MEAN-PERIODIC FUNCTIONS

We show that any mean-periodic function f can be represented in terms of exponential-polynomial solutions of the same convolution equation f satisfies, i.e., ’, f 0 (E’(IRn)). 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. BERENSTEIN AND B. A. TAYLOR

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 con- cerned 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 Sec- tion 4), and a new "summability method" for summing formal expansions of mean- periodic functions which, in contrast to the original method of L. Schwartz [64], applies in any number of variables.We also mention some applications of our work on interpolation from varieties of codimension greater than 1 to certain systems of equations (Theorem 9).MEAN-PERIODIC FUNCTIONS 201 2. We begin by discussing some familiar convolution equations.Consider first a homogeneous ordinary differential equation with constant coefficients, m I a'DJ f(x) 0 j=O 3   (l) where D i dx' a .Euler was the first to consider the problem of finding J all solutions to this equation.In [25] he showed that they are given by finite sums of the form f(x) where the summation extends over all roots a of the algebraic equation aj (-a) j 0 and the c a are polynomials of degree <ma multiplicity of a.
Euler also studied periodic functions and proposed to write them in terms of exponentials.A periodic function (of period 2n, say) satisfies the equation f(x+T) f(x-n) 0 and is represented by an izfinite series f(x) where the frequencies a satisfy the equation ina -ina e e 0.
For the history of the theory of these series, the Fourier series, see [28,78].For the theory there is the comprehensive treatise [77].The main points we wish to recall may be summarized as follows.
THEOREM i: (i) The Fourier series (3) of a smooth (C) periodic function f converges to f.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 N > 0.
(iii) 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.If f has also the period 2p (in addition to being periodic of period 2n) the only a for -iax which c 0 are those for which e is also periodic of period 2p.In particular, if p is not commensurable with , we have f m const.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 E f (IRn)   denote the space of all infinitely differentiable functions on IRn, endowed with the usual topology of uniform convergence of the functions and their derivatives of all orders on compact subsets of IRn.
' such that * O.If we want to emphasize the equa- tion satisfied by f we will say that f is mean-periodic with respect to .
The equation defining periodic functions is of the form where 6 denotes the Dirac point mass measure at a. Simi- -n a larly, if 6 (j) is the j-th derivative o[ 6a a a 6 (j)(g) (-i)Jg (j)(a) and if m ajiiSa (j)' then (. , f)(x) -0 is the homogeneous differential equation j---O (i).The frequencies a which appear in the expansions (2) and (3)  x,e which is the 7o,riar transfer of ,.The examples suggest that one should be able to expand a mean-periodic function f in terms of the simplest solutions to k -lax the convolution equation it satisfies, namely the monomial-exponentials x e having frequencies determined by the equation ix ix ,()e 0 ,e and k m multiplicity of the root .That is, we expect that , 0 will where the c are polynomials with the same restrictions in their degree imposed above.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 * 0 (see [44, vol. where the oj R. (The a. are polynomials with complex coefficients.)In the 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 are of the form (5) when u represents a difference-differential oper- ator were those of Schmidt [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 define the difference operator by (z) If k @, the rational numbers, then all the roots of (a) 0 are of the form p 6 Z and qk(q Z) and each of them is simple except when a 0 which is a double root.If we pick k a Liouville number, as done by Leont'ev, these roots tend to pair up, at least for infinitely many values of p and q.In fact, with a convenient choice of k, one can find subsequences Pk' qk of the integers such that Ip k qkkl _< (l+IPkl)-k as k , 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 and coined the term mean-periodic function.This terminology arose from the fact that if f satisfies x+ , (x) then f is periodic and its average (mean) over a whole period is zero [15].
Finally, in 1947, L. Schwartz succeeded in proving the following version of Theorem I for functions of one variable (f 6 E(IR)).
THEOREM 2 (see [64]): Let f be mean-perlodlc with respect to and let V vC) {= 6 : () 0} be the zero set of .Then f has a unique ornal series expansion (5) and there are disjoint finite subsets V k (groupings)   such that V is actually convergent in E to f, after a certain abelian sun,nation procedure is performed.(For some details about this suBmation procedure, see next section. -iax It can be interpreted as f(x) where both the -0 k-i a k ' series and the limit are meaningful in the topology of E.) From this theorem it follows that if f satisfies several convolution equations, the only non-zero coefficients c (x) in (5) are those indexed by frequen- cies = 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.
In the fifties, J-P.Kahane (in his thesis [40]) and A. F. Leont'ev extended Schwartz' results to analytic mean-periodic functions.An account of Leont'ev's work in this area accompanied by a detailed bibliography appears in [48].In [21], Ehrenpreis showed that for a wide class of distributions the abelian summation process is not necessary.More precisely, we call a distribution E E' sZcZy corasn if there are constants A,B,e > 0 such that for any x E IR max{l(y) l: y IR, Ix-yl -< A log(2+ Ixl)} >_ e(l+ Ixl) -B. ( It turns out that every for which is a non-zero exponential-polynomial is slowly decreasing.On the other hand, if E C O then it is not slowly decreas- ing (see [21, 5]).
THEOREM 3 [21] If is slowly decreasing, there exist finite groupings V k of V() so that for any f E E satisfying * --0 the corresponding series converges to f in .
So far, we have considered only mean-periodic functions in E, but it is, of course, possible to replace by other spaces.Schwartz actually defined con- tinuous mean-periodic functions; i.e., f C(n) is mean-periodic if there is a non-constant Borel measure with compact support such that * f 0. However, such f can be approximated very well by functions (n) f X , f f, where X (x) -nx(--) is, as usual, a C sequence tending to 0' and these g functions also satisfy the same equation, *f by the commutativity and associativity of the convolution product.Thus, the 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.By considering instead of E, the space A(n) {f entire holomorphic in n} and convolutors A' (n) (analytic functionals, basically Radon measures with com- pact support in Cn) one has the analytic mean-periodic functions.Generally, the reasoning involved carries over from to these other cases and only occa- sionally will we point out some differences as remarks.Accordingly, functions f will always be smooth and distributions always have compact support unless otherwise stated.
REMARK i: There is a technical condition similar to (9), defining slowly de- creasing analytic functionals.It is easy to see that every 0 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.(The con- vergence of (i0) being this time in A(C), 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 for which the series (5) converges in E without any group- ings (i.e., for which we have #(V k) i in (i0) for all k).Ehrenpreis and Malliavin gave partial answers to both questions in [24].We shall give the com- plete answer in the next section.
For the more subtle properties of mean-periodic functions, such as their rela- tion to almost-periodicity, quasianalyticity, and related questions in the theory 3. We continue to deal with the case of one variable.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 without groupings.To end the section we indicate how to prove Theorem 2.
Given n numbers a I, ,a n, associate to them a norm in n as follows.
Let d denote d max{l, lak-ajl, i _< k,j _< n}.We assume first that all the are distinct.Then if b E n we define 1/2 Ilbll =.d--j= (11) where the vector c (Cl,...,c) is related to b by c Db, D the non- n The transpose of this matrix appears as the matrix of the linear map (bl,...,bn) It is not hard to see what to do when some of the coincide.For example, J consider the sequence al,a2,a2,a2,a3, and the map (bl, b 5 As in Theorem 3, to the grouping V k of roots we will associate the partial sum of (5 where m indicates the multiplicity of the root .. Set n n m I + + r,mr norm (Ii) of this vector b computed using the sequence i''" "'i'2''" r where each root appears repeated according to its multiplicity and d d k max{l,lj-gl, I _< j, g <_ r}.
We can now state the main result of this section.Set p(z) llm z + log(l+Iz I) to simplify the writing.
THEOREM 4 [7]: Let be slowly decreasing.Then the zero set V of can be divided into disjoint finite groupings Vk, V { 6 : () 0} U Vk, in such a way that any mean-periodic function f with respect to has a Fourier expansion (I0) convergent in E. Furthermore, the coefficients of the expansion, as defined by ( 14), satisfy the estimates llblI(k) for any N > 0.Here k denotes any point in V k.Conversely, any series (i0)   whose coefficients satisfy (15) defines a function f in E such that * 0.
SKETCH OF THE PROOF: 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 F: is a linear isomorphism between E' and the space of entire functions ' ': {: 3A 0 such that where Izl (z.l)1/2 (IZll+...+IZnl2) 1/2 and Im z (Ira z I,.. ,Ira Zn )" The con- volution product corresponds to the usual multiplication of functions under this isomorphism.The space ' is the inductive limit of the Banach spaces , ^! EN: { entire: sup I (z) Z Ehrenpreis proved that the Fourier transform map F is also a topological isomor- phism [23].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 E: * 0}.
The ideal generated by , J {, v v 6 E'} is closed in ' because is slowly decreasing [21].It annihilates M (recall <,v,f>= v**(0)), and one can see that is precisely the dual of the space E'/J (see [38]).Using the Fourier transform isomorphism, one sees that this last space is isomorphic to the quotient space '/, where is the ideal generaged by in '.On the other hand, if is any function in E' and [] its class modulo J, then [] is completely determined by the" restPcton of to V. (By restriction we mean the double-lndexed sequence () (a) where a E V has multiplicity m.Such a double-lndexed sequence {aa,3} constl- tutes an crnalytc 2uneton on V.) The proof is completed by characterizing those analytic functions on V which are restrictions of E '. (Here is where the interpolation theory mentioned in the introduction enters.)Once this is accom- phlshed, an element f M defines a continuous linear functional F on this space, and so can be identified with a double-indexed sequence {be }.This de- fines the series (i0) we are looking for, in fact, -izx since the restriction to V of e F(6 is given by the sequence x (-ix) -lax

! e
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 '/J as a certain space of analytic func- tions on V (see [7] for the details).
REMARK 2: In the case of analytic mean-periodic functions there is no res- triction on (see Remark i) and Theorem 4 holds if we take p(z) zl in (15).
Note that in the last theorem we gave no indication of the size of the group- ings V k, i.e. of the numbers r k and n k appearing in the definition of the norm ll-ll(k).Actually such an estimate exists (see [7]) but r k and n k are unbounded in general.In the case of difference-dlfferential operators, it can be seen that d k can be taken <i and n k is always bounded, in fact, if is given by ( 6) and v j indicates the degree of the polynomial aj, one gets the estimate 2 (l+v) (14 ), . (l+vj), maxl (see [7]).In the case of the example (7) the zeroes cj are ali simple (with only the exception a 0), n k r k-2 but in general r k # i as can be seen from Theorem 5 below.
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 k i (see Theorem 5), then the norms "II (k) are easy to compute.
In fact, in this case the multiplicity m k of k coincides with n k and f(x) is absolutely convergent over compact sets, etc. and for any N > 0 Ibk,I 0(exp(-Np( k))) as k . ( In the case of the Fourier series the roots are k k E Z, r k n k i and ( 16) is exactly (ii) of Theorem i since p(k) log(l+Ikl).The estimate (16) is the same obtained by Ehrenpreis-Malliavin [24], where only the case r k --i was discussed.
It is easy to derive interesting and simple estimates from (16) even when r k > i.For example, if we assume d k i and all the multiplicities m 1 mr 1 (r=rk), we can write the grouping V k as ak),...,a(k)r and (16) implies that for j i ,r Ibj -j for every N > 0 (see [7]).
THEOREM 5 [7]: Let be slowly decreasing.A necessary and sufficient condition that every grouping V k can be taken to contain a single point of V (i.e.r k i) is that for some , C > 0 exp(-C] Im al) where a runs after all the points of V and m is the multiplicity of 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, m k i.
REMARK 3: In the analytic case ( 17) is replaced by exp(-Clal) m Within the class of dlfference-differential operators, it is easy to see that if (6) has all frequencies .rational (or mutually commensurable) then there is no need for groupings (i.e.r 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 multiplicities m k are bounded in the case of exponential polynomials, (17) shows that a necessary condition for r k _= i is that for some 5, A > 0 exp (-AI Im lk-ajl >_ 5   A (k# j). ( 19) (14 lakl) It follows easily from the estimates in [5, 31] or even from [17] that (19) is also sufficient for r ki.Furthermore, from the work of Pdlya et al [17, 46], it follows that (19) is equivalent to the existence of , B > 0 such that Ehrenpreis has conjectured that if all the e. are algebraic then (20) holds [23].
Particular cases of this conjecture have been proved [22, 57].This discussion also shows that Theorem 5 includes the results from [72]. (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 summation procedure.Our method is similar to that used by Schwartz in [64, 66]   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 variable (where this uniqueness does not hold).Roughly the idea consists in noting that: (i) in dealing with the analytic case there is no restriction at all on (other than # 0) for convergence with groupings, and (ii) given # 0 and there is a constant c > 0 such that for x E max{l(y) I: y E ]R and Ix-yl _< (l+elxl) } >_ Ce (21) (compare with ( 9)).Thus although not every E' is slowly decreasing, they fail by very little.This fact has been used in other contexts [33, 37]; here the idea is to use (ii) to reduce the situation to one similar to (i).
Let f E satisfy the equation , O, and choose L 0 > 0 so that supp c_ (_Lo,LO) In what follows R > 0 will be a sufficiently large constant (eventually we let R-+ ).Call the function that coincides with f in (-R,R) and is identically zero outside this interval fR; then , R(X) 0 when x (-R+L0,R-Lo). ( Since fR is bounded and of compact support, its Cauchy transform is defined as y-3+ The limit makes sense in (Ixl<R), i.e., is uniform over compact subsets of (-R,R) and the same is true for derivatives.
The distribution acts as a convolutor also on functions holomorphic in the upper half-plane by means of the formula ,(z) I (z-t)(t)dt <t,(z-t)> where the "integration takes place over the real axis, the function * is also holomorphic for Im z > 0. Similarly for Im z < 0 or even if is defined only on a strip parallel to the real axis.If is defined on a verti- cal strip or half-strip, then , still makes sense as long as the width of this strip is bigger than L 0 and z is restricted conveniently.
Since ,(x) *f(-x), it is easy to see that (22) implies lira [(*F )(x+iy) (, F R)(x-iy)] 0, Ixl < R L 0, y-3+ ( 24) and the limit is uniform on compact subsets of the above interval.Hence the F R are holomorphic functions F and * continuation of each other across the segment (-R+L0,R-L0) of the real axis, and they define a holomorphic func- tion G R in the vertical strip {z C: IRe z < R L0}.Because of the esti- mate (21) one can find H R holomorphic in IRe z < R such that ,H R GR, when IRe z < R-L 0. Hence , (F H R) 0, for IRe z < R-L 0 and Im z > 0 or Im z < 0 respectively.The same kind of argument used in Theorem 4 shows that for R big enough there are groupings V k independent of R and L I > 0, also independent of R, such that F(z) HR (Z)   ,R(z)e -iz k a k with uniform and absolute convergence in compact subsets of IRe z < R-LI, Im z > 0 or Im z < 0 respectively, where the notation is self-explanatory.
Since we also have (23), we can apply a diagonal process and obtain polynomials c a such that g 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.Using these results, one sees easily that the when g O.
3. 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 and the unknown function f is also sought in E. Clearly, one has to assume the compatibility condition ,h v* g, and we do so henceforth.To simplify the arguments we assume there is no "resonance" in this system, that is, { E c: .() In order to apply Theorem 4 we assume that one of the distributions, say , is slowly decreasing. (All these assumptions were already made in [21].)Under the above assumptions we wish to find necessary and sufficient conditions on the pair g,h in order that the system (25) have a solution in E. Before settling this, let us show how functional analysis provides the answer to the related question, for which ,v is it possible to solve (25) for al g,h which satisfy (26)?
First of all, note that the lack of resonance together with the Spectral Analysis Theorem show that (a) if a solution f to (25) exists at all, It is unique, and (b) the algebraic ideal ((,v)) generated in E' by and is dense in '. (If f   annihilates ((,)) then it necessarily satisfies * v* 0 and hence f 0 by (a); since is reflexive, (b) follows.) If ((,v)) were closed then by (b) there would be distributions 0,v0 ' Conversely, (28) implies that ((,v)) E' and hence it is closed.It is easy to check that f 0" g + 0*h solves (25), hence (28) is sufficient for the solvability of (25) for eve.ry pair that satisfies (26).We now show that (28) is also a necessary condition.The set K of pairs (g,h) satisfying ( 26) is a closed subspace of the Frchet space .Let M denote the I-i map If image(M) K then,by the open mapping theorem [38], M' ' ' E' is onto, where M' is the transpose map to M. Now, <M'(,),f> <(,),Mf> <,* f> + <,v * f> **(0) + ** f(0) <q)*.+ k*,f>, hence M'(,) ,, + ,.Since M' is onto (28) follows.
(z) sin z and (z) i sin z z, where is chosen as in (8) above.Note that we can reduce the pair of equations ( 25)-( 26) to the equivalent with the compatibility condition This is possible since being slowly decreasing implies G, E E is onto * (ae-Gee which leads to a triangular system of equations in the coefficients of the polynomials .This linear system is solvable (uniquely) because all the terms in the diagonal equal (), which doesn't vanish since (e) 0 and we have assumed ( 27) holds.Applying the conditions (15) to the coefficients of F oh- tained this way, we obtain a set of necessary and sufficient conditions in the coefficients of H for the solvability of the system (30).These conditions be- come simpler when all the roots of () 0 are simple, since then (32) defines the numerical coefficients by If i form one of the groupings of roots of () 0 and, for instance r maxl j -kl _<i then, as explained before Theorem 5, a necessary condition on the is given by I I k-j l(j)10(exp(-Np(j))) j k#j for all N > 0 (j=l ,r).For example, when (z) sin z and (z) --sin-z, we are really dealing with periodic functions F and H and one gets z k the condition on the Fourier coefficients qk of H, for every N > 0 ({=0} ).This gives, lkl 0(l+Ikl) -N' dist(k,mk)) as k-, m which imposes a severe restriction on the Fourier coefficients of H if k is chosen as in (8).
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.After the work of Schwartz [64], it wasn't very clear which shape, if any, Fourier representation would take in several variables.It was suggested in [64] that it might be easier to deal with the Spectral Analysis Theorem (and its companion, the Spectral Syn- thesis Theorem).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 l(Z) 6(z) O} but i "'''6 6 E'(n) (n> 2) for which {z n {f 6 E: i* 6" 0} {0}.(In the analytic case, one can get by with two equations.)On the positive side, Malgrange had shown in his thesis that the exponential-polynomial solutions to * 0 were, in fact, dense in the space of all solutions.This confirmed the possibility of a representation (see [49, 50] and [19, 37] for related results), but gave no hint as to what the repre- sentation looked like.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 every smooth function has a Fourier representation of the following kind.Given (n) there is a Radon Measure cn and this integral converges in E (see [4,23]).The point here is that we are allowing arbitrary complex frequencies to occur in (33).Going back to the wave equation in 2 D'Alembert showed that the general smooth solution of (34 with fl' f2 smooth functions of one variable (see [44]).
where t denotes the variable in .Inserting (36) into (35) we get -i (Xl-X2) w I -i (Xl+X2) w f(x) e dVl(W + e dr2 (w) Let z 6 2 represent the dual variable to x 6 2, then the above expression can be written as f (x) I{Zl+Z2=0} e "Zdvl(z) + i-z2=O} e "Zdv2(z) z-z=O} -ix e "Zdv(z), (37)   where the last integral converges in E and the Radon measure dv is supported by the algebraic variety V {z 2 2 2 z I z 2 0}.This is precisely the alge- braic variety V() {z 2 (z) 0}, where is the distribution arising from the convolution equation (34).(It is easy to see that the choice of measure dv is not unique [59].) This representation of the solutions of the wave equation was generalized to the case of arbitrary linear differential operators with constant coefficients in [22].As usual, if P is a polynomial in n variables with complex coefficient and Dj = x.' then P(D) represents a partial differential operator.3 THEOREM 6: Let 'V be the algebraic variety of the zeros of P, V {z cn: r P(z) 0} Then there are finitely many algebraic varieties V c V, U V. V, J j=l J and differential operators 0. with constant coefficients such that if P(D)f j=l V. 3   3 where the dr.are Radon measures in n with support in V. and all the inte- grals converge in E.Moreover, if all the irreducible factors of P are dis- tinct then one has the representation 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.In [23, 59] this theorem is actually proved for systems.For instance if PI(D) P (D) 0 then (38) holds with these m two differences (i) V {z 6 cn: pl(z Pro(z) 0} and (ii) the 8j do no longer have constant coefficients.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.The point here is that the algebraic nature of the variety V plays a very strong role.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 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 convo- lution operators which look like 8m m-I f + l(X') , n n where x' (n-l).The proof follows closely Xn_I) and I m 6 E' that of Ehrenpreis' Theorem 6 and avoids trouble by restricting consideration to a single equation.When [4] appeared, the remaining difficulties were twofold: 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].To cope with the neces- sary "groupings" we gave in [8] a definition of slowly decreasing (for E'(Rn)) slightly different from the usual one [21].Instead of giving the formal definition, let us just mention a few examples of such distributions (a) For n i, the definition is the same as before, i.e. condition (9).
(e) In the analytic case, every E A' (n), @ O, is slowly decreasing.
One can then prove THEOREM 7 [8] Let be slowly decreasing V {z n (z) 0}.There exists a locally finite family of closed subsets Vj of V, differential opera- tors ij and a partition of the Index family J {21} into finite subsets Jk such that if , f --O, then k j k V. j where the series converges in E and dr. are Radon measures with support in Vo.
REMARK 3: Both Theorems 6 and 7 contain also information about the mass of the measures dr. which corresponds to the estimates (15) of Theorem 4.
As mentioned at the end of Section 4, it is also possible to eliminate the condition that is slowly decreasing.
THEOREM 8 [8]: Let # 0 be a distribution in E' then, with the same notation as in Theorem 7, we have f(x) lira ..e dv (z) where the dvj,g are Radon measures with support in Vj the limit are taken in the sense of E.
and both the series and 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 jointly slowly decreasing distributions '''''r" Once this is done, Theorem ?
holds with V {z C n i r (z) (z) 0}.The definition itself is quite technical; and the only point worth emphasizing here is that it requires V to be a complete intersection variety of codimension r.Thus our result is not strong 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 Jacobi interpolation function, which plays the role of the Newton polynomial Q from (13).This Jacobi interpolation function was so named because it appears implicitly in [39].For other insights into the contents of [39] we suggest [30].
If f E(2) satisfies the system of equations * 2 * 0, then f admits a series representation which converges in E and is of one of the follow- ing two kinds, -ia-x -i-x () f(x) (x)e + e a J a V\J where V {a 2 () 2(a) 0}, J is a finite subset of V (independent of f) and the polynomials c (x) (a J) have degree less than the multiplicity of a. (It is implicit that V is a dis- where the c (x) in the first sum have the same meaning as in (i) and those with a 6 V 1 are polynomials of degree _<1.
The two cases arise according to whether the system of algebraic equations a I + bl + clq + dl 0 a 2 + b2 + c2 + d2 q 0 has a pair of simple roots in C 2 or a single double root.
The proof consists simply of checking that i' 2 are jointly slowly de- creasing and applying the general theory from [8].Delsarte had proved a weaker version of this theorem in [16] under the additional assumption that the roots of L 2 V were all simple while in [56] Meyer gave an version of this theorem.A similar expansion holds when the square is replaced by a convex polygon, which improves on a recent result of Yger [74].
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 ull ne@.gborhood of the variety V {z E n: l(Z r(Z) 0} (even for the case r i) and hence is sub- stantially weaker than our results.(The case r i is very easy and was done in [2].) 6.The equivalent to Theorem 5 is conspicuously missing from the previous sec- tion; the reason is that we have only partial results in this direction.Typical examples are the following (see [8]).
THEOREM i0: Let be a difference-differential operator of the form (4) such that the discriminant of the pseudopolynomial (see (41)) does not vanlsh identically.Then if , 0 we have 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: Let E E' be such that at every point of V {z C n for some constants e, A > 0. Then the conclusion of Theorem l0 holds.
Even under the assumption that V is a manifold (as in Theorem Ii) Spectral Analysis Theorem hold, especially in relation to the study of distribu- tions 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.

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, hence we can find G such that ,G g and take F f-G, H h-v*G in (30)-(31).It follows that and are mean-periodic with respect to , so we can apply Theorem 4

THEOREM 9 :(
Let S denote the unit square in the plane with vertices A i). Let gl,g 2 LI(S) be extended as zero outside S. Assume further that the eight complex numbers al,a2,...,dl,d 2 satisfy the condition A(A,B)A(A,C)A(C,D)(B,D) crete set and that for a V\J, cC.)(ii)With V and J as above, V\JV I U V 2, V I V 2 and V 2is the disjoint union of the doublets {a,} (a#), and - of all continuous linear functionals on E) then the convolution of with compact support in IR n (i.e., space x <!',f (f) is the duality pairing of [' with [.One importance of the con- volution product resides in the fact that the most general linear continuous oper- ator M" [ [ which commutes with translations can be represented by a convolu- tion, i.e., if M 3, ch.