I ntrnat. J. ath. & ah. Sci.
Vol. 3 No. 2 (1980) 199-235
199
MEAN-PERIODIC FUNCTIONS
CARLOS A. BERENSTEIN
Department of Mathematics
University of Maryland
College Par, Maryland 20742
U .S .A.
B.A. TAYLOR
Department of Mathematics
University of Michigan
Ann Arbor, Michigan 48109
U.S.A.
(Received September 25, 1979)
ABSTRACT:
We show that any mean-periodic function
f
can be represented in
terms of exponential-polynomial solutions of the same convolution equation
satisfies, i.e.,
’,
f
0
(E’(IRn)).
f
This extends to n-variables the work of
L. Schwartz on mean-periodicity and also extends L. Ehrenpreis’ work on partial
differential equations with constant coefficients to arbitrary convolutors.
We
also answer a number of open questions about mean-periodic functions of one variable.
The basic ingredient is our work on interpolation by entire functions in
one and several complex variables.
200
C. A.
KEY WORDS AND PHRASES.
BERENSTEIN AND B. A. TAYLOR
leai-peridic fu;zcio, Ip,CaC6,n by e fuci,ns.
1980 MATHEIATICS SUBJECT CLASSIFICATION CODES.
$102, 4202, 4602.
I.
Primy 42A84, Secodcoy 3902,
The theory of mean-periodic functions, initiated by Jean Delsarte in 1935 is
one of the latest, and most important, incarnations of the old subject of convolution equations, an area which in one guise or another has been a central object
of interest to mathematicians over the past two hundred years.
It is a subject
which touches many different branches of mathematics, ranging from number theory
to applied mathematics.
In recent years, major contributions to the theory have
been made by L. Schwartz, J.-P. Kahane, L. Ehrenpreis, and B. Malgrange, among
others.
The present survey is an attempt to place some of these contributions in
mathematical and historical perspective and to give applications of our recent
work [7,8] on interpolation by entire functions which answers a number of old
questions and (as usual) raises new ones.
While only a few proofs have been
given, we have tried to provide complete references or, at least, sources for
references.
The particular aspect of the theory of mean-periodic functions we are concerned with in this paper is their representation as sums or integrals of exponentials.
The main new result is the existence of such a representation for func-
tions of more than one variable (Theorems
7, 8).
Other results that are new even
for functions of one variable are the estimates of coefficients in exapansions of
mean-periodic functions (Theorems 4, 7, ii) and applications (Theorem 5 and Section
4), and a new "summability method" for summing formal expansions of mean-
periodic functions which, in contrast to the original method of L. Schwartz
applies in any number of variables.
We also mention some applications of our work
on interpolation from varieties of codimension greater than
of equations (Theorem 9).
[64],
1
to certain systems
201
MEAN-PERIODIC FUNCTIONS
2.
We begin by discussing some familiar convolution equations.
Consider first a
homogeneous ordinary differential equation with constant coefficients,
m
f(x)
I a’DJ
3
(l)
0
j=O
I d
i dx’
D
where
a
J
.
Euler was the first to consider the problem of finding
all solutions to this equation.
In [25] he showed that they are given by finite
sums of the form
I
f(x)
c (x)$
where the summation extends over all roots
aj (-a)
and the
c
a
a
j
(2)
of the algebraic equation
0
<ma
are polynomials of degree
iax
multiplicity of
a.
Euler also studied periodic functions and proposed to write them in terms of
exponentials.
A periodic function (of period
f(x+T)
and is represented by an
izfinite
f(x-n)
a
0
series
I
f(x)
where the frequencies
say) satisfies the equation
2n,
c e
-iax
(3)
satisfy the equation
e
ina
e
-ina
0.
For the history of the theory of these series, the Fourier series, see [28,
202
C.A. BERENSTEIN AND B. A. TAYLOR
78].
For the theory there is the comprehensive treatise [77].
The main points we
wish to recall may be summarized as follows.
THEOREM i:
f
(i)
converges to
f.
(C)
The Fourier series (3) of a smooth
periodic function
Further, the series can be differentiated term by term any
number of times and the resulting series converges to the corresponding derivative
of
f.
(ii)
The coefficients
c
a
of the above series satisfy the estimates
(4)
for every
(iii)
N > 0.
Conversely, any series (3) whose coefficients
c
a
satisfy (4) is the
series of some smooth periodic function.
(iv)
The coefficients are uniquely determined by
f.
Note that these four properties are also satisfied by any finite sum (2).
Observe further that (iv) can be strengthened to the following.
the period
which
c
2p
0
(in addition to being periodic of period
are those for which
particular, if
p
e
-iax
2n)
If
has also
f
the only
is also periodic of period
is not commensurable with
,
we have
f m const.
a
2p.
for
In
This unique-
ness property is easily seen to be a consequence of the convergence of the series
(3) in
The periodic functions and the solutions of constant coefficient differential
equations are examples of mean-periodic functions, a notion introduced by Delsarte
and Schwartz
[15, 64].
Following the notation of Schwartz [63], let
denote the space of all infinitely differentiable functions on
n,
IR
E
f(IRn)
endowed with
the usual topology of uniform convergence of the functions and their derivatives
of all orders on compact subsets of
with compact support in
IR
n
(i.e.,
n.
IR
Recall that if
’
’(IRn),
is a distribution
the topological dual
MEAN-PERIODIC FUNCTIONS
E)
space of all continuous linear functionals on
E(Rn)
f
with
is defined
203
then the convolution of
(formally) by
I
(* f)(x)
f(x-t)(t)dt
and precisely, by
f (t)
where
f(t-x)
x
(f)
<!’,f
<b, X>
(x
(* f)(x)
denotes the translate of
[’
is the duality pairing of
f
by
[.
with
x,
(t)
f(-t), and
One importance of the con-
volution product resides in the fact that the most general linear continuous oper-
[
M"
ator
[
which commutes with translations can be represented by a convolu-
M (f(x+t))
tion, i.e., if
(Mf)(x+t)
x
Mf(x)
then
(* f)(x)
for some
[’.
(unique)
DEFINITION.
A function
’
zero distribution
tion satisfied by
f
f
is called
such that
O.
*
we will say that
f
mean-periodic
if there is a non-
.
If we want to emphasize the equa-
is mean-periodic with respect to
The equation defining periodic functions is of the form
(
6
with
larly, if
m
j---O
(i).
5
6
where
-n
(j)
a
ajiiSa(j)’
, f)(x)
is the
then
The frequencies
a
f(x/n)
0
denotes the Dirac point mass measure at
6a
derivative o[
j-th
(.
f(x-n)
, f)(x)
-
0
6a
(j)(g)
6a
(-i)Jg (j)(a)
a.
Simi-
and if
is the homogeneous differential equation
which appear in the expansions (2) and (3) are the zeros
of the entire holomorphic function,
(z)
which is the 7o,riar
transfer
e
ix. z
of
d(x)
,.
x,e
-ix.
The examples suggest that one should be
C. A. BERENSTEIN AND B. A. TAYLOR
204
f
able to expand a mean-periodic function
in terms of the simplest solutions to
the convolution equation it satisfies, namely the monomial-exponentials
k -lax
x e
having frequencies determined by the equation
,e
and
k
ix
,()e ix
.
multiplicity of the root
m
0
That is, we expect that
,
will
0
lead to a series
f(x)
()=0
c
where the
above.
(5)
c(x)e -ix
are polynomials with the same restrictions in their degree imposed
Before drawing any conclusion we consider another example.
A particular kind of convolution equation which includes the two cases
studied by Euler are the difference-differential equations.
These were first con-
sidered by Condorcet in 1771 who found the exponentials with frequencies solving
()
0
as particular solutions to
*
(see [44, vol. 3, ch. 8] for the
has as Fourier transform an exponentia
In this case the convolutor
details).
0
polynomial.
-i00. z
m
(z)
j--1
where the
oj
R.
(The
a.
(6)
aj(z)e
In the
are polynomials with complex coefficients.)
19th century, Cauchy and Poisson among others worked on this kind of problem (see
[44, 58]), but the first significant attempts to prove that all solutions of
*f
0
(5) when
are of the form
ator were those of Schmidt
u
represents a difference-differential oper-
[61], Polossuchin [60], Schrer [62] and Hilb [34].
The difficulty is that the series (5) is in general not pointwise convergent as
shown by
Leont’ev [47].
His
example is the following.
Let
\
IR,
\ #
0,
and
MEAN-PERIODIC FUNCTIONS
define the difference operator
205
by
(z)
4 sin z sin
(7)
z,
explicitly,
f(-
* f(x)
If
k
@,
p 6 Z
and
double root.
x) + f(
--x)
f(
+
(a)
the rational numbers, then all the roots of
qk(q
Z)
and each of them is simple except when
If we pick
k
a Liouville number, as done by
tend to pair up, at least for infinitely many values of
a convenient choice of
f(--y-x).
x)
k,
one can find subsequences
p
Pk’
0
are of the form
a
0
Leont’ev,
and
qk
q.
which is a
these roots
In fact, with
of the integers
such that
Ip k qkkl
_<
(l+IPkl)-k
k
as
,
(8)
and these values of the frequencies cause trouble for the unrestricted convergence
of the series (5) (compare with Theorems 4 and 5 below).
Other important work on
difference-differential equations appeared in [12, 17, 68, 69, 73].
An account of
these and other references can be found in [58, Section i] and in the treatise [i,
especially
chapters 4 and 6].
Meanwhile, in 1935, Delsarte, inspired by the work of Fredholm, Pincherle and
Volterra (see for instance [70]), had considered more general convolutors
coined the term mean-periodic function.
if
f
This terminology arose from the fact that
satisfies
, (x)
x+
f(t)dt
X--N
then
f
is periodic and its average
and
-
O,
(mean) over a whole period is zero [15].
C. A. BERENSTEIN AND B. A. TAYLOR
206
Finally, in 1947, L. Schwartz succeeded in proving the following version of
(f 6 E(IR)).
Theorem I for functions of one variable
THEOREM 2 (see [64]):
V
vC)
ornal
{= 6
: ()
series expansion
such that
V
0} be the zero set of
Then
is actually convergent in
V
k
(groupings)
E
to
f,
=(x)
-x
k
after a certain abelian sun,nation procedure
(For some details about this suBmation procedure, see next section.
It can be interpreted as
f(x)
c
lira
-0 k-i
a
k
(x)e -iax
’
series and the limit are meaningful in the topology of
From this theorem it follows that if
tions, the only non-zero coefficients
=
has a unique
U V and
i k
k=l a
cies
f
(5) and there are disjoint finite subsets
[
is performed.
.
and let
be mean-perlodlc with respect to
f
Let
f
c (x)
where both the
E.)
satisfies several convolution equain
(5) are those indexed by frequen-
which are common roots of the Fourier transforms of those convolutions.
In particular we have
COROLLARY:
(Spectral Analysis Theorem)
If
f
satisfies the convolution
equations
and
then
f -: 0.
This is a generalization of the fact that a non-constant function cannot have
two incommensurable periods.
207
MEAN-PERIODIC FUNCTIONS
In the fifties, J-P. Kahane (in his thesis [40]) and A. F. Leont’ev extended
An account of Leont’ev’s
Schwartz’ results to analytic mean-periodic functions.
work in this area accompanied by a detailed bibliography appears in [48].
Ehrenpreis showed that for a wide class of distributions
corasn if
there are constants
max{l(y) l:
IR,
y
It turns out that every
slowly decreasing.
the abelian summation
More precisely, we call a distribution
process is not necessary.
A,B,e > 0
Ix-yl -<
Ixl)}
>_
E’ sZcZy
E
x E IR
such that for any
A log(2+
In [21],
e(l+
Ixl) -B.
(9)
is a non-zero exponential-polynomial is
for which
then it is not slowly decreas-
E CO
On the other hand, if
ing (see [21, 5]).
THEOREM 3 [21]
V
k
V()
of
is slowly decreasing, there exist finite groupings
If
so that for any
f E
E
satisfying
--0
*
the corresponding
series
f (x)
converges to
f
in
Z
k
.
a
EZ
vk
c (x) e
a
-iux
(i0)
So far, we have considered only mean-periodic functions in
course, possible to replace
by other spaces.
tinuous mean-periodic functions; i.e.,
non-constant Borel measure
such
where
f
-nx(--)
is mean-periodic if there is a
is, as usual, a
C
(n)
0.
*f
with compact support such that
can be approximated very well by functions
X g (x)
but it is, of
Schwartz actually defined con-
C(n)
f
E,
f
sequence tending to
,f
X
0’
However,
f,
and these
functions also satisfy the same equation,
*f g
*X g
*f
(*f)*X g
O,
by the commutativity and associativity of the convolution product.
Thus, the
C.A. BERENSTEIN AND B. A. TAYLOR
208
theory of continuous mean-periodic functions can be reduced to the case of smooth
functions and is not studied separately.
Similarly, if
E
is replaced by
t,,
the space of all distributions, then one obtains mean-perlodlc distributions.
E,
considering instead of
convolutors
A’ (n)
Cn)
pact support in
the space
A( n)
{f
entire holomorphic in
By
n}
and
(analytic functionals, basically Radon measures with comone has the analytic mean-periodic functions.
the reasoning involved carries over from
to these other cases and only occa-
sionally will we point out some differences as remarks.
will always be smooth and distributions
Generally,
Accordingly, functions
f
always have compact support unless
otherwise stated.
REMARK i:
There is a technical condition similar to (9), defining slowly de-
creasing analytic functionals.
0
It is easy to see that every
is automat-
ically slowly decreasing in this sense, hence the conclusion of Theorem 3 holds
with no restrictions in the case of analytic mean-periodic functions.
vergence of (i0) being this time in
A(C),
(The con-
see [7,23].)
Neither of Theorems 2 and 3 provides information analogous to parts (il) and
(iii) of Theorem I.
Nor do they address the question of characterizing these
slowly decreasing
E
for which the series (5) converges in
ings (i.e., for which we have
#(Vk)
i
in (i0) for all
Malliavin gave partial answers to both questions in [24].
k).
without any group-
Ehrenpreis and
We shall give the com-
plete answer in the next section.
For the more subtle properties of mean-periodic functions, such as their relation to almost-periodicity, quasianalyticity, and related questions in the theory
of Dirichlet series, see [i, i0, 20, 23, 26, 41, 42, 51, 52, 53, 55, 65, 66,
3.
We continue to deal with the case of one variable.
67].
Under the assumption that
is slowly decreasing, we shall state a theorem that extends Theorem i and give
a sketch of its proof.
Under the same assumption on
we characterize convergence
MEAN-PERIODIC FUNCTIONS
To end the section we indicate how to prove Theorem 2.
without groupings.
Given
Let
d
a
numbers
n
denote
209
I,
,an,
lak-a jl,
Then if b E n
max{l,
d
are distinct.
associate to them a norm in
i _< k,j _< n}.
n
as follows.
We assume first that all the
we define
1/2
Ilbll
j=
where the vector
(Cl,...,c)
n
c
=.d--
(11)
is related to
by
b
Db,
c
D
the non-
singular matrix
i
i
0
0
(a3-c I) (c3-a2)
(an-a I) (an-C2)
0
0
0
(Cn-Cl) (Cn-C2) (an-a 3)
i
i
The transpose of this matrix appears as the matrix of the linear map
(Q(a I)
,Q(an ))
where
Q
is the
(12)
(bl,...,b n)
Newton interpolation polynomial
n
Q(z)
j=l
bj
l-<k<j
(z-Ck)
b
I
It is not hard to see what to do when some of the
consider the sequence
Q(2)’Q’(c2)’
Q"(a 2)
2!
al,a2,a2,a2,a3,
’Q(a3));
+
J
and the map
the corresponding
5 x 5
b2(z-a I)
+
coincide.
(bl,
matrix
b
5)
D
(13)
For example,
(Q(Cl),
is given by
C. A. BERENSTEIN AND B. A. TAYLOR
210
D
i
i
0
0
I
0
0
2-i
1
(3-el) (3-2)
0
0
0
0
0
0
0
(3-i) (3-2)
As in Theorem 3, to the grouping
V
of roots
k
V()
in
(r=r k)
we will associate the partial sum of (5)
I
cV
c (x)e
a
ix
k
where
m
m
b
r’
r
(-ix)
!
I
bj
j=l=O
(b
b
I 0
ml-l’
I
norm (Ii) of this vector
b
2 0
b
n
n
max{l,lj-gl,
I
+ log(l+Iz I)
z
THEOREM 4 [7]:
d
d
r
k
Set p(z)
to simplify the writing.
Let
be slowly decreasing.
can be divided into disjoint finite groupings
Vk,
in such a way that any mean-periodic function
f
Fourier expansion
denote the
g <_ r}.
We can now state the main result of this section.
llm
+
I +
i’’" "’i’2’’"
where each root appears repeated according to its multiplicity and
_< j,
m
r
llbll(k
and let
computed using the sequence
b
(14)
3
Set
3
Cn
r,mr _I
i.x
e
..
indicates the multiplicity of the root
3
g
(I0) convergent in
E.
V
Then the zero set
{ 6
V
: ()
with respect to
of
U
0}
Vk,
has a
Furthermore, the coefficients of the
expansion, as defined by (14), satisfy the estimates
llblI(k)
for any
N > 0.
Here
k
0(exp(-NP(k)
denotes any point in
k
as
Vk.
whose coefficients satisfy (15) defines a function
(15)
(R),
Conversely, any series (i0)
f
in
E
such that
*
0.
MEAN-PERIODIC FUNCTIONS
SKETCH OF THE PROOF:
211
The idea goes back to work of Ehrenprels on partial
differen1al equations which we shall discuss in Section 6 [4, 23, 59].
As every-
where throughout the theory, the fundamental tool is the Fourier transform,
of distributions of compact support.
We also need to use the Paley-Wiener-
Schwartz theorem which states that the map
tween
E’
and the space of entire functions
’:
where
Izl
{: 3A
0
is a linear isomorphism be-
’
such that
(IZll+...+IZnl2) 1/2
(z.l)1/2
F:
and
(Ira z I,.. ,Ira
Im z
Zn )"
The con-
,
volution product corresponds to the usual multiplication of functions under this
isomorphism.
’
The space
^!
EN:
{
is the inductive limit of the Banach spaces
sup
entire:
Z
I (z)
Ehrenpreis proved that the Fourier transform map
phism [23].
F
is also a topological isomor-
It is this last result that permits the reduction of questions about
mean-periodic functions to problems in function theory.
To outline the proof, let
M
be the closed subspace of all mean-periodic
,
functions with respect to
{f
The ideal generated by
slowly decreasing [21].
one can see that
,
J
{,
v
It annihilates
E:
*
v 6 E’}
0}.
is closed in
M (recall <,v,f>=
is precisely the dual of the space
E’/J
’v**(0)),
because
(see [38]).
is
and
Using
the Fourier transform isomorphism, one sees that this last space is isomorphic to
C. A. BERENSTEIN AND B. A. TAYLOR
212
the quotient space
is the ideal generaged by
where
E’
is any function in
other hand, if
[]
’/,
[]
and
the" restPcton
is completely determined by
in
its class modulo
of
to
’.
J,
On the
then
(By restriction we
V.
mean the double-lndexed sequence
()
(a)
where
a E V
tutes an
has multiplicity
m.
2uneton
V.)
crnalytc
on
3 < m,
0
Such a double-lndexed sequence
E
’.
(Here is where the
interpolation theory mentioned in the introduction enters.)
M
f
phlshed, an element
constl-
The proof is completed by characterizing those
which are restrictions of
analytic functions on V
{aa,3}
Once this is accom-
defines a continuous linear functional
F
on this
{be }.
space, and so can be identified with a double-indexed sequence
This de-
fines the series (i0) we are looking for, in fact,
<5
f(x)
since the restriction to
(-ix) e-lax
V
I
z
of
V =0
e
-izx
F(6 x
(-ix)
be
-ix
is given by the sequence
The fact that the series (5) obtained from this converges absolutely,
!
after groupings, in
and the required estimates (15) are consequences of the
above mentioned characterization of
tions on
m-i
F (e -izx)
x’ f>
’/J
as a certain space of analytic func-
(see [7] for the details).
V
REMARK 2:
In the case of analytic mean-periodic functions there is no res-
(see Remark i) and Theorem 4 holds if we take
triction on
zl
p(z)
in (15).
Note that in the last theorem we gave no indication of the size of the groupings
Vk,
norm
ll-ll(k ).
i.e. of the numbers
r
k
and
n
k
appearing in the definition of the
Actually such an estimate exists (see [7]) but
unbounded in general.
r
k
and
n
k
are
In the case of difference-dlfferential operators, it can be
213
MEAN-PERIODIC FUNCTIONS
d
seen that
can be taken
k
given by (6) and
v
j
<i
is always bounded, in fact, if
n
k
and
indicates the degree of the polynomial
aj,
is
one gets the
estimate
where
.
v
zeroes
general
cj
r
k
n
k
_<
maxl
(l+vj),
2 (l+v)
(14 ),
+ log
v
v
(see [7]).
In the case of the example (7) the
a
are ali simple (with only the exception
0),
n
k
r
k-
2
but in
as can be seen from Theorem 5 below.
# i
In the next section we give an application of the estimates (15).
Here we
will compare them with classical ones for Fourier series given by Theorem i (ii).
If all the
r
i
k
"II (k)
(see Theorem 5), then the norms
In fact, in this case the multiplicity
m
k
of
(-ix)
f(x)
!
k [j=O
-ikx
’
is absolutely convergent over compact sets, etc. and for any
Ibk, I
0(exp(-Np( k)))
as
In the case of the Fourier series the roots are
k
(16) is exactly (ii) of Theorem i since
p(k)
and
n
k
coincides with
k
bk
are easy to compute.
N > 0
.
k
(16)
k E Z,
log(l+Ikl).
r
k
n
and
i
k
The estimate (16) is
the same obtained by Ehrenpreis-Malliavin [24], where only the case
r
discussed.
k
-
i
was
It is easy to derive interesting and simple estimates from (16) even when
r
k
m
1
> i.
For example, if we assume
mr
1
(r=rk),
d
k
i
and all the multiplicities
we can write the grouping
V
k
as
ak),...,a(k)r
C. A. BERENSTEIN AND B. A. TAYLOR
214
and (16) implies that for
,r
i
j
Ibj
for every
N > 0
-j
(see [7]).
THEOREM 5 [7]:
be slowly decreasing.
Let
(i.e.
r
k i)
can be taken to contain a single point of
V
condition that every grouping
k
,
is that for some
a
exp(-C] Im al)
C
(i+ lal)
V
runs after all the points of
root of the equation
(a)
and
^(m)
(As usual
--0.
V
C > 0
I (m) (a)l
where
A necessary and sufficient
(17)
is the multiplicity of
m
d
(z) z=a .)
(a) denotes
m
m
as a
dz
While Theorem 5 is not stated explicitly in [7] it is an immediate conse-
quence of Example 17 and Remark 4 there.
Ehrenpreis-Malliavln [24] give a condition immediately seen to be equivalent
to
(17) in the only case they considered,
REMARK 3:
m
k
i.
In the analytic case (17) is replaced by
exp(-Clal)
(18)
m
Within the class of dlfference-differential operators, it is easy to see that
if (6) has all frequencies
no need for groupings (i.e.
.
r
-
rational (or mutually commensurable) then there is
k
i).
On the other hand, the example (7) shows
that this is not always the case (the roots get too close together because of the
transcendental nature of the relation between the frequencies.)
Since the
MEAN-PERIODIC FUNCTIONS
multiplicities
m
k
215
are bounded in the case of exponential polynomials, (17) shows
that a necessary condition for
r
i
_=
k
is that for some
A > 0
5,
exp (-AI Im
lk-ajl
>_
5
(14
lakl)
(19)
(k# j).
A
It follows easily from the estimates in [5, 31] or even from [17] that (19) is
also sufficient for
it follows that
r
-
k
Furthermore, from the work of Pdlya et al [17, 46],
i.
(19) is equivalent to the existence of
I k-jl
(l+lkl)-B
>_
Ehrenpreis has conjectured that if all the
,
B > 0
(20)
(k# j)
are algebraic then
e.
such that
Particular cases of this conjecture have been proved
[22, 57].
also shows that Theorem 5 includes the results from [72].
(20) holds [23].
This discussion
(Certainly, the con-
sequences of Theorem 5 for difference-differential equations when (19) holds have
been part of the folklore of the subject and motivated Ehrenpreis’ conjecture.)
It has recently been shown by W. A. Squires (private communication) that the
condition,
slowly decreasing, is not necessary for the validity of Theorem 4.
In general, we can remove it at the expense of bringing into the picture a further
Our method is similar to that used by Schwartz in [64, 66]
summation procedure.
to prove Theorem 2; but, since we avoid reliance on the uniqueness of the coef-
ficients of the Fourier expansion, we can handle the case of more than one var-
iable (where this uniqueness does not hold).
that:
Roughly the idea consists in noting
(i) in dealing with the analytic case there is no restriction at all on
(other than
# 0)
for convergence with groupings, and (ii) given
there is a constant
max{l(y)
I:
c > 0
y E
]R
such that for
and
Ix-yl
_<
# 0
and
x E
(l+elxl) }
>_
Ce
(21)
C. A. BERENSTEIN AND B. A. TAYLOR
216
(compare with (9)).
E’
Thus although not every
is slowly decreasing, they
This fact has been used in other contexts [33,
fail by very little.
37]; here the
idea is to use (ii) to reduce the situation to one similar to (i).
Let
c_
supp
E
f
(_Lo,LO)
In what follows
R- + ).
(eventually we let
(-R,R)
in
and is identically zero outside this interval
> 0
from the pair
Im
0
> 0
so that
f
in
then
fR;
x (-R+L0,R-Lo).
when
i
2i
([13, 14, 54]).
0
and
Im
< 0
+
F
F
by
R,
lim
y-3+
R
I
R
-R
(22)
-t
The function
FR(x-iy)]
(Ixl<R),
f(t) dt
We get two analytic functions
respectively.
[FR(x+iy)
The limit makes sense in
(-R,R)
0
is bounded and of compact support, its Cauchy transform is defined as
fR
C,
Im
L
and choose
will be a sufficiently large constant
R > 0
FR()
for
O,
Call the function that coincides with
, R(X)
Since
,
satisfy the equation
f(x),
F
R
can be recovered
fR
Ixl
+
F R,
(23)
< R.
i.e., is uniform over compact subsets of
and the same is true for derivatives.
acts as a convolutor also on functions holomorphic in
The distribution
the upper half-plane by means of the formula
,(z)
where the "integration
also holomorphic for
I
(z-t)(t)dt
<t,(z-t)>
takes place over the real axis, the function
Im z > 0.
Similarly for
Im z < 0
or even if
*
is
is
217
MEAN-PERIODIC FUNCTIONS
,
cal strip or half-strip, then
L
this strip is bigger than
,(x)
Since
lira
and
0
*f(-x),
[(*F
still makes sense as long as the width of
is restricted conveniently.
z
it is easy to see that
(, F R)(x-iy)]
)(x+iy)
y-3+
is defined on a verti-
If
defined only on a strip parallel to the real axis.
(22) implies
Ixl
0,
< R
and the limit is uniform on compact subsets of the above interval.
F
functions
mate
G
F
R
are holomorphic continuation of each other across
in the vertical strip
R
(21) one can find
H
R
Hence the
of the real axis, and they define a holomorphic func-
(-R+L0,R-L 0)
the segment
tion
*
and
(24)
L 0,
{z
C:
IRe
holomorphic in
,HR
GR,
when
L0}.
< R
z
IRe
z
< R
IRe
z
< R- L
Because of the esti-
such that
0.
Hence
, (F
for
IRe
z
< R- L
Im z > 0
and
0
H
R)
Im z < 0
or
argument used in Theorem 4 shows that for
independent of
R
and
L
I
F(z)
> 0,
0,
big enough there are groupings
R
also independent of
HR(Z)
k
a
R, such that
,R(z)e
-iz
k
with uniform and absolute convergence in compact subsets of
Im z > 0
or
Im z < 0
IRe
z
< R-
LI,
respectively, where the notation is self-explanatory.
Since we also have
y-0+lim
The same kind of
respectively.
H R) (x+iy)
(F+
R
(F R
HR> (x-iy)
f (x)
V
k
C. A. BERENSTEIN AND B. A. TAYLOR
218
EClx
in
ca
< R)
from (23), we can apply a diagonal process and obtain polynomials
such that
g
f(x)
6,
e-3+ k
k
where the convergence of the series and of the limit is in
E().
This gives a
proof of Theorem 2 which avoids involving the uniqueness of the coefficients or
the Spectral Analysis Theorem.
when
3.
g
Using these results, one sees easily that the
O.
We give here a simple application of the estimates (15) from Theorem 4.
This
example is also borrowed from [7] but we provide more details here.
We wish to study the problem, raised in [21], of solving the over-determlned
system of convolution equations
(25)
where
g,h E
f
and the unknown function
is also sought in
E.
Clearly, one
has to assume the compatibility condition
,h
and we do so henceforth.
"resonance"
v*
(26)
g,
To simplify the arguments we assume there is no
in this system, that is,
{ E c:
.()
C(n)
o}
.
In order to apply Theorem 4 we assume that one of the distributions, say
slowly decreasing.
(All these assumptions were already made in [21].)
(27)
,
is
Under the
above assumptions we wish to find necessary and sufficient conditions on the pair
219
MEAN-PERIODIC FUNCTIONS
E.
in order that the system (25) have a solution in
g,h
Before settling this,
let us show how functional analysis provides the answer to the related question,
for which
is it possible to solve (25) for
,v
g,h
al
which satisfy
(26)?
First of all, note that the lack of resonance together with the Spectral
Analysis Theorem show that (a) if a solution
((,v))
unique, and (b) the algebraic ideal
dense in
*
If
’.
(If
0
v*
((,v))
f
0
f
(25) exists at all, It is
to
E’
generated in
((,))
annihilates
and hence
f
by
and
then it necessarily satisfies
by (a); since
is reflexive,
were closed then by (b) there would be distributions
(b) follows.)
’
0,v0
such that
*0
Conversely, (28) implies that
f
to check that
0" g
solvability of (25) for
+
also a necessary condition.
E’
onto, where
M’
The set
.
K
of pairs
f----+
M’(,)
M
is the transpose map to
,,
+
,.
satisfying (26) is a
denote the
I-i map
(*f,v*f).
M.
<(,),Mf>
**(0)
hence
Let
(g,h)
then,by the open mapping theorem [38],
<M’(,),f>
It is easy
We now show that (28) is
pair that satisfies (26).
M:
K
and hence it is closed.
solves (25), hence (28) is sufficient for the
closed subspace of the Frchet space
If image(M)
(28)
6.
v*v 0
((,v))
0*h
eve.ry
+
is
Now,
<,* f>
+
M’
’ ’
E’
is
+ <,v * f>
** f(0)
<q)*. + k*,f>,
Since
M’
is onto (28) follows.
If follows from [34, 40] that (28) is equivalent to the existence of constants
C. A. BERENSTEIN AND B. A. TAYLOR
220
such that
exp(-Cllm zl)
(l+lzl) c
I(z) l+ I(-)1 >-
for all
(29)
z.
Going back to our original problem, it is clearly interesting only when (29)
fails, e.g.
above.
(z)
sin
z
and
(z)
i
sin
z
z,
is chosen as in
where
(8)
Note that we can reduce the pair of equations (25)-(26) to the equivalent
0
,F
(30)
v, F
H
*H
O.
with the compatibility condition
being slowly decreasing implies
This is possible since
[21], hence we can find
in
(30)-(31).
(31)
G
,G
such that
It follows that
g
and take
E
E
G,
F
f-G,
is onto
H
are mean-periodic with respect to
and
h-v*G
,
so we can apply Theorem 4, obtaining
and
Hx)
both convergent in
.
[
k
(-x)e
k
Applying the second equation from (3), we get
*
(ae- iax)
Gee
i ax
which leads to a triangular system of equations in the coefficients of the
32
.
polynomials
221
MEAN-PERIODIC FUNCTIONS
This linear system is solvable (uniquely) because all the terms
(),
in the diagonal equal
assumed (27) holds.
(e)
which doesn’t vanish since
0
and we have
Applying the conditions (15) to the coefficients of
F
oh-
tained this way, we obtain a set of necessary and sufficient conditions in the
for the solvability of the system (30).
H
coefficients of
come simpler when all the roots of
the numerical coefficients
If
i
maxlj -kl
r
()
0
are simple, since then (32) defines
by
()
form one of the groupings of roots of
_<i
These conditions be-
then, as explained before Theorem 5,
and, for instance
0
a necessary condition on the
is given by
I
for all
N > 0
--sin- z,
j
k#j
l(j)10(exp(-Np(j)))
(z)
For example, when
,r).
(j=l
Ik- j
sin
z
and
(z)
and
H
and one gets
which imposes a severe restriction on the Fourier coefficients of
H
if
z
we are really dealing with periodic functions
F
k
the condition on the Fourier coefficients
N > 0
({=0}
lkl
chosen as in (8).
of
0([(k)[(l+[k[) -N)
[Vlk[
for every
qk
).
H,
as
k
as
k-
This gives,
-N’
0(l+Ikl)
dist(k,mk))
m
,
k
is
C.A. BERENSTEIN AND B. A. TAYLOR
222
5.
We proceed now to discuss mean-periodic functions functions of several vari-
ables.
It was the question of the existence of a Fourier representation for such
functions which motivated both our work
[8] and our reevaluation of the one vari-
able situation, which in turn led to the Theorems 4 and 5 above.
of Schwartz [64], it
wasn’t very clear
would take in several variables.
which shape, if any, Fourier representation
It was suggested in [64] that it might be easier
to deal with the Spectral Analysis Theorem
thesis Theorem).
After the work
its companion, the
(and
Spectral Syn-
Roughly speaking this corresponds to the uniqueness of the re-
presentation of a mean-periodic function in the case of one variable.
However, it
turned out that the Spectral Analysis Theorem fails when the number of variables
n
2.
In fact, in 1975, D. I. Gurevich constructed in [32] six distributions
i "’’’6
{f 6 E:
6
E’(n)
(n> 2)
6"
i*
with two equations.)
0}
n l(Z)
{z
for which
6(z)
but
(In the analytic case, one can get by
{0}.
On the positive side, Malgrange had shown in his thesis that
the exponential-polynomial solutions to
space of all solutions.
0
*
were, in fact, dense in the
This confirmed the possibility of a representation (see
[49, 50] and [19, 37] for related results), but gave no hint as
sentation looked like.
O}
to what the repre-
In 1960, Ehrenprels attacked this question in the case of
partial differential operators with constant coefficients.
To start with we shall
see what can be done about the wave operator in two variables.
Since
E
is an AU space, it follows that
representation of the following kind.
dv
in
C
n
every
smooth function has a Fourier
(n)
Given
there is a Radon Measure
such that
(X)
and this integral converges in
e
cn
-ix- Z
dr(z),
E (see [4, 23]).
(33)
The point here is that we are
allowing arbitrary complex frequencies to occur in (33).
Going back to the wave
223
MEAN-PERIODIC FUNCTIONS
equation in
2
a2
02
xI2
x 22
(34)
O,
f(x)
D’Alembert showed that the general smooth solution of (34) is of the form
f(xl,x2)
measures
dVl,
dv
2
e
3
t
-i
Let
z 6
2
-itw
dr.(w)
.
e
(Xl-X2) w
(36)
1 2
j
3
C
denotes the variable in
f(x)
Hence there are two
such that
in
f.(t)
where
(35)
f2(xl+x2)
smooth functions of one variable (see [44]).
fl’ f2
with
+
fl(Xl-X2)
Inserting (36) into (35) we get
dVl(W
I
+
x 6
represent the dual variable to
-i
e
(Xl+X2) w
2,
dr2 (w)
then the above expression
can be written as
f (x)
I{Zl+Z2=0}
"Zdvl(z)
e
+
e
i- z2=O}
where the last integral converges in
by the algebraic variety
braic variety
V()
{z
{z
V
2
(z)
from the convolution equation (34).
dv
is not unique
2
E
"Zdv2(z)
e
2
I
0},
z
2
2
where
0}.
"Zdv(z),
(37)
z-z=O}
and the Radon measure
z
-ix
dv
is supported
This is precisely the algeis the distribution arising
(It is easy to see that the choice of measure
[59].)
This representation of the solutions of the wave equation was generalized to
C. A. BERENSTEIN AND B. A. TAYLOR
224
the case of arbitrary linear differential operators with constant coefficients in
and
P
As usual, if
[22].
=
Dj
THEOREM 6:
0}
P(z)
then
x.’
3
Let ’V
is a polynomial in
f
a partial differential operator.
P,
be the algebraic variety of the zeros of
V
Then there are finitely many algebraic varieties
and differential operators
then
represents
P(D)
variables with complex coefficient
n
0.
J
c V,
cn:
{z
V
r
V,
U V.
j=l J
P(D)f
with constant coefficients such that if
0
3
has a representation
r
f(x)
j=l
where the
dr.
are Radon measures in
grals converge in
E.
[
V.
n
0 .e
3
-ix. z
(38)
dr. (z)
3
with support in
V.
and all the inte-
Moreover, if all the irreducible factors of
P
are dis-
tinct then one has the representation
f(x)
e
-ix- Z
(39)
dr(z),
V
for some Radon measure
dr.
The proof follows the same lines as Theorem 4 above except that the function
theory needed is much harder.
systems.
For instance
two differences (i) V
no
if
{z
In [23, 59] this theorem is actually proved for
PI(D)
6 cn: pl(z
longer have constant coefficients.
P
m
(D)
0
Pro(z)
then (38) holds with these
0}
and (ii) the
8j
do
In particular, they show that the Spectral
Analysis Theorem holds for partial differential operators, in contrast to the
counterexample for general overdetermined systems found (later) by Gurevich.
point here is that the algebraic nature of the variety
role.
V
The
plays a very strong
There is very little more we can offer by way of comment except to encour-
age the reader to look into the original sources.
Recently some simplifications
225
MEAN-PERIODIC FUNCTIONS
of parts of the proofs have appeared [18, 75, 76].
In [2, 4] an extension of Theorem 6 was given for a subclass of those convowhich look like
lution operators
8m
M* f(x)
m
0x
f
l(X’) ,
+
n
where
x’
(x
Xn_ I)
I
and
m-I
f
m-i
0x
+
+
m 6 E’ (n-l).
I
Mm (x’
(40)
* f’
n
The proof follows closely
that of Ehrenpreis’ Theorem 6 and avoids trouble by restricting consideration to a
When [4] appeared, the remaining difficulties were twofold:
single equation.
first, the concept of groupings had to be extended to several variables and,
second, in the case of systems the Spectral Analysis Theorem had to be built into
the system.
We finally solved this problem in [8]; an announcement of some of our
results for the case of a single equation appeared in [3].
slowly decreasing (for
sary "groupings" we gave in [8] a definition of
E’(Rn))
To cope with the neces-
slightly different from the usual one [21].
Instead of giving the
formal definition, let us just mention a few examples of such distributions
For
(b)
Every difference-differential operator
n
_
i,
condition (9).
the definition is the same as before, i.e.
(a)
(in
n
variables) is slowly
decreasing.
(c)
is a difference-differential operator,
If
supp v
cv(supp ),
cv(supp )
(d)
E’(IRn)
and
then
+
.
Here
v
cm(n)
is slowly decreasing.
v
denotes the convex hull of the support of
there is a number
m
supp v _c
),
If
v
o
cv(supp
(depending on
then
+
)
such that if
Furthermore,
and
is slowly decreasing.
v
is of the form (40)it is always slowly decreasing.
In this case
is a distinguished pseudopolynomial,
(z)
z
m
n
+ 1 (z)z
m-i
n
+
+%(z’),
(4)
C. A. BERENSTEIN AND B. A. TAYLOR
226
z’
,Zn_ I)
(z I
and
E’ (n-l)
E
l’"’’m
Note that after an affine
change of coordinates a non-zero polynomial becomes a distinguished
pseudo-polynomlal.
(e)
E A’ (n),
In the analytic case, every
@ O,
is slowly decreasing.
One can then prove
THEOREM 7 [8]
Let
exists a locally finite family of closed subsets
tors
ij
J
and a partition of the Index family
such that if
,
f --O,
Vj
n
{z
V
be slowly decreasing
V,
of
{21}
(z)
0}.
There
differential opera-
into finite subsets
Jk
then
k
E
where the series converges in
j
k
and
V.
dr.
j
are Radon measures with support in
Vo.
REMARK 3:
the measures
Both Theorems 6 and 7 contain also information about the mass of
which corresponds to the estimates (15) of Theorem 4.
dr.
As mentioned at the end of Section 4, it is also possible to eliminate the
is slowly decreasing.
condition that
THEOREM 8 [8]:
Let
notation as in Theorem
# 0
7, we have
f(x)
lira
be a distribution in
.
0 k
where the
dvj,g
.e
k V.
j
3
are Radon measures with support in
the limit are taken in the sense of
E’
dv
Vj
then, with the same
j,s
(z)
and both the series and
E.
For overdetermined systems, we must restrict ourselves to systems for which
the Spectral Analysis Theorem holds.
This is done by defining what we mean by
227
MEAN-PERIODIC FUNCTIONS
jointly slowly decreasing distributions
holds with
{z
V
C
n
r
i (z)
’’’’’r"
(z)
0}.
Once this is done, Theorem ?
The definition itself is quite
technical; and the only point worth emphasizing here is that it requires
a complete intersection variety of codimension
V
to be
Thus our result is not strong
r.
enough to cover the general situation studied by Ehrenpreis and Palamodov in the
case of differential operators (see [23, 59]); on the other hand, we are dealing
with analytic varieties instead of algebraic varieties.
We refer the reader to
[8] for the exact details and pertinent examples; here let us simply observe that
one of the main ingredients in the proof is the construction of a function, the
Q
Jacobi interpolation function, which plays the role of the Newton polynomial
from (13).
This Jacobi interpolation function was so named because it appears
For other insights into the contents of [39] we suggest [30].
implicitly in [39].
Here is an application of our result about systems; other examples appear in
[8].
THEOREM 9:
(0,0),
B
Let
(i,0),
zero outside
S.
S
denote the unit square in the plane with vertices
(0,i)
C
D
and
(i,i).
Let
LI(S)
gl,g 2
Assume further that the eight complex numbers
A(A,B)A(A,C)A(C,D)(B,D) # 0,
where, for instance,
b
I
I
A(A,B)
alb 2 a2b I.
a
Define
i,2
E’ (JR2)
j
b
2
2
by
aj5 A + bj5 B
+
cj6 C + dj6 D
+g j,
j
be extended as
al,a2,...,dl,d 2
satisfy the condition
a
A
1,2.
C. A. BERENSTEIN AND B. A. TAYLOR
228
If
2)
E(
f
satisfies the system of equations
*
E
admits a series representation which converges in
0,
2*
f
then
and is of one of the follow-
ing two kinds,
()
(x)e
f(x)
-ia-x
+
e
a J
V
where
a V\ J
2 ()
{a
f)
(independent of
2(a)
crete set and that for
With
V
and
J
a
ca(x)e
a
as above,
-ia-x
+
c (x)
those with
[
aVl
j
where the
a 6 V
J
c
V\J,
c
C.)
V\J
V
V
I U 2,
a
is a finite subset of
(x)
c
(a
(It is implicit that
a.
disjoint union of the doublets
f(x)
0},
and the polynomials
less than the multiplicity of
(ii)
-i-x
{a,}
V
V
.
I
(a#),
(x)e -ia-x +
2
J)
V
have degree
V
and
is a dis-
V
2
is the
and
(c e -ia-x +c
a
{a,}V 2
e-i-x)
in the first sum have the same meaning as in (i)
are polynomials of degree
1
and
_<1.
The two cases arise according to whether the system of algebraic equations
a
a
has a pair of simple roots in
I +
+
2
C
2
bl
+ clq +
0
dl
b2 + c2 + d2q
or a single double root.
The proof consists simply of checking that
i’ 2
creasing and applying the general theory from [8].
version of this theorem in
V were all simple
0
are jointly slowly de-
Delsarte had proved a weaker
[16] under the additional assumption that the roots of
while in
[56] Meyer gave an L 2
version of this theorem.
similar expansion holds when the square is replaced by a convex polygon, which
improves on a recent result of Yger [74].
A
229
MEAN-PERIODIC FUNCTIONS
As a final observation in this section we note that while a Fourier representation for a general class of systems of convolution equations was proposed in
[23, Chapter 9], this representation uses a
{z E
n: l(Z
r(Z)
0}
ull ne@.gborhood
(even for the case
(The case
stantially weaker than our results.
r
i
r
i)
of the variety
V
and hence is sub-
is very easy and was done
in [2].)
6.
The equivalent to Theorem 5 is conspicuously missing from the previous sec-
Typical
tion; the reason is that we have only partial results in this direction.
examples are the following (see [8]).
THEOREM i0:
be a difference-differential operator of the form (4)
Let
such that the discriminant of the pseudopolynomial
identically.
Then if
,
0
(see (41)) does not vanlsh
we have
f(x)
e
-iX- Z
dr(z)
V
for some measure
dv
concentrated on
V,
the integral being convergent in
This theorem is a sharpening of the one proved in
[2; 4, Chapter 4] where
under the same hypothesis only a representation of the form (38) was obtained.
Observe that the second part of Theorem 6 is a particular case of this theorem.
Theorem i0 can be extended to distributions
for which
is a distinguished
polynomial whose discriminant satisfies certain lower bounds (see [8, 4]).
THEOREM ii:
(z)
0},
Let
E
E’
be such that at every point of
V
{z
we have
]grad (z)
for some constants
e,
A > 0.
e
e,xp(-A Im z[)
(+Ixl) A
Then the conclusion of Theorem l0 holds.
Even under the assumption that
V
is a manifold
(as in Theorem Ii)
C
n
C. A. BERENSTEIN AND B. A. TAYLOR
230
we don’t know the necessary conditions for a representation of the form (39).
This representation corresponds to the situation in Theorem 5 where all the multi-
plicities are one.
The problem of finding necessary and sufficient conditions on
slowly decreasing
in order that the conclusion of Theorem i0 holds or, more
generally, for either representation (38) or (39) from Theorem 6 to hold, is one
of the most interesting open questions left in the theory of mean-periodic func-
Among the very few cases for which we know the answer,
tions of several variables.
is the case of discrete varieties (see
THEOREM 12:
Let
A’
{
(C2),
.(z)
such that
A(z 2)
(z 2)
the distinguished pseudopolynomial
(Zl
(z 2)
[8]); another is the following.
0.
q)
(z2)) (Zl %(z2)),
Then the conclusion of Theorem i0 holds if
and only if the map
has a closed range when considered as a map from the space
{F entire:
iF(z2)
_<
A
exp(B(l(z21 + Iz2])),
for some
A,B > 0}
into itself.
Summarizing, hegeneral theory of mean-periodic functions of several variables stands now, where the one variable theory stood in the sixties.
Still mis-
sing is work of the caliber of [23, 59] which explored the applications of Theorem
6 to the theory of partial differential equations.
ter
3] deals with an application of these theorems
also a sharpening of these results in [6]).
Only one chapter of [4, Chapto convolution equations
(see
We expect that further developments
will come through new results in interpolation theory, in applications to other
areas like those in [29] and in the study of general classes for which the
231
MEAN-PERIODIC FUNCTIONS
Spectral Analysis Theorem hold, especially in relation to the study of distributions in symmetric spaces or Lie groups (see
[9]).
As S. Bochner has argued in [ii, p. 65-66] surveys are usually biased in the
sense that in trying to present the gist of the ideas of our predecessors one does
not always do justice to their work.
He goes on to say that nevertheless this ap-
proach has "a very good effect on the whole."
Though we cannot expect to escape
this general indictment, our hope is that the present work will contribute to fur-
ther developments in the area of mean-periodicity.
That alone should be coumted
as a positive effect.
ACKNOWLEDGMENT:
It is a pleasure to acknowledge support received from the
National Science Foundation under Grants MCS 78-00811 and MCS 76-06727 during the
preparation of this manuscript.
We are also grateful to Professor L. Zalcmn for
helping us to improve on the readability of the present paper; of course, we alone
are to blame for any shortcomings.
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