JOAQUIM ORTEGA-CERD.~ AND KRISTIAN SEIP* ..... the rectangles Rj will be bounded above and below by some constants since Ar is bounded above and ...
BEURLING-TYPE DENSITY THEOREMS FOR WEIGHTED L p SPACES OF ENTIRE FUNCTIONS By JOAQUIM ORTEGA-CERD.~ AND KRISTIAN SEIP*
1
Introduction
In [ 1], Berndtsson and one of us identified Beurling-type density conditions for sampling and interpolation in certain generalized Fock spaces. The purpose of the present work is to solve an essential problem left open in that paper. The setting is as follows. Let a subharmonic function ~b be given, whose Laplacian satisfies 0 < m < Ar < M (m, M positive constants) for all z E C. We denote by ~ (1 < p < ~ ) the set of all entire functions f for which f e - r belongs to LP(C). Our objective is to prove that the density conditions which in [1] were shown to be sufficient, are also necessary, for a sequence to be sampling or interpolating for ~-~. (We defer the definitions of sampling and interpolating sequences to the next section.) The arguments which have previously allowed such strict density theorems (see [12, 13, 9]) all rest on a scheme developed by Beurling in his study of sampiing and interpolation of bandlimited functions [2]. Some of these arguments seem indispensable and will be used again. We shall need some additional new tricks, but the main novelty of this paper is probably our way of adapting Beurling's approach to a setting seemingly different from the previous cases, which all involved holomorphic spaces with a suitable shift invariance. A prime example is the classical Fock space, which we obtain by setting r = ~lzl 2 (~ > 0). In [12], a key role was played by the translation operator Tr defined for every ~ E C by ( T c f ) ( z ) = (T~ f ) ( z ) = e~2r162
f ( z - ~).
*Part of this work was done while the second named author was visiting Centre de Recerca Matemhtica, Barcelona during September 1997 under support of the Comissionatper a Universitats i Recerca. The firstnamedauthoris supportedby the DGICYTgrantPB95-0956-C02-02 and grant 1996SGR00026. 247 JOURNALD'ANALYSEMATHt~MATIQUE,Vol.75 (1998)
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J. ORTEGA-CERD,kAND K. SEIP
A crucial property of Tr used in Beurling's scheme is that it acts isometrically in Clearly, the spaces ~'~ do not enjoy this sort o f group invariance, but we have found that the translation group can be brought into action in much the same way. Namely, we define a translation operator Tr for every ( E C acting isometrically from 5r~ to a different space 5r~. The two functions r and r162are related by the equation Az(r () r162 = 0, -
-
and we have uniform bounds on the growth of the functions r162 In the classical case, it so happens that we may choose r = r162for every (. This "translation invariance" is what allows us to adapt Beurling's scheme (as modified in [12]) to the present setting. By giving a complete description of sampling and interpolation in U~ in terms of densities only, our paper completes a line of research which was started in [12] and [14], and continued in [9] and [1]. However, it leaves us with the problem of understanding why some Banach spaces of entire functions allow strict density theorems, and others do not, like certain spaces of Paley-Wiener-type (see [ 10] and also [3] and [8]). This is a question which awaits a careful study, although the two papers [9] and [10] shed some light on it.
2
D e f i n i t i o n s a n d m a i n results
We set A = 02/Oz02, which differs from the standard convention for the Laplace operator A by a factor 4. We write f ~< 9 whenever there is a constant K such that f 0 such that K satisfies
IK(z, C)l
R and [Vx(z)] _< 2. Set h = x f , and choose a j such that [[h[(F- aj)[[%,p _< 2~, [[h[[r > 1/2, and e-r
(~) < 2e-r
for Izl < R. Then h has the desired properties, but it is not holomorphic. We may, however, correct it by solving the 0 equation Ou = f o x . Because
c[fOx[Pe-peaj da < (4E)p, we m a y apply Theorem A, and so there is a solution u with Ilull%,~ < ~. We split F - aj into two sequences/~ and ~', where f~ consists of the points from F - aj which lie in the corona R - 2 < Izl < R + 1, and f~' = (F - ay) \ 3. Since u is
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J. ORTEGA-CERD, h, AND K. SEIP
holomorphic outside R - 1 < Izl < R, we may apply the first inequality of (3.3) to obtain Ilul~'ll+o~ p < ~. On the other hand, fn-3 0, we introduce the following two finite dimensional subspaces of ~ . Denote by Ws the subspace generated by the functions k(w, s), s E D(z, R + p) N S, and by Ws the subspace generated by the k(w, i), i 6 D(z, R) N I. (One should think of R as much bigger than p.) We define Ps and P1 to be the orthogonal projections of ~ onto Ws and WI, respectively. Consider the operator T = PzPs defined from WI to WI. We are going to estimate the trace of this operator in two different ways. To begin with, tr (T) < rankWs < #{D(z, R + p) fq S}. On the other hand, tr ( T ) =
Z
(T(k(w,i)),P1k*(w,i)),
iEIAD(z,R)
where {k*(w, i)} is the dual basis of k(w, i) in H. Since T = PxPs, then for any
i E IMD(z,R), (T(k(w, i)), Pik*(w, i)) = (k(w, i), k*(w, i)) + (Ps(k(w, i)) - k(w, i), Pik*(w, i)), whence tr (T) >_ # { i 6 I N D(z, R)}(1 - sup I(Ps(k(w, i)) - k(w, i), k*(w, i))1). i
Recall that we normalized k(w,i) so that IIk(w,i)ll -~ 1. Therefore IIk*(w,i)ll ~- 1 too. Thus if we can show that IIPs(k(w,i)) - k(w,i)ll < e for a sufficiently big p,
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J. ORTEGA-CERDA AND K. SEIP
we have proved that for every e > 0 there exists a p such that for all large R, the following inequality holds:
(1 - e ) # { I f-I D(z, R)} _< # { S fq D(z, R + p)}. This estimate implies the desired inequalities for the densities. It remains to be shown that IIPs(k(w, i)) - k(w, i)II < ~. To this end, we note that since S is a sampling sequence, k(w, i) = ~ s e s k( s, i)k(w, s)e -r . Therefore,
IIPs(k(w,i))- k(w,i)[I p
p
where the last inequality follows from Theorem B. The sum on the right side of this inequality is smaller than e if p is big enough, since S is a uniformly separated sequence. [] We now obtain nonstrict density conditions for sampling and interpolation. The inequality in the interpolation case will not be used later, because we shall give a direct proof of the strict inequality in the next section. But the proof is the same, and it comes for free. C o r o l l a r y 1. If F is a uniformly separated sampling sequence for ~ , D~(F) > 2/7r. IfF is an interpolating sequence for ~ , then D~-(F) < 2/7r.
then
P r o o f . Assume that F is a sampling sequence for )r~. For every e > 0, we may construct a sequence A with D~-(A) = D~ (A) = 2/(7r + e) as follows. Partition the plane C into strips of the form k - 1 < Re z < k, k E Z. Each of the strips can be partitioned into rectangles R3 such that fnj Ar = (Tr + e)/2. The sides o f the rectangles Rj will be bounded above and below by some constants since Ar is bounded above and below. Thus if we take as A a uniformly separated sequence consisting of one point from each of these rectangles, it is easy to check that it has the desired density. Now A is an interpolating sequence because D~ (A) < 2/7r. (This is Theorem 2b of [ 1].) Thus by the comparison lemma, we have just proved that D~(F) > D~-(A) = 2/(7r + e). Since e was arbitrary, we have the nonstrict inequality. For F an interpolating sequence, we proceed analogously. This time we use Theorem 2a of [ 1] saying that A is interpolating for f ~ whenever D~- (A) < 2/r. []
P r o p o s i t i o n 4. If F is a sampling sequence for ~ sampling for )r~_~lzl~ for all sufficiently small e > O.
(1 0 and f E f ~ , o ,
e-C(z) f(z) = E f(7)g(z, 7)e -r 7EF
and ~ Ig(z,'Y)l < K uniformly in z. This is so by a duality argument, because {f(7)}Tcr ~ f(z)e -6(z), with f E 9r~ '~ is a bounded linear functional on a closed subspace o f g~r whose norm we can bound independently of z. (We repeat here an argument from [13], p. 36.) We apply this representation formula to the function f(w)e :~-2~1zl2, and get
e-C(z) f(z) = E f('y)e-r
g(z, "y)
"/cF 2 for every f E ~-~-,Izl:" B y the Cauchy-Schwarz inequality, we obtain
If(z)1%-2~(z) 0 such that any other sequence F' = {%J} which is 6P-close to F (i.e. Lemma
17n - %'1 < 6Pfor all n) is interpolating with Mr, < M r / ( 1 - 5Mr). P r o o f . Using (3.4) we prove that i f F and F' are tSP-close (with 6p smaller than the separation constant o f F), then 0o
Ilf(7~)lPe-pr
-If(TJ)IPe-pr
~
6Pllfll~,~.
'n,=O
In order to prove that 1-" is interpolating, we pick an arbitrary sequence o f values {a'n} such that ~] la'lpe -pr a n' and
< 1. We must construct a function f such that
< K. We know that there exist gn such that gn(Tn) = g,~(Tm) = 0 for n ~ m and the function 9 = ~ g n verifies
f('Yn') e+r
=
IIg]lr
< Mr. Moreover, we have that
!
I
-r
Ilfll~,p
oo
Ig(Tn')IP:Pr
~ ' I~'1~:~r z) _
n~-O
~ 5PM~.
Hence we can pick An o f modulus 1 in such a way that if we define fl = ~ Angn, it still verifies [If1 [Jr
