数理解析研究所講究録
1001 巻 1997 年 22-38

22

On the ”grouping” phenomenon for
holomorphic solutionsof infinite order differential equations
Daniele C. Struppa
Department of Mathematical Sciences
George Mason University
University Drive, Fairfax, VA 22030, USA.

23
1. Int.roduction
Purpose of this paper is to clarify some interpolation questions related to the space$\mathcal{H}’(\Omega)$of analytic functionals carried by$\mathrm{a}\cdot \mathrm{C}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{a}\mathrm{c}\mathrm{t}}}$set $K$ contained in the convex

open set. In particular, I will address the issue of”grouping of terms” whicharises when one looks for exponential series representations of solutions, in thespaceof holomorphic
onofof infinite order differential.
equations.
$\Omega\subseteq C$

function..s$\mathcal{H}(\Omega)$$\Omega,$$\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}\mathrm{e}}\mathrm{m}.\mathrm{S}$Virtually all of the material contained in this paper originates from [

$2|,$

[

$5|$

and [6](but see also the recent monograph [1]); on the other hand, we are presenting thismaterial in a unified fashion as a background to a joint work with T. Kawai [7]on the application of these ideas to some very classical overconvergence results ([3],[4] . Of some interest, hopefully, is the final construction of a large class of examples$)$

of infinite order differential$\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\dot{\mathrm{a}}$tors for which ”grouping” is necessary.

Acknowledgement. The author whishes to thank the R.I.M.S. of Kyoto University for the hospitality during the writing of this paper, as well as Professor Kawai,
for his invitation to lecture at the workshop ”Exact $WKB$ Analysis and Fourier
Analysis in the Complex Domain”.

2. Basic Definitions
Let

$p$

a subharmonic function, $p:\mathcal{O}arrow R$ , which satisfies the following conditions:(i) $p(z)\geq 0,$ $\log(\iota+|z2|)=O(p(z))$ ;(ii) there exists$C>0\mathrm{s}\mathrm{u}\mathrm{C}\mathrm{h}$.

that if $|z-\zeta|-\leq 1$ , then$p(()\leq Cp(_{Z})+c$ .

Such a function will be said to be a weight, and following H\"ormander, see [1], we
define the space$A_{\mathrm{p}}=A_{p}(C)=${$f\in \mathcal{H}(C)$: for some $A,$ $B>0,$ $|f(Z)|\leq Ae^{Bp(z)}$ }.

24Remark 1. The main example we are interested in, is the case in which $p(Z)–|z|$ .
In this case $A_{p}(C)$ is the space $Exp(\sigma)$ of entire functions of exponential type,which is isomorphic (via Fourier-Borel Transform) to the space$\mathcal{H}’(C)$of analytic

functionals (Paley-Wlenecr theorem).
Remark 2. For every $B>0$ , one can consider the Banach space$A_{p,B}= \{f\in \mathcal{H}(\sigma) :||f||_{B}:=\sup_{\sigma}|f(z)e^{-B(z})p|<+\infty\}$

By considering an increasing, diverging sequence

introduces an$\mathrm{L}\mathrm{F}$$\{B_{n}\}_{n\geq 1}$.

of positive numbers, one-topology (inductive limit of a sequence of Fr\’echet spaces) on$A_{p}$

by the equality$A_{p}= \mathrm{i}\mathrm{n}\mathrm{d}\lim_{n}Ap,B_{n}$.

Note that such topology is obviously independent of the sequence$\{B_{n}\}$. In this

topology, closure is equivalent to sequential closure.
ultaneously with$\mathrm{S}\mathrm{i}..\mathrm{m}${$A_{p,0}=A_{p,0}(C)=$$A_{p}$, one may consider the space:$f\in \mathcal{H}(C)$$\forall\epsilon>0\exists A_{\epsilon}>0$such that$|f(Z)|\leq A_{\epsilon:}e^{6}$$P(z)$}.

Functions in this space are said to be of minimal type with respect to $p(z)$ . If,
as before, $p(z)=|z|$ , thenis the space Exp0
of functions of infraexponen$A_{p,0}$$(\sigma)$tial type (order 1 and type ), which is isomorphic, (again via the Fourier-Borel$0$transform) to the space$(\mathcal{H}(\{0\}))’$

equivalently, to the space
Remark 3.

$A_{p,0}$$B_{\{0\}}$of hyperfunctions supported by the origin.

is a Fr\’echet space whose norms are given by.

$||f||_{m}= \max_{C}\{\{|f(Z)|\exp(-\frac{p(z)}{m})\}$One immediately sees that

$A_{p,0}$is an FS-space.

Remark 4. If, instead of the space$\mathcal{H}(\Omega)$of analytic functionals carried by the origin or,$\mathcal{H}(C)$of entire functions, one considers the spaceof holomorphic functions on a convex open set$\Omega\subseteq \mathcal{O}$, the spaces$A_{p}$

and$A_{p,0}$

25
are not sufficient anymore to describe analytic functionals, and some modifications
are needed. Indeed, the spacecontained in$\Omega$$\mathcal{H}’(C)$of analytic functionals carried by some compact

is isomorphic to the space $Exp(\Omega)$ of entire functions for which there

exists a compact set$I\acute{\mathrm{t}}\subseteq\Omega$and a constant $A>0$ such that$|f(Z)|\leq A\exp(H_{f\dot{\mathrm{i}}}’(z))$

where$H_{K}(Z)= \sup_{w\in \mathrm{A}}Rer(z\cdot w)$

is the so-called supporting function of$I\mathrm{i}^{\Gamma}$. The arguments are given in the next

sections for the spaces $A_{p}(C)$ and $Exp(C)$ , need to be suitably modified, if one
wants to apply them to the spaces $Exp(\Omega)$ . Unfortunately, in this lat,ter case, theestimates and the results are not so clear as in the$Exp(\Omega)$$A_{p}$

case. Note, as well, thatis not an algebra anymore.

3. Interpolation

A multiplicity variety in$z_{k}\in \mathcal{O},$$m_{k}\in J\mathrm{V}$$C$

is a collection $V=\{(z_{k}, m_{k}), m_{k}\geq 1\}$ of pairs with, and $|z_{k}|arrow+\infty$ . Weierstrass’ theorem ensures that, given amultiplicity variety , there is always an entire function
The space $A(V):=\mathcal{H}(C)/I(f)$ of holomorphic functions on
$V$

$f$

such that $V=V(f)$ .

$V$

can be identified withthe space$A(V)=\{\{a_{k,l}\} :a_{k,l}\in C, 0\leq l\leq m_{k}-1, l\in J\mathrm{V}\}$.

There is a natural surjective restriction map$\rho:\mathcal{H}(C)arrow A(V)$defined by$\rho$:$g arrow\{\frac{1}{l!}\frac{\partial^{l}g}{\partial z^{l}}(_{Z_{k}})\}k,l$

The fundamental problem which arises in the theory of mean-periodic functions(i.e. functions which are solutions of a system of convolution equations) is theidentification offor it.

$\rho(A_{p}(\sigma_{))}$. To this purpose, we begin by defining a first ”candidate”

26
Definition. The space $A_{p}(V)$ is defined as the set of all sequencessuch that, for some $A,$ $B>0$ and all $k\geq 1$ ,$\sum_{0\leq l\leq mk}|ak,l|\leq A\exp(Bp(z))$$\{a_{k},\iota\}$in $A(V)$.

Remark 5. Using Cauchy’s inequalities for holomorphic functions, and the property of weight functions, it is immediate to deduce that$\rho(A_{p}(\mathcal{O}))\subseteq A_{p}(V)$.

Remark 6. With reference to Remark 4, one can also define the space $Exp\Omega(V)$in $A(V)$ for which there exists a compact
as the space of all the sequencessuch thatand a constant$\{a_{k},\iota\}$$A_{\epsilon}>0$$K\subseteq\Omega$$\sum_{0\leq l\leq mk}|ak,\iota|\leq A\exp(H\kappa(zk))$

Also in this case it is easy to verify that
Definition. A multiplicity variety
the weight ) if$V$.

$\rho(E_{X}p(\Omega))\subseteq Exp_{\Omega}(V)$.

is said to be interpolating (with respect to$p$$\rho(A_{p}(C))=A(pV)$ .

This definition (and a similar one which could be given for $Exp(\Omega)$ ) immediately
raises two questions which we will address in the rest of the paper:(a) find conditions for $V$ to be interpolating;.
(b) describe$\rho(A_{p}(\sigma_{))}$

While a complete solution to (a) and (b) does not exists for the most general case,things are fairly simple when one considers the case $p(z)=|z|$ . In fact, one has thefollowing result (see [1] and [2] for many related results as well):Theorem 1. Let $f\in Exp(\sigma)$ . Then $V(f)$ is interpolating if and only if there areconstants $A,$ $B>0$ such that$| \frac{1}{m_{k}!}\frac{\partial^{m_{k}}f}{\partial z^{m_{k}}}(z_{k})|\geq Ae-B|z_{k}|$

(1)

27
for all $k=1,2,$$\ldots$.

Remark 7. Theorem 1 is not true (without additional assumptions) for weights
which are not radial and satifying thecondition $p(2z)=O(p(Z))$ . For
every weight , however, condition (1) is still sufficient.
$\mathrm{d}\mathrm{o}\dot{\mathrm{u}}\mathrm{b}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}$$p$Remark 8. The reason why $p(z)=|z|$ behaves differently is the fact that, in, we have the Lindel\"of theorem which claims that if $f\in Exp(C)$ and$Exp(\mathcal{O})$$g,$is entire, then$\frac{g}{f}$is still in $Exp(C)$ . This can be rephrased by saying that the ideal$I(f)$ is closed in $Exp(\sigma)$ .

(unless conditions on$Exp(\sigma)$$\frac{g}{f}$$f$This property is not true anymore for different weights

are imposed), nor it is true that any non-principal ideal in

is closed.

An (almost) immediate consequence of Theorem 1 is the following necessary condi-tion for interpolation, which we will be using shortly:

Theorem 2. Let $f\in Exp(C)$ . Then if $V(f)$ is interpolating, there are positive
constants $A,$ such that, if we set $d_{k}:= \min\{1, \min_{i\neq k}|z_{i}-z_{k}|\}$ , then$B$$d_{k}^{m_{k}}\geq Ae^{-B|z|}k$.

(2)

Remark 9. Once again, this theorem is not valid in arbitrary$Exp(\Omega))$ , unless additional conditions are imposed on.

$A_{p}$

spaces (or in$f$

In order to describe $\rho(Exp(\sigma))$ for the case in which the variety

is non-interpolating, we need to discuss an important property of entire functions of exponential
growth, which is essentially the basis for Lindel\"of theorem.
$V$

Theorem 3. Let $f\in Exp(C)_{f}f(\mathrm{O})=1$ . Then there exists a sequence of radiisuch that for some positive constants $A,$$\{R_{n}\}_{y}R_{n}\uparrow+\infty_{f}R_{n+1}\leq 4R_{nf}$$\min|f(Z)|\geq A\exp(-BRn)$ .

$|z|=R_{n}$

$B$

28
The proof of this result is essentially a consequence of the minimum modulus
theorem for holomorphic functions. The result is crucial is establishing the closure

in $Exp(\sigma)$ , and also plays a crucial role in constructing
the image $\rho(Exp(C))$ . Since, in the sequel, we will be considering infinite orderdifferential equations, we now consider an element $P(z)\in Exp0(\sigma)\subseteq Exp(C:).$, andof the ideal generated by$f$

we take a sequenceof radii such that$\{R_{n}\}$$\min|P(z)|\geq A\exp(-BRn)$ ,$|z|=R_{n}$as given by$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}\sim.3.,$$\mathrm{F}:$in.ally, we d.e.note by$V=\{\{\alpha_{k},m_{k}\}, k=1,2, \ldots, m_{k}\geq 1\}$the multiplicity variety associated to $P(z)=0$ , and we assume that the roots
have been ordered so that...

$\leq|\alpha_{k}|\leq|\alpha_{k+1}|\leq\ldots$$\alpha_{k}$.

Let$V_{n}=\{(\alpha_{k}, m_{k}) :be the subset of$V$for which$\alpha_{k}\in C_{n}$k_{n}\leq k<k_{n+1}\}$, where$C_{1}=\{z :.

|z|<R_{1}\}$and
R_{n-1}<|z|<R_{n}\}$ ,$c_{n}=\{z :Now, given any function$\varphi$

holomorphic on$C_{n}$$n\geq 2$.

, one can construct the interpolating

function$\varphi_{n}(z)$$:= \frac{1}{2\pi i}\int_{\partial C_{n}}\frac{\varphi(\zeta)}{P(\zeta)}\cdot\frac{P(()-P(z)}{\zeta-z}d\zeta$

defined and holomorphic on$C_{n}$.

One immediately sees that$\varphi_{n}(z)=\varphi(Z)-\frac{P(z)}{2\pi i}\int_{\partial C_{n}}\frac{\varphi(\zeta)}{P(\zeta)}$.

$\frac{d\zeta}{\zeta-z}=$29$=\varphi(z)+P(z)\psi_{n}(z)$with$\psi_{n}$holomorphic on$C_{n}$. As a consequence we have that$\rho_{n}(\varphi_{n})=\rho_{n}(\varphi)=\{a_{k},\}(n_{l})\cdot=\{a_{k}l :\alpha_{k}\in C_{n}\}$.

If we now define, on $A(V_{n})$ the norm$||a^{(n)}||_{n}:=||a_{k}^{(},|n)|ln \inf=\{||\varphi||_{\infty} :we see that,’$\mathrm{f}\mathrm{o}\mathrm{r}$\varphi\in A(C_{n})_{)}\rho_{n}(\varphi)=\{a_{k,\iota}^{(}\}n)\}$,any $\varphi\in Exp(C)$ ,$||\rho n(\Psi)||n\leq A\exp(B|z|)$for some positive constants $A,$$B$,

$z\in C_{n}$, which do not depend on,$n$. Now, if$\underline{a}\in A(V)$,$\underline{a}$can be written as a sequence$\underline{a}=\{a_{n}\}_{n},$$a_{n}\in A(V_{n})$,and we can define$A_{*}(V)= \{\underline{a}=\{a_{n}\}\in A(V) : \exists D>0\mathrm{s}.\mathrm{t}.

||\underline{a}||_{D}:=\sup_{n}||a_{n}||_{n}e^{-DR_{n}}<+\infty\}$and the following crucial result can be established [2]:Theorem 4. There is a topological isomorphism between $\rho(Exp(\sigma))$ and $A_{*}(V))$
where $A_{*}(V)$ is endowed with its natural inductive limit topology. In particular,$\frac{Exp(C)}{I(P)}\cong A_{*}(V)$4.. Ex.

The$\dot{\mathrm{w}}\mathrm{i}\mathrm{l}\mathrm{l}$.

$\mathrm{p}_{\mathrm{o}\mathrm{n}\mathrm{e}}.$$\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}1$.

representation$\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}- \mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}$material which we have$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{C}\mathrm{u}\mathrm{S}\mathrm{s}\dot{\mathrm{e}}\mathrm{d}$in$\mathrm{t}\dot{\mathrm{h}}\mathrm{e}$previous sections,

now allow us to prove the representation theorem for holomorphic (entire) so-lutions for an infinite order differential equation. We recall, in this respect, that30
the space of infinite order differential operators is, in fact, isomorphic to the space
Exp0
of entire functions of infraexponential type. Specifically, any infraexpofor a differential operator $P(d/d_{Z)}$ , andnential type function $P(z)$ acts as a$(\sigma)$.

$\mathrm{s}\mathrm{y}\mathrm{m}\mathrm{b}_{0}1$

we have the following result:
Theorem 5. Let$P\in Exp_{0}(\mathcal{O})$$V=\{z :, and letP(z)=0\}=\{(\alpha_{k},m_{k}) :|\alpha 1|\leq|\alpha 2|\leq\ldots\}$.

Then there exists a sequence a sequence of indices $k_{1}<k_{2}<\ldots$ such that every
entire solution of$f$$P(D)f=0$

can be written an$f(z)= \sum_{\geq n1}(_{k_{n}\leq\iota}\sum_{k<k_{n+}}P_{k}(_{Z)}e^{\alpha}kz)$,, and where the sequenceis a polynomial of degree less thanof such polynomials satisfies growth conditions necessary to the convergence of thegrouped series on the right hand side of (3).

where$P_{k}$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{f}.\cdot$

First, we$\{P_{k}\}$$m_{k}$

consider. t.he map$\beta$$:= \rho\cdot \mathcal{F}:\mathcal{H}’(C)arrow\frac{Exp(C)}{I(P)}$and $Exp.(.C)$ . It is immeis the Fourier-Borel isomorphism between
diate to see that is a continuous surjective map between DFS-spaces. Then, bya well known result from functional analysis, its transpose map

where$\mathcal{H}’(C)$$\mathcal{F}$$\beta$$\beta^{t}$:$( \frac{Exp(C)}{I(P)})’arrow \mathcal{H}(C)$has closed image, and is a topological isomorphism onto its own image. Moreover,$Im(\beta^{t})=(ker\beta)\perp$ and since $F(ker\beta)=I(P)$ , we obtain (since
is an isomor$\mathcal{F}^{\cdot}$

phism) that $ker\beta=P(D)(\mathcal{H}’(\sigma_{))}$ , and so, finally,$(ker\beta)^{\perp}--\{f\in \mathcal{H}(\mathcal{O}) :P(D)f=0\}$31
is the kernel in$\mathcal{H}(C)$of the infinite order differential operator $P(D)$ . Since we knowthat $Exp(\mathcal{O})/I(P)$ is topologically isomorphic to $A_{*}(V)$ , our proof will be concluded
by simply describing the dual $(A_{*}(V))’$ of $A_{*}(V)$ . To do so, let$\underline{b}=\{b_{n}\}_{n\geq 1}$

with $b_{n}\in(A(V_{n}))’$ for any$\prime n$, and with norms$\underline{b}\in(A_{*}(V))’$$||\cdot||_{n}’$; thenin these dual

spaces given by$||b_{n}||_{n}’:= \max\{\langle a_{n}, b_{n}) :||a_{n}||_{n}\leq 1\}$

,

where
$(a_{n}, b_{n}):= \sum_{k,l}a_{k}^{()},\iota kb^{(n)}n,\iota$

with (hopefully) obvious meaning of the symbols. On the other hand,is an element of, then th. re exists $C>0$ such that$\mathrm{i}\mathrm{f}\underline{a}=\{a_{n}\}_{n\geq 1}$$A_{*}(\dot{V})$$.\mathrm{e}$$\sup_{n}||a_{n}||_{n}e^{-CR_{n}}<+\infty$

and therefore if$\underline{b}$,is an element of $(A_{*}(V))’$ , then we have that for every $D>0$ ,$|| \underline{b}||’:=\sum_{n\geq 1}||b|n|_{n}’e^{DR_{n}}<+\infty$

and now the duality bracket is obviously defined by$\langle\underline{a},\underline{b}\rangle=\sum_{\geq n1}$

Consider now a solution

$f$

(an’

$b_{n}$).

of the differential equation$P(D)f=0$ ;then there exists a unique$S\in \mathcal{H}’(\sigma)$$\underline{b}\in(A_{*}(V))’$such that$f=\beta^{t}(\underline{b})$. Then, for every,$\langle S,$$f)=\langle S,$ $\beta^{t}(\underline{b}))=\langle\beta(S),\underline{b}\rangle=\langle\rho(F(S)),\underline{b}\rangle$,

with the series at the right side which (by the definition of the topology in$\mathcal{H}’(C)$)

converges uniformely for every bounded family of analytic functionals . In particular, for in a fixed compact set in , we can take
. Then$S$$z$$\mathcal{O}$$S=\delta_{z}$$\rho(F(\delta_{z}))=\rho(e^{z})\omega=\{\frac{z^{j}e^{\alpha_{k}z}}{j!}\}_{k\geq 1,0\leq}j\leq mk$

32
so that$( \delta_{z}, f\rangle=f(Z)=\sum_{1n\geq}(_{k_{n}\leq\cdot 0}k<k_{n}.+1,\leq j<mb_{k,j}\sum_{k}\frac{z^{j}e^{\alpha_{k}z}}{j!}\mathrm{I}$with uniform convergence of the series on each compact set in$C$.
$\bullet$

Now that we have completed the proof, we would like (in order to prepare

for the examples in the next section) to see whether it is possible to make theestimates on the coefficientsa little more explicit. For the case in which
is an interpolating variety, the situation is quite simple, since $A_{*}(V)=A_{p}(V)$( $=Exp(V)$ in our case), and therefore there are no more groupings. Indeed, if isinterpolating, it possible to choose the circles
in such a way that (discounting
multiplicities), only one zero of $P(z)$ falls between
and, for every . In
this case we see that$\{b_{k,j}\}$

$V$

$V$

$\{R_{n}\}$$R_{n}$$((Exp(V))l=\{\underline{b}=\{b_{k,j}\}s.t$. $\forall D>0,$$R_{n+1}$$n$$\sum_{k,j}e^{B||}(\alpha_{k}\sum_{jo\leq<mk}|b_{k,j}|)<+\infty\}$which is as explicit as we can get, and gives us the necessary and sufficient condition
for the convergence of the series (with no groupings).
In the case in which the variety is non-interpolating, groupings may become
necessary, and the precise description of the bounds becomes much subtler. The
method which Berenstein and Taylor used in [2] (see also [1]) consists in using
Newton’s divided differences as a way to compute explicitely a choice for an inter-

polating function on . The method is a bit complicated, but our purposes will
be satisfied if we quote the result, at least for the case of groupings whose points
all have(some- further modifications would be necessa..ry for thegeneral case).
$V$$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{C}\sim \mathrm{i}\mathrm{t}\mathrm{y}.\mathrm{o}\mathrm{n}\mathrm{e}$Assume we have, in$V_{n}$, the distinct simple points$(n)$$\alpha_{1}$

,

$(n)$
$\alpha_{2}$

,...,

$(n)$
$\alpha_{k_{n}}$.

33Construct thesquare matrix $J=J^{(n)}$ (where theeliminated troughout its entries for the sake of readibility):$t_{n}\cross t_{n}$has been$\mathrm{s}\mathrm{u}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{x}.(n)$

$J=$

;

it should be noted that the construction of this matrix can be described explicitely
in terms of the divided differences of different orders at the points in. Given$V_{n},$

this matrix, the estimates on

element of $(A_{*}(V))’$ if for$\underline{b}=\{b_{n}\}$$\mathrm{a}\mathrm{n}\mathrm{y}_{e}D>0$

become quite explicit. Specifically,, and every choice of points$\sum_{n\geq 1}(_{j=1}\sum^{t_{n}}|c(n)|j\mathrm{I}e^{D|\beta_{n}|}<+\infty$

where the$\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}_{\mathrm{o}\mathrm{r}}\underline{c}^{()}n$is defined by$[1]$$\underline{c}^{(n)}=J\cdot b_{n}$$\beta_{n}\in V_{n}$$\underline{b}$

is an

,

,

.

5. Examples of the necessity of groupings.

In this final section I will show how to construct a large class of examples of differential operators of infinite order such that the exponential representation of their

entire solutions requires necessarily the use of groupings. Examples of this kind(but with different growth conditions) have been given by Schwartz and by Leont’ev(see [8] or [1] and the references therein). Our simple, but explicit, construction has
been, however, the starting point for the forthcoming work of Kawai and the author[7], in which we examine a new treatment of Bernstein-Ostrowski overconvergencephenomena [3], [4] based on the properties of infinite order differential operators.
In principle, the idea for the construction for such an example is simple. Since$ker(P(D))=(A_{*}(V))’$ , and $(A_{*}(V))’=Exp(V)$ if and only if
is interpolating,then one may assume that the ”grouping” phenomenon will occur whenever thevariety is not interpolating, and it is therefore sufficient to construct a variety ,whose points are the only (simple) zeroes of a function of infraexponential type,
$V$

$V$

$V$

$P$

34
and such thatis not interpolating. However, the condition of

being interpolating is only sufficient to guarantee that no groupings are necessary. In fact, while wehave no examples to the contrary in the case of varieties of infraexponential type,we know that if $p(z)=|Imz|+\log(1+.|z|)$ , it is possible to find (and characterize)
classes of varieties which, even though they are not.interpolating, they still define
convolutions operators whose solutions can all be represented as exponential series,
whose convergence does not require grouping procedures. These varieties are called”weakly interpolating”, [2], and, for the case of $p(z)=|z|$ , it is known that thenotion of interpolating an.d weakly interpolating varieties coincide.
To construct the variety we need, consider a sequenceof zero density(i.e.
) so that we know that there exists $P_{1}\in Exp_{0}(\sigma)$ such that
$V(P_{1})=\{\alpha_{k}\}$ . We now choose, for every, a quantity such that for any $A,$ $B>0$ ,. This can be easily done by taking, for example,
Now
consider the new sequence of points
$V$

$V$

$|\alpha_{k}|\uparrow+\infty$$\alpha_{k},$$\lim_{narrow+\infty}\frac{n}{\alpha_{n}}=0$$k$$\epsilon_{k}$$|\epsilon_{k}|<Ae^{-B|\alpha_{k}|}$$\epsilon_{k}=e^{-|\alpha_{k}|^{2}}$$\beta_{k}:=\alpha_{k}+\epsilon k$.

Obviously$\lim_{narrow+\infty}\frac{n}{\beta_{n}}=narrow+\lim_{\infty}\frac{n}{\alpha_{n}+\epsilon_{n}}=0$

and so we know that a second function$\{\beta_{k}\}$$P_{2}$,

exists in Expo$(\sigma)$, such that $V(P_{2})=$. Now the function$P=P_{1}\cdot P_{2}$is still in Expo $(C)$ , which is an algebra, and $V(P)=\{\alpha_{k}\}=\{\beta_{k}\}$ . On the other
hand, we may use Theorem 2 to immediately see thatis not an interpolating
variety, since condition (2) is violated for. I claim that the operator whose
symbol is $P$ is such that groupings are necessary to represent all the solutions
to $P(D)f=0$ . To see this, we will assume that each sequenceand$V$$d_{k}=\epsilon_{k}$$\{\alpha_{k}\}$$\{\beta_{k}\}$is, by itself, interpolating. This request is not strictly necessary, but will makeour computations easier. In order to ensure thatis interpolating, it will beenough (see [1], for a discussion of this well known result of Levinson) to ask that,$\{\alpha_{k}\}$35
for example,$|\alpha_{k}-\alpha_{h}|\geq 2|k-h|$. In this way we are also guaranteed that an analogous
estimate holds for the sequence, which is therefore automatically interpolating.

for all indexes

$h$

and

$k$

$\cdot$$\{\beta_{k}\}$This shows that if we apply Th.eorem 5 to the equation$P(D)f=P_{1}(D)(P_{2}(D)f)=0$ ,then the groupings produced by the proof (but which, a priory, may still not be necessary) are composed of two roots each, namely we are grouping the pairs

for any value of$\{(\alpha_{k}, \beta_{k})\}$(th. reader may want to compare this example with the one given
by Bernstein in the second chapter of [3] . Thus every entire solution of$k$$\mathrm{e}$$)$$P(D)f=0$

can be represented by the grouped series$f(_{Z})= \sum_{k=1}^{+\infty}(A..ke^{\alpha}kz+Bke^{\beta k)}z$

where the coefficients
andmust satisfy suitable growth conditions to ensure.
that the series is convergent in the topology of$A_{k}$$B_{k}$$\mathcal{H}(C)$In order to determine the exact nature of these growth conditions, we proceed toconstruct the matrices
for this specific case (note that here we have $t_{k}=2$ for$J^{(k)}$all values of $k=1,2,$$\ldots$). We immmediately see that it is$J^{(k)}=$and that the corresponding element$b_{n}$

an element of $(A_{*}(V))’$ if and only if

is given by $b_{k}=(A_{k}, B_{k})$ and$carrow=J^{(k)}\cdot b_{k}$$\underline{b}=\{b_{k}\}$issatisfies (4). A simple matrix

multiplication shows that
$c=(\sim A_{k}+B_{k}, (\alpha k-\beta_{k})B_{k})$36
and therefore the estimates (4) give (see Remark 10 below)(5)$\lim_{karrow+\infty}\frac{\log|A_{k}+B_{k}|}{|\alpha_{k}|}=\frac{!\mathrm{o}\mathrm{g}|Ak+Bk|}{|\beta_{k}|}=-\infty$

and$\lim_{karrow+\infty}\frac{\log|\alpha_{k}-\beta k||\beta k|}{|\alpha_{k}|}=\frac{\log|\alpha_{k}-\beta k||\beta k|}{|\beta_{k}|}=-\infty$(6).

Remark 10. Note that (4) can be rewritten as$\sum_{n\geq 1}||\underline{\mathrm{c}}^{(n}|)|e^{D}|\rho_{n}|<+\infty$

for any $D>0$ and any choice of points

This condition is obviously.

$\beta_{n}\in V_{n}$equivalent to$||\underline{c}^{()}n||e^{D|\beta|}n=O(1)$.

can be interpreted
Since all norms are equivalent in finite dimensionale spaces,
norm; if we now take the logarithm, we obtain (5) and (6).
as either or$||\underline{c}^{(n)}||$$l^{1}$$l^{\infty}$

Remark 11. Condition (6) looks misleadingly asymmetric. However, since$|A_{k}|\leq|A_{k}+B_{k}|+|-B_{k}|$,we see that (6) actually is equivalent to a condition in which$|A_{k}|$.

Now note that since$|B_{k}|$is replaced by.
$|\alpha_{k}-\beta_{k}|=e^{-|\alpha_{k}|^{2}}$, we obtain$\log|\alpha_{k}-\beta_{k}|=-|\alpha_{k}|^{2}$,and therefore condition (6) becomes$\lim-|\alpha_{k}||\beta_{k}|=-\infty$,$karrow+\infty$

which is satisfied (for example) by any sequence$\{\beta_{k}\}$bounded away from zero.

37
At this point we see that the choice $\{A_{k}=1\},$ $\{B_{k}=1\}$ satisfies both (5) and (6);as a consequence the series$f(z)= \sum_{=k1}^{+\infty}(e^{\alpha z}.k-e^{\beta_{k}})z$

is an entire solution to$P(D)f=0$ .

However, if we were to remove the condition of ”grouping” in the series, we would
lose convergence immediately. Indeed, if the series were convergent without group-

ings, its general term would be bounded at every point. But now, by simply taking$z=1$ , we have, with $A_{k}=1$ ,$|A_{k}e^{\alpha_{k}z}|=|e^{\alpha_{k}}|arrow+\infty$as $karrow+\infty$ .
This simple argument shows that groupings cannot be eliminated a priori from thestatement of Theorem 5. In the terminology of [3] (but see also [7]) we say that theabscissa of convergence of the series(for $a_{k}=(-1)^{k+1},$,
is zero, while its abscissa of overconvergence is.
$\sum_{k=1}^{+\infty}ake^{\gamma z}k$$\gamma_{2k}=\beta_{k})$$\gamma_{2k-1}=\alpha_{k}$$-\infty$

REFERENCES[1] C.A. Berenstein and R. Gay, Complex Analysis and Special Topics in Harmonic

Analysis, Springer Verlag, New York, 1995.
[2] C.A. Berenstein and B.A. Taylor, A new look at interpolation theory for entire

functions of one variable, Adv. in Math. 33 (1979), 109-143.
[3] V. Bernstein, Lecons sur les progr\‘es re’cents de la the’orie de s\’eries de Dirichlet,

Gauthier-Villars, Paris, 1933.
[4] P. Dienes, The Taylor Series, Dover, New York, 1957.
[5] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Inters-

cience, New York, 1970.

38[6] L. Ehrenpreis and P. Malliavin, Invertible operators and interpolation in A U-

spaces, J. Math. Pures et Appl. 5 (1974), 165-182.
[7] T. Kawai and D.C. Struppa, in preparation.
[8] A.F. Leont’ev, The representation of functions by generalized Dirichlet series,

Russian Math. Surveys 24 (1969), 101-178.