of A(x)
Since the field
are not equivalent.
(3.4)
2B
r
Then for r=l
R
(3.5)
(y)r(X).
It is easy to deduce from the definition of A(x) that the matrices A(x) satisfy D’Alembert’s functional equation
A(x+y) + A(x-y)
(3.6)
2A(x)A(y).
It follows from (3.6) that
B (x+y) +
r
Br(x-y)
2B
r
(X)Br(Y)
r=l
(3.7)
,R,
so that Lemma i is proven.
Lemma 2.
For r=l,...,R
b(r)(x)
IX r(x) + r(-X)]/2’
where X is a multiplicative homomorphism of
r
Xr
is not identically one, then
into
, Xr(X+y)
Xr(X)r(y).
If
322
A.L. RUKHIN
r
(x)
G (-x)]
r
r
[G (x)
r
where G (x) are lower triangular matrices of dimension m with all diagonal eler
r
ments equal to
t
Qr’ gr
Qr’ Qr
Gt(x)Qrr with invertible
Gr(X)Gr(Y); QrGr (x)
Xr(X)’ Gr(x+Y)
m.
(
It follows from (3.7) that
Proof of Lemma 2.
b(r)(x+y)
+
2b(r)(x)b(r)(y).
b(r)(x-y)
All solutions of this D’Alembert’s functional equation are known to be of the
form (cf. Kannappan [7])
b
where
Xr
(r)
r
Xr(-X)]/2
B2(x0
)-Ir
[Br(2Xo)-I]/2
ular lower triangular matrix G
B
where P
2(xO)-I
r
r
E
0
such that
such that G
B2(xo)-I.
r
r
pr
Indeed
O.
Thus one can put
m
-I
(-l)i+l(2i-l) ’’..
+
Gr [(Xr(X0)-Xr(-X0))/2][I+Pr/2
i=2
Clearly
Gr
Br(X)
commutes with all matrices
2
and
i
QrGr
i!
.t
GrQr
Now let
Gr(X
G-l[Br(X)(Gr-B
(x0)3 + B r (x+x0)]
r
B (x)-Gl [B (x)B (Xo)-Br(X+XO) ].
r
r
r
r
It is easy to check (cf. [5]) that
r(x)Gr(y)
G (x+y)
r
G
QrGr (x)
Gtr(x)Qr"
and
Evidently
Gr(X)
and
Gs(x)
# 1 and the
Moreover, one can find
is nonsingular.
2
Xr(2XO)
[(Xr(XO)-Xr(-XO))/212[I+Pr
is a nilpotent matrix,
r
into
Xr(X)Xr(Y).
is not identically one there exists x
matrix
q
is a multiplicative homorphism of
Xr(X+y)
If
+
[Xr(X)
(x)
are inequivalent for r
#
s and
P]
a nonsing-
323
FUNCTIONAL EQUATION FOR CUMMIFfATIVE GROUPS
+
G (x)
r
-I
2B (x)-G
r
r
G (-x)
r
[2Br (X)Br (Xo)-Br (X+Xo)-Br (-X+Xo)
2Br (x)
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)
r
(x+y) +
r(Y-X)
2B
r
(X)r (y)
so that
r (x-y)
B (y)
r
r (x)-Br (x) r (Y)"
Using again (35) we see that
2B
r (x)[r(X+y)
+
r
(y)r(2X)
2Br(Y)[Br(x-Y)r(X+y)
(x-y)]
2B
r
+
Br(X+Y)r(X-y)].
Now one deduces from (3.7)
B
r
[B (x) + B (x-2y)]/2
(Y)Br(x-y)
r
r
and
Br(Y)Br (x+y)
[Br(X)
+ B r (x+2y)]/2.
Thus
[Br (x)-Br (x-2y) ]r (x+y)
-[Br (x)-Br (x+2y) ]r (x-y).
It is easy to check that
[Gr(Y)-G r (-y)][Gr(x-y)-Gr(rX+y)]/2
Br(X)-B r(x-2y)
and
-B (x) + B (x+2y)
(y)-Gr(-y)][Gr(x+y)-Gr(-X-y)]/2.
K the matrix G (x)-G r(-x) is nonsingular.
Xr(2X)
r
r
It follows
K [Gr(x+y)-G (-x-y)] (x-y)
[G
(x-y)-Gr(-X+y)]Er(X+y).
r
r
r
r
r
Let K
{x:
r
Thus if y
i}.
x+y
that the relations
[G
r
[G
r
K
r
and x-y
(x+y)-Gr(-X-y) ]-i r (x+y)
19other words for m
K
with some vector
x+2y, x, y
r
If there exists x
r
imply
-i
[G r(x-y)-G r(-x+y)
r
x+y
(3.8)
with y
K
r
and x-y
We prove now that every element z
0
(x-y).
[G r(z)-G r(-z)]r
r if z has the form z
K
K
r
r(Z)
z
r
If x
Kr such that Xr(XO)
#
i we put z
K
r
K
r
or
has this form.
(z+x0)-x 0.
324
A.L. RUKHIN
Clearly z +
xov Kr
show that z
x+y
x0/2 Kr.
and
K
with x, y
one has
r(X)
i, then we
Indeed in this case it suffices to take
r
z/2.
y
.
Thus (3.8) holds for all z
z. 6
Kr
If for all X E
Kr,
Let z 6
r
(z+x) +
r
Kr,
x
K
We prove now that (3.8) is valid for all
r
+ z
then x
Kr
Kr.
and x-z
Therefore
+
Gr(-Z-x)-Gr(X-Z)] r
2Br(X) [Gr(Z)-Gr(-Z) ]r"
(z-x)
[G r (x+z)-Gr (-x-z)
From this relation and (3.5) it follows that (3.8) holds if there exists
x
K
2x
such that the matrix B (x) is nonsingular.
r
r
K
If 2x 6 K
r
follows
x Kr
r
for all
for all x.
x
Thus
zq
Assume that
Xl(X)
Lemma 3.
then because of the condition
Xr(X)
Thus (3.8) is true for all
The latter condition is met if
2
it
i for all x contrary to our assumption.
and Lemma 2 is proven.
i and let q
m
I.
Then
1(x)
is a polynomial
of degree 2q-l, which has the form
q-i
k
l(X)
0
k(x,x).
M
(2k+l)! l(X)"
Here M(x,y) are matrices of dimension q under the following conditions:
M(Xl+X2,Y)
Mq(x,x)
O,
M(Xl,Y)
M2(x,y)
+
M(x2,Y)
M(x,x)M(y,y),
91 (x+y) =91 (x) + 91 (y)’9
Proof of Lemma 3.
1
QiM(x,x) Mt(x,X)Ql
M(x,Y)l(X) M(x,x)l(y) and
M(y,x),
M(x,y)
q"
We have
I + N(x), where
Bl(X)
Nq(x)
0, q
m
I.
Thus
i (x+y)
+
1(x-y)-2gl(X)
2N(y) 1(x)
and
N(x+y) + N(x-y)-2N(x)
2N(y) + 2N(x)N(y).
The latter identity can be rewritten
[e(y/2)-e(-y/2)]2N(x)
Easy induction shows that for k
2N(y)[l + N(x)].
1,2,...
[L(y/2)-L(-y/2)]2(x)
2k(y)[I
+ N(x)].
Thus in particular
[L(y/2) ]-L(-y/2) ]2q-2N(x)
2q-iN q-I (y),
(3.9)
FUNCTIONAL EQUATION FOR CUMMUTATIVE GROUPS
325
which implies
[l-L(y)]2q-lN(x)
i.e., N(x) is a polynomial of degree 2q-2.
0;
Because of the result mentioned in
Section 2,
N(x)
where for k
N2q_2(x)
+
+
l,...,q-i
[n(y/2)-L(-y/2)]2kN2k(X
i.e.,
Nz(x)
N2k(X)
is a homogenous polynomial of
(2k)!N2k(Y);
degree 2k, N2k(nx)
n2kN2k(X).
These
polynomials are defined by the formulas
q-2
and for k
N2k(X)
[L(x/2)-L(-x/2)]2q-2N(’)’
i
N2q-2(x)
(2q-2)!
,i
[L(x/2)-L(-x/2)]2k[N(’)-N2q_2(’)-...-N2k+2(’)].
1
We prove at first that
q-i
Z
N2k(X)
where the coefficients
j=k
djk[2N(x)]J/(2j).’,
can be found in the following way.
djk
triangular matrix formed by
djk
If D is the lower
P -I where the elements
k < j, then D
have the form
i
jP-k
(Clearly
Pjk
2k)’
2k
i=0
2k
(i)(-l)
i
(i-k)
2j
=0ifk>j).
Indeed,
N2q- 2(x)
so that d
q-i q-i
[L(x/2)-L(-x/2) ]2q-2N(-)
(2q-2)!
i
[2N(x)
(2q-2)
q-I
i.
Also,
[L(x/2)-L(-x/2)]2kN2j (0)
2k
Z
i=0
2k
i
2k)
(-1) N2j ((k-i)x)
i
2k
i
i
(-i) (i-k)
(2k)!PjkN2j (x).
2j
N2-’3 (x)
Pjk
of P
326
A.L. RUKHIN
Thus
I
(2k)
N2k(X)
[L(x/2)-L(-x/2)
ql
[2N(x) ]k/(2k)!-
j=k+l
2k[N(0)-N2q_2 (0)p
"-N2k+2 (0)
(3.10)
kN2j (x),
and it follows by induction that
J
djk=- i=k+l dji Pik
J3
But these identities mean D
-D(P-I) + I or DP
> k
I.
We prove now that
[L(y/2)-L(-y/2)]2N2k(X)
2[N2k(Y)
+ },
j<k
N2j (Y)N2 (k_j) (x) ].
Indeed
[L(y/2)-L (-y/2)
2
[2N(x)
k l(Z)
[e(y/2)-e(-y/2)]2
,
2k
i--0
2k
i=0
k-i
"
i=0
k-i
i=0
k-i
i
2k)
(-i) 2N((i-k)y) l(z+(i-k)x)
i
2k
i
)(-l)i2N((i-k)Y)[lz+(i-k)x)+l (z-(i-k)x)
(2k)(-l)i4N((i-k)y)N((i-k)X)gl(Z).
1
2k
(-1)i2N((i-k)Y) 1 (z)
i
, (2k)
2k
i=0
2k
(2k)(_l)i
(z+(i-k)x)
1
i
i
(-i) 12N ((i-k) Y)N ((i-k) x)
i (z)
(2k)
i (-1)12N((i-k)Y)gl(Z).
Therefore
[e(y/2)-e(-y/2)
2
[2N(x)
2(2k)!
ipj+ikN2jy)N2i (x)+2(2k)
J,
and
[e(y/2)-L(-y/2)
]2N2k(X)
q-1
j=k
djk[L(y/2)-L(-y/2)
2 [2N(x)
(2j)!
]J
PjkN2j (y)
FUNCTIONAL EQUATION FOR CUMMUTATIVE GROUPS
327
-I
I djk Pn+ijN2N (y)N2 i (x)+2
=2j=k
nl
2[
N
i<k
2i
(y)N 2 (k-i) (y) +
j=k
i
djk PijN2i(Y
N2k (y)]
Using (3.10) repeatedly we can now establish the following formula
[L(y/2)-L(-y/2)]22k(.)
2
k
jl+...j k
[2N
k
N2Jl (Y)...N2j
2(y)]k,
(3.]_1)
which gives the basic result:
i
N2k(Y)
[L(y/2)-L(-y/2)
(2k)!
]2]q2k(-)
Note that there exists a function M(x,y) on
2N2(x)=M(x,x)
(y)
k
2N (Y)
2
(2k)!
with values in
q
such that
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)
It is easy to check that for k
2N(y)E l(x).
1,2,...
[e(y/2)-t(-y/2)]2kE l(x)
[2N(y)]kE l(x).
Thus
[ey/2)-t(-y/2) ]2qE l(x)
and
El(X)
0,
is a polynomial of degree 2q-l.
Analogously to previous considerations,
l(X) --92q_l(X)
where
2k+l(X)
}2k+l (nx)
n
2k+l
E(x)
+91(x),
is an homogenous polynomial of degree 2k+l,
g2k+l (x).
Note, that if 2x
function
+
2y, then
2i(x/2)
92k+i (x)
2k+l (y)’
k
0,i
q-l.
Thus the
is defined.
Similarly to (3.10) we prove
q-i
2k+l(x)
c
j=k
jk
[(2j+l)’]-i [2N(x)] j E(x),
where the lower triangular matrix C formed by the coefficients
form C
-1.
V
Here V is the matrix with elements
(3.12)
Cjk,
k < j has the
328
A.L. RUKHIN
2k
2
Vjk
Clearly v
Jk
(2k+l)!
0 if k >
0
i
2j+l
(2k)
(-i) (i-k+ll2)
i
J.
Also
[L(y/2)-L(-y/2) ]212N(x)
2[L(y/2)-L(y/2)]2
j
(-i)
i
i=0
i 1 ((i-J) x+x/2)
2j
2
i=0
27
4
i=0
(i.3)(-l)12N((i_j+i/2)Y)1((i_j+i/2)x)
2j
(i)(-1 )i
n ,k
N2n(Y)k+l(x)(i-j+l/2)2n+2k+l’z
so that
[L(y/2)-L(-y/2)]2 2k+l(X)
2
jk_ CkVn+i
.
n,1
T2n y )2i+i (x)
2 N2 (k-i) (Y)2i+l (x).
Using this identity repeatedly one obtains
[L(y/Z)-L(-y/2) 2k %k+l(X)
2
k
ii+.
[2N
.+ik+ik+l=kN2il (y)...N2ik(Y) 2ik+l+I (x)
2(y)]kl(x)
and
[L(y/2)-L(-y/2)
]2k+l2k+l (x)
[2N 2 (y)
]kl(y).
k=0,1
q-i
(3.13)
Therefore
2N (x)
2
92k+I (x)
(2k+l)!
k
i (x)’
and
l(X)
-i
k=0
[2N2(x)]
(2k+l)
k
l(X)
The relation (3.13) for k=l implies that
M(x,y) ql (y)
Since
l(X)
M(y, y) ql (x).
is an additive homomorphism.
is a polynomial of degree one, 4
1
Thus
we proved Lemma 3.
Proof of Theorem i.
Now we are able to obtain the desired formula for the
FUNCTIONAL EQUATION FOR CUMMIPrATIVE GROUPS
solution of (1.2).
Clearly
fl(x)
f2(x)
f(x)
where
[f(x) + f(-x)]/2,
fl(x)
329
+
f2
[f(x) -f(-x)]/2.
It follows from (1.2)
that
fl(x+Y)-fl(x-Y)
<[h(x)-h(-x)]/2,k(y)>
<w(x),k(y)>
and
f2(x)
Thus
fl(x)
<h(0),k(x)>
<h,k(x)>
is a symmetric solution of the functional equation of Lemma i so that
the vector functions k(x) and (x) possess all properties given in the Lemmas.
Therefore
f(x)-f(0)
fl(x)-fl(O)+f2(x)
<h(x/2) ,k(x/2) >
<w(x/2) ,k(x/2)>+<h,k(x)>
<Q(x/2),(x/2)> + <Th,(X)>
R
<Qrr (x/2) ’r (x/2)>+<fr’r (x)>
r=l
<
ml-i Mk(x/2 x/2)
fl’ E (2k+l)
k--0
ml-I [Mt (x/2,x/2)
i
+
<
R
Z
r=2
E
(x)>
ql(x),
(2k+l)!
ml-i Mi(x/2, x/2)
(x)>
(2i+l)
i--0
k=O
{<[Gt(x/2)-Gtr(-X/2)]Qrr
r
[Gr(X/2)-Gr(-X/2)] r >
+
<fr’ [Gr(X)-Gr(-X) ]>
ml-I Mt (x,x)]k
(x)>
<
(2k+l)! fl
+
<
ml-I [Mt(x,x) ]k
(x)
(2k+2)! Q
k=0
k=0
R
+
[<c ,G
r=2
We used here the formula
r
r(x) r>+<dr’ G r(-x) r >+<c r-d r
r
>
x)>
330
A.L. RUKHIN
I
(x/2,x/2)
(2k+l)
=0
]
2
q-i
2
k=0
(x,x)
(2k+2)’
which follows from the properties of M(x,x).
Mt(x,x)
The formula (3.1) follows with H(x,x)
Gt(x).
r
and F (x)
r
We prove now that every function f of the form (3.1) satisfies the equation
(1.3).
Note that for k > i
k
k
H (x+y, x+y)-H (x-y, x-y)
2i:k_i odd(ik)[H(x,x)+H(y,y)]i[2H(x,y)] k-i
2i:k-iE odd,
2
<
i+l k
(ki) (j)[H(x,x)] j+(k-i-l)/2 [H(y,y)] i-j+(k-i-l)/22k-iH(x,y
i
j<_i
2k
[H(x,x) ]i [H(y,y)
(2 i+l
k-l-i
(3.14)
H (x y).
The last identity follows from the formula
i:k-i odd
2j+k-i=2p-i
2k
(ki)()2k-i
(2p_ 1
a2Pb 2k-2p
which is easily obtained by comparison of coefficients of
(a2+b2+2ab)k-(a2+b2-2ab)k
and
in expansions
(a+b)2k-(a-b) 2k.
Also,
(x+y,x+y)
+
Hk(x-y,x-y)
2
.2k. i
i<k
[2i)H
(x x)H
k-i(y,y)
Using this formula, (3.14) and properties of the function H one obtains
f(x+y)-f(x-y)
2<
ml-I Hk(x,x)
(2k)
k=0
fl
+
ml-i
ml-iHk
k0
(2k+l)
Q19(x)’
E
t
[H (y y)
]i
(2i+i)
i=0
9(y)>
R
+
E
r=2
<F (x)c
r
t(-y)]
>.
r-Fr (-x)d r [Ft(y)-F
r
r
r
(3.15)
Thus (1.2) holds and the statements of the Theorem 1 about vectors
C(x)f
I
+
S(X)Ql.(x ), Fr(X)Cr-Fr(-X)d r
and
[Ft(x)-Ftr(-X)]r
,r
from the assumed linear independence of functions
hj ,kj,
j=l
xE
St(x)9(x),
r=2,...,R follow
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
FUNCTIONAL EQUATION FOR CUMMUTATIVE GROUPS
respect to commuting matrices A(x) from (3.4).
REMARK i.
If
q
Theorem i is proved.
is a topological group and f
(or g) is assumed to be a contin-
uous (or only a measurable) function, then the condition
1 (or 2) can be replaced by the following one:
331
2 Q
the subgroup 2
of the Theorem
q
is dense in
In-
cidentally, this condition means that the dual group does not have elements of
order two.
Theorems i and 2 are true if the field
REMARK 2.
into corresponding vector spaces over
In this case all homomorphisms from
.
should
into vector spaces over a finite extension of
be replaced by homomorphisms from
the field
is not algebraically closed.
is the field of reals, this extension coincides with
Of course if
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
tension of the initial field
The general form of a solution of (i.I) easily follows from Theorems
REMARK 3.
i and 2.
.
where X is a multiplicative homomorphism into a simple ex-
Namely, if m and p denote the maximal number of linearly independent
j (x), 8j (Y)-Sj
functions among
(-Y) and among
j (x), 8j (Y)+j (-Y),
j=l
n, re-
spectively, then
[f(x)+g(x)]/2,
l(X)
2(x)
[f(x)-g(x)]72
where the forms of f(x) and g(x) are given in Theorems i and 2.
4.
DISCUSSION.
It follows from the proof of Theorem 1 that every solution f of (1.2) has the
fo rm
(4.1)
<L(x) f,A>.
f(x)
Here L is a cyclic representation of the group Q in the space V with a cyclic vector f, and the space V
m.
The element A of the dual space V
representation L
<h, A>
,
spanned by the vectors [L(x)-L(-x)]f, x E Q has dimension
L (x)
t
L (-x).
h(0) for all h from V.
L (x)A, x. 6
Q
is a cyclic vector for the contragradient
(Indeed we define A in the following way:
Then <h, L (x)A>
must span the whole space V .)
h(-x) and the vectors
Clearly the representation L
under these conditions is defined uniquely up to equivalence.
A natural question
A.L. RUKHIN
332
is whether the representation L is finite dimensional.
Bounds for the dimension
The same question can be formulated for
of L in terms of m are also of interest.
the functional equation (1.3)
It was proved in [13] that for both equations the space V is finite dimensional if
Q
is a compact group.
In the non-locally compact case the situation for the
equations (1.2) and (1.3) is different.
Here is an example of a solution to (1.3)
with infinite dimensional representation L.
lxl 12
be an infinite dimensional Hilbert space, g(x)
Let
g(x+y) + g(x-y)
11x
2(
lYl 12),
12 +
so that (1.3) holds, and the dimension of the subspace
[L (x)+L (-x) ]g, x6 Q
is two, and
Then
V+ spanned by
the vectors
g(x) is a polynomial of degree two.
However
4 < x,y
g(x+y)-g(x-y)
and the space V
is an infinite dimensional one.
dimensional space as well.
>,
Therefore V
V(g) is an infinite
Thus not every polynomial solves (1.2) or (1.5).
Note that in this example the homomorphism
that if g is an odd function, g(-x)
g(x+y)-g(x-y)
so that g also satisfies (1.2).
of Theorem 2 is zero.
Also note
-g(x), and g satisfies (1.3), then
g(x+y) + g(y-x)
Thus both spaces
the dimension of V does not exceed 2p.
<u(y), v(x)>,
V+and
V_
have dimension p, and
Of course the same remark refers to equa-
tion (1.21.
Now let f be a solution of (1.2).
Then F has the form (3.11, and
2<C(Y)fl,St(x)(x)>
F(x+y) + f(x-y)
+ 2<H(x,y)T(x)Q I (x),
rt(y) g (y)>+2<T(X)Ql
R
+
2<T(ylQIqqy),
qy)>+
<F (x)f +F (-x)d
r
r r
r
r=2
(x), (y)>
>.
[Ft(xl+Frt(-x)]
r
r
The proof of (4.2) is analogous to that of the identity (3.15).
Note that the second term in (4.1) has the form
<H(x,y)T(x)Q
19(x),Tt(y)(y)>
<H(x,x)T(x)Q
m
I (y),Tt(y) (y)>
1
i,j=l
ij (x) qi(Y)Nj (Y)’
(4.21
333
FUNCTIONAL EQUATION FOR CUMMUTATIVE GROUPS
where
ij (x)
are elements of the matrix
H(x,x)T(X)Qland 9i(y)
Tt(y)(y).
dinates of the functions #(y) and
Therefore the dimension of the space
R
m21 +
I +
m
2 +
r=2
V+
j (y)
are coor-
does not exceed
m21 +
mr
and
m / 2.
Thus
m
_<
V(f)
dim
+ 2m +
2
_<
m
2
+
2m
+ 2,
and the next result follows.
Every solution f of the equation (1.2) has the form (4.1) with
2
a finite dimensional representation L, dim L < m + m + 2. The representation L
Theorem 3.
is 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:
(i)
g(-x)
(ii)
d im Hom
-g(x),
x
q,
q,
Under condition (i) dim L
1,2,
for n
Pn <
<_
2p; under condition (ii) dim L
<_
p(pp+2).
The proof of Theorem 4 under condition (ii) follows from the following for-
mua
valid for any solution of
(1.3)
t
t
2<H(x,y)S(X)Qla I ,S (Y)al>+2<S(y)(y) ,C (x)al>
g(x+y)-g(x-y)
R
+
r=2
<[Fr(x)gr-Fr(-X)b r
>
[Ftr(Y)-Ft(-y)]a
r
r
This identity implies, that the dimension of the subspace V
is less or equal to
R
PlPPl + Pl + r=2 Pr
p
+
PI p Pl
Therefore
dim
V(g) <_ p +
Pl
ppl+l
< p
(Pp+2),
and Theorem 2 follows.
Assume now that
is a topological group and continuous solutions of equations
(1.2) and (1.3) are considered.
group of Q.
The
(x)
0 for all x belonging to a compact sub-
Therefore the first term in the formula (3.1) vanishes if
is a com-
pact 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.
334
A.L. RUKHIN
In this case, the power series for, say,
393] for one dimensional result.)
[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
[ [ckXk(x)+dkXk(-x) ],
f(x)
k=l
of
Xk(X)Xk(Y),
Xk(X+y)
where
q.
m are different multiplicative homomorphlsms
k=l
The same is true for equation (1.3).
As another application of Theorems i and 2 notice that every solution of (1.2)
(However, as we noticed, not every exponen-
or (i. 3) is an exponential polynomial.
Indeed, the proof of Theorem 1 shows that
tial polynomial can be a solution.)
<T(X)Qlg(X),9(x)> are polynomials in components of (x) of degree
I is
gr(X) (I+C r(x)) where gr(X) is a multiplicative homomorphism,
m
<S(x)f I, (x)> and
2m and
Fr(X)
r
the identity matrix, and the matrix C (*) is a nilpotent one, C (x)
r
r
<Fr(X)fr ’r >= r(x)<l
where
Pr(X)
+
Cr(x)fr,r >
is a polynomial of degree m
in this work.
grant MCS 77
Thus
gr(X)Pr (x)
In the case
ditlons on the coefficients of these polynomials (See
ACKNOWLEDGEMENT.
0.
=
m
R
one can indicate con-
[16].).
The author is grateful to Professor R.C. Penney for his interest
This work was supported in part by National Science Foundation
19640
and in part by National Science Foundation grant MCS
7802300.
REFERENCES
!.
ACZEL, J. Lectures on functional equations and their applications, Academic
Press, New York (1966).
2.
ACZEL, J.
3.
COROVEI, I.
4.
ENGERT, M.
Functions of Binomial Type Mapping Group ids into Pings,
Zeitschrift 154 (1977), pp. 115-124.
Math.
The cosine functional equation for nilpotent groups, Aequationes
Math. 15 (1977), pp. 99-106.
Math. 32
Finite dimensional translation invariant subspaces, Pacific J.
(1970), pp. 333-343.
5.
HEWITT, E. and ROSS, K.A.
6.
HOSSZU, M.
Abstract Harmonic Analysis, Vol. i, Berlin-GotingenHeidelberg, Springer-Verlag (1963).
Some remarks on the cosine functional equation, Publ. Math.
Debrecen 16 (1969), pp. 93-98.
FUNCTIONAL EQUATION FOR CUMMUTATIVE GROUPS
335
-I)
The functional equation f(xy) + f(xy
groups, Proc. Amer. Math. Soc. 19 (1968), pp. 69-74.
7.
KANNAPPAN, PI.
8.
LAIRD, P.G.
9.
LIJN, G. van der.
i0.
McKIERNAN, M.A. "Equations of the form H(x o y)
Math. 16 (1977), pp. 51-58.
Ii.
McKIERNAN, M.A.
12.
O’CONNOR, T.A.
13.
PENNY, R.C. and RUKHIN, A.L. D’Alembert’s functional equation on groups,
Proc. Amer. Math. Soc. 77 (1979), pp. 73-80.
14.
REJTO, L. B-algebra valued solution of the cosine equation, Studia Sci. Math.
Hung. 7 (1972), pp. 331-336.
15.
RUKHIN, A.L.
16.
RUKHIN, A.L.
+/-7.
STONE, J.J.
18.
SUPRUNENKO, D.A. and TYSHKEVICH, R.I.
New York, London, 1968.
2f(x)f(y) for
On characterizations of exponential polynomials,
Math. 80 (1979) pp. 503-507.
Pacific J.
La definition fonctionelle des polynomes dans les groupes
abeliens, Fund. Math. 33 (1945), pp. 42-50.
"The Matrix Equation a(x o y)
Aequationes Math. 15 (1977), pp. 213-223.
fi(x)gi(y),"
Aequationes
a(x)+a(x)a(y)+a(y),"
A solution of D’Alembert’s functional equation on a locally
compact Abelian group, Aequationes Math. 15 (1977), pp. 235-238.
The families with a universal Bayesian estimator of the transformation parameter, Symposia Mathematica, Instituto Nationale di Alta
Mathematica, Vol. 21 (1977), pp. 19-26.
Universal Estimators of a Vector Parameter, submitted to
Journal of Multivariate Analysis.
Exponential polynomials on commutative semigroups, Appl. Math.
and Stat. Lab. Technical Note, No. 14, Stanford University 1960.
Commutative Matrices, Academic Press,