On interpolation and sampling in Hilbert spaces of analytic functions.

In this paper we give new proofs of some theorems due to Seip, 
Seip-Wallsten and Lyubarskii-Seip on sequences of interpolation and 
sampling for spaces of analytic functions that are square integrable with 
respect to certain weights. The results are also given in a somewhat more 
general setting.


Introduction
In a series of recent papers Seip [S], Seip-Wallsten [S-W] and Lyubarskii-Seip [L-S] have studied sets of Interpolation and sampling for various spaces of analytic functions of one variable. Part of these results concern Hubert spaces of functions that are square integrable against certain weights, and another, closely related part deals with similar spaces with uniform norms. The methods used in these papers are based on classical-type but intricate constructions of one-variable nature that to some extent go back to Beurling [B].
In [O] Ohsawa has suggested the use of L 2 -techniques for cF to prove results of the above type. In particular, Ohsawa gives a proof of the sufficiency part of the theorem of Seip-Wallsten concerning Interpolation in the space of entire functions in C satisfying As in the approach initiated by Bombieri, Hörmander and Skoda (see [H]), the main difficulty in such a proof is the construction of a (pluri)-subharmonic function with prescribed singularities at the points where one wishes to interpolate. For this Ohsawa uses part of the constructions of Seip-Wallsten, ultimately going back to Beurling, and he poses äs a problem to give a more elementary proof. One purpose of this note is to show how that can be done. As it turns out, the method we use also works for more general weights and therefore also implies the sufficiency part of the theorem by Lyubarskii-Seip. Furthermore, we shall show how the positive direction of the sampling theorem can be obtained in a similar manner, and we shall also permit somewhat more general growth conditions than Lyubarskii-Seip (see Theorem 2). J ) First author supported by the NFR. Second author partially supported by the DGICYT grant PB92-08084-C02-01 and also by the Comissionat per Universitats i Recerca de la Generalität de Catalunya.
In [O] Ohsawa also gives a new proof of the theorem of Seip about Interpolation in Bergman spaces of the disk. This part of Ohsawa's work contains two essential ingredients. The first one is, s in the case of entire space, the construction of a subharmonic function with prescribed singularities. In [O] no details of this construction is given, so in section 3 of this paper we shall show how the construction in entire space can be adapted to the disk case. For this we use a so called invariant convolution in the disk, introduced by Ulrich [U]. The other ingredient in Ohsawa's proof is a generalized Version of H rmander's L 2estimates for 5", inspired by a theorem of Donelly and Fefferman [D-F]. Ohsawa deduces this L 2 -estimate from a more general Version involving vector bundles over Kahler manifolds. Since this terminology probably is not so well known among specialists in one complex variable, we shall also take this opportunity to give a direct proof for the disk (see also [Bei] for a related argument). In section 3 we shall also prove the positive part of the sampling theorem, and show that both the Interpolation and the sampling theorems of Seip [S 2] hold for more general weights.
The theorems of Seip et al. also concern spaces that are defined by other norms than L 2 , notably uniform norms. In section 4 we show that the methods of this paper also give results of this type, even in our somewhat more general setting. The general lines of the proofs are the same s in the L 2 case, with the difference that the L 2 -estimates of H rmander are replaced by similar estimates in uniform norms from [Be2] and [Be3].
One comment is in order. The results of Seip et el. are attractive partly because they are so precise and give necessary and sufficient conditions for Interpolation and sampling. This paper gives different proofs of the positive directions of these theorems, but we have no new ideas about proofs of the converse directions. In particular, we do not know if the conditions in Theorem 2 and 4 are also necessary for sampling and Interpolation, although this seems likely, and perhaps can be proved along the lines of Beurling [B], and Seip [S], [S2].

Interpolation and sampling in C
Let φ be a subharmonic function in C, and let F 2 , be defined by By definition, a sequence Γ = {z,.} c C is sampling for F% if there are constants A and B such that for any h e F% The sequence Γ is called interpolating if for any sequence {c,·} such that we can find an heF£ such that h(Zj) = Cj. Introducing the notation /^ for the space of space of all sequences ξ = {^} such that ||£|| 2 = ^\^\ 2 β~φ (ζ^ < oo we see that Γ is interpolating iff the natural restriction map from F% to /^ is surjective, and sampling iff it is bounded and injective, with closed r nge. In [S] is defined the notion of upper and lower density of a sequence in the following way: D + (F) = limsup sup«(rnA(z,r))/7rr 2 and = liminf inf«(rnA(z,r))/7ir 2 . r-*oo zeC (If E is a set, n (E) denotes the number of elements in E.) Finally a sequence is called uniformly separated if the infimum of the distances between distinct points is strictly positive. The main results of Seip [S] and Seip-Wallsten [S-W]  In Lyubarskii-Seip [L-S], an analogous notion of density depending on an angle is introduced and shown to characterize interpolating and sampling sequences for F% when φ is subharmonic, of class C 2 and positively homogeneous of degree 2 (i.e. φ(ίζ) = t 2 φ (ζ) for t > 0).
Unwinding the definitions of D + and D ~ we see that D + < y if and only if for some δ > 0 and all sufficiently large r and all z it holds that (1) *(ΓηΔ(ζ,Γ))/πΓ 2 <}>-(5 ?
and that D ~~ > y if and only if for some δ > 0 and all sufficiently large r and all z it holds that (2) «(ΓηΔ(ζ,τ))/πΓ 2 >7 + (5. It is not hard to see that both these conditions are equivalent to saying that (1) and (2) hold for some large r. For a general subharmonic function φ we now define a similar notion. Here, s well s in the rest of this paper we let the Laplace operator Δ be defined s Δ = d 2 / 'dz dz, a convention which differs from the Standard one by a factor 4.
Definition. The sequence Γ is dense with respect to Δ φ if for some r < oo and δ > 0 it holds that «(ΓηΔ(ζ, r))/r 2 for all z.
Γ is thin with respect to Δ0 if for some r < oo and δ > 0 it holds that w(rnA(z,r)X/A· 2 < Δ φ (ζ) -δ for all z.
We are now ready to formulate the first version of our main result.
Theorem 1. Suppose φ is subharmonic in C and that Δ φ is uniformly bounded. Then a uniformly separated sequence Γ is In the light of what we just said it is clear that when φ = α|ζ| 2 this is just a rephrasing of the sufficiency part of the theorem of Seip-Wallsten. Moreover, one can check that when φ is positively homogeneous of degree 2 we get the result of Lyubarskii-Seip (note that such a function always has a uniformly bounded laplacian if it is of class C 2 ).
Theorem l has s a consequence the following, perhaps more natural, theorem.
Theorem 2. Suppose φ is subharmonic in C and that Δ φ is uniformly bounded. Then a uniformly separated sequence Γ is (a) interpolating for F£ if for some r < oo and δ > 0 it holds that « (Γη Δ (z, r))/ r 2 < -^-2 f Δ<£(0-<5 for all z nr K-z|<r and (b) sampling for F£ iffor some r < oo and δ > 0 it holds that Assume that the condition in Theorem 2 (a) (or (b)) holds for a certain value of r. Let l and be the averages of φ over disks with radius r. φ τ is again subharmonic, and the conditions mean precisely that Γ is thin (or dense) with respect to Δ φ,. From Theorem l we conclude that Γ is interpolating (sampling) for F£. But it is easily seen that, if the laplacian of φ is uniformly bounded, then φ -φ,= O (r 2 ). Since r is a fixed number, this proves Theorem 2, given Theorem l .
To prepare for the proof of Theorem l let v = £ 2j be the measure consisting of a pointmass at each point in our sequence, which we assume from now on is uniformly separated. Let E = l /π log | z | 2 be the fundamental solution of the Laplace operator (with our convention ΔΕ = <5 0 ). We now define an auxiliary function This function is certainly well defined if Γ is finite. Notice that the value of v at z then depends only on the points in Γ with \Zj -z\<r 9 since for the other points the two terms in the definition of v cancel by the mean value property for harmonic functions. Therefore we can define v for arbitrary sequences by a limiting procedure. An alternative way to define v is to first let "r = (Xr -<*o) * E -This function satisfies (and of course is characterized by) the properties Aw r = 1/Tir 2 -(5 0 in \z\ < r, and u r = du r /dn = 0 on |z| = r. Moreover u r = 0 in \z\ > r. Explicitly, u r is given by if |z| < r and u r = 0 otherwise.
In particular u r has compact support so we can define v by v =u r * v .
Since E is subharmonic it follows from the submeanvalue property that u r ^ 0 so we always have v ^ 0. Of course it holds that Assume now that Γ is thin with respect to Δ φ. This means precisely that for some large enough r and some positive δ nv * Xr in Aj. We next combine the /)'s using a partition of unity.
Let#e C c°°( C)be suchthatg = l for |z| < ε 0 /2,# = Ofor \z\ > ε 0 and \dg\ ^ C eo . Put Then f(Zj) = c j so / interpolates the right values. Finally we shall modify / to get a holomorphic interpolating function by solving a ^-equation. Note first that We then apply H rmanders J-theorem [H], which implies that we can find a solution U to fiU = fif satisfying Since Δ φ ^ δ and fif vanishes when zeA(z J5 ε 0 /2) for some z j the right hand side is dominated by some constant times Consequently Moreover U(Zj) = 0 for each Zj since e~v ~ i/\z -Zj\ 2 near z,·. Let

h=f-U .
Then H(ZJ) = c jy and J|A| 2 e~*<oo since both/ and U satisfy this estimate. The proof of Theorem l (a) is therefore complete.
We now turn to the sampling part of Theorem l . For a moment, let v be an arbitrary positive measure on C. Assume v * χ Γ ^ C and that πν * Xr > Δ</> + δ which is the analog of the density condition for general measures. Put v = ( v -v * χ Γ ) * E * and φ = (πν + φ) .
As before ψ ^ φ but this time we have Δφ ^ v -<5 .
-+ 00 -> D By the differential inequality for (7^ 0 so it follows that (3) This is already an inequality of sampling type. We would like to choose s before v = ]T j but we cannot do that directly since ψ would then be identically equal to -oo on the support of v, so the inequality would be of no value.
Instead we shall take v to be a smoothed Version of a sum of Dirac measures. Let Γ be a sequence which is dense with respect to Δ</>. Let v be defined s where 0 < i < l (we make no distinction between an absolutely continuous measure and its density with respect to Lebesgue measure). Since Γ is dense we can choose t so close to l that for some large r. Then ψ = ην + φ will satisfy the estimates Summing over j we find Choosing ε small enough we can absorb the second term on the right in the left hand side. This proves that Γ satisfies the left of the inequalities in the sampling condition. The other inequality is trivial since we have assumed that Γ is separated, so Theorem l (b) is now completely proved.

Interpolation and sampling in D
Let φ be a subharmonic function in D, and let JF^2(B) be defined by The main results of [S 2] concerning weighted Bergman spaces in the disk are the following:
Similarly, if μ is a measure on the disk and g is a function, we define We will need the following easy properties of the invariant convolution. If /, g, h are measurable functions on the disk, and morover h is radial, then (/»*) = (**/) and (/* )*g=/*(A*g).
The first property is just a change of variables and the second holds since For any 1/2 < r < l, we define^! ifi/2<|C|<r, where c r is chosen in such a way that Hi r || L i (d ) = 1. Observe that c r /log-> l s r -»l, thus D + (Γ) < γ if and only if there exists some δ > 0 such that for all z e D and for all r < l large enough it holds where v (z) = π £ (l -l^l 2 ) 2^^) · Analogously, D~(F) > γ if and only if there exists zier some δ > 0 such that for all r < l large enough it holds We will define now a corresponding notion for arbitrary subharmonic functions. We define the invariant Laplace operator s = (l -\z\ 2 ) 2 d 2 jdzdz.
Definition. The sequence Γ is dense with respect to Αφ if for some r < l and δ > 0 it holds that for all z.
Γ is thin with respect to Αφ if for some r < l and δ > 0 it holds that for all z, where s before v (z) = π £ (l -|z i | 2 ) 2 (5 2 .(z) .
Then the following theorem holds.
We will need s before an auxiliary function Observe that the value of v (z) is determined by the points in Γ such that ρ (z, Zj) < r, since (/* £ r ) =/when / is harmonic. Moreover, since ξ, is radial, we have that That follows from the fact that v * E is subharmonic and ξ τ is radial with H^rll L i (dA ) = l· Now define ψ = φ + Ό.
Since v ^ 0, we have ψ (ζ) ^ φ (z). Moreover, it follows from the definition that -r 2 t?-log for all z such that Q(Z,ZJ) < ε 0 , if ε 0 is chosen so that Q(z j ,z k ) > 2ε 0 for all j Φ k. We are ready to solve the Interpolation problem now. Let {cj} be a sequence such that For eachy* let Gj be a holomorphic function such that GJ(ZJ) = 0 and if Q(z 9 Zj) ^ 2ε 0 . Such a function G,· exists since Αφ is uniformly bounded. To see this, we can first'apply an automorphism of the disk sending Zj to 0, and then the same proof s in section 2 works. Take #eC c°° such that g = l for |ζ|<ε 0 /2, g = 0 for |ζ|>ε 0 and This function interpolates the values Cj in the points z,·, and moreover We finally have to modify the function / to get a holomorphic function. We need the following variant, due to Ohsawa, of H rmander's theorem.
Theorem E (Ohsawa). Lei ψ be any subharmonic function in the disk such that Δ φ > δ > 0. Then there is a solution U to the equation 3 U = g such that If the sequence z,· satisfies the hypothesis in Theorem 4 (a), Thus, by the above 5-theorem, there is a solution to 5 U = 3/ such that Since 3/vanishes when Q(Z,ZJ) < ε 0 it follows that and Consequently, since we find Moreover t/(z,) = 0 since ^ ψ · l -zf, z·-z. near z^. If we then take we get that h(zj) = c j9 h e H (D) and < +00 .
Hence the Interpolation part of theorem 3 is proved.
The hypothesis on v from Theorem 4 (b) gives that and moreover it holds that Furthermore (6) and one may also verify that (7) |t; for any z such that ρ (z, z^) < ε.
Let Ae/^2 (B), and put t/= |A| 2 e~v. Since log|A| is subharmonic it follows that and therefore, Take a family of cut-off functions g r such that g r ^ 0, g = l if |z| < r, g = 0 if and such that g r is uniformly bounded by a constant independent of r. Then, by (6), and lim f (l -\z\ 2 )g r (z)&UdX(z) = lim J Δ((1 -|z| 2 )g r (z)) Udl(z) ^ 0 .
r-* l 0) r-+l p Therefore lim J (l -|z| 2 )g r (z) υ&ψάλ ^ 0 . -Thus, Taking into account (7), we obtain^ i \h\ 2 e-«dm(z) ^ J Consider ΗοΙοιηοφΜο functions G^ defined for all z such that ρ (ζ, z,·) ^1/2 with the properties: Gj(zj) = 0 and for all z such that ρ (ζ, ζ ; ·) ^ l / 2. As before, the existence of such functions follows since we have assumed Αφ is bounded. Fix j and call gj = he~G j , then ρ(ζ,ζ </ ·)<ε j ρ(ζ,ζ>)<ε -j Clearly, since m({z 9 Q(z,Zj) ^ ε}) ^ Ce 2 (l -|z,.| 2 ) 2 it follows that Summing over j we find If we take ε small enough, we may absorb the second term on the right in the left band side, and we obtain the left inequality in the sampling condition. The other inequality is clear since Γ is a separated sequence.

The non-Hilbert case
The Interpolation theorems we have been discussing also have natural analogs in spaces that are not defined by L 2 -norms (see [Sl], [S 2] and [S-W]). We shall now briefly indicate how the methods of the previous sections can be adapted to uniform norms.
First we treat the case of entire space. Let, s in section 2, φ be a subharmonic function with uniformly bounded laplacian in C, and put ΒΡ φ = {/eJy(C); sup \f\ 2 e'+£ C] .
A subsequence Γ of C is interpolating for ΒΡ φ if for any sequence {q} such that there is a function / in ΒΡφ such that /(z,·) = c jf We then have the following theorem.
Theorem 5. IfT is a separated sequence which satisfies the hypothesis of Theorem 2 (a), then Γ is interpolating for ΒΡ φ .
As in section 2 we first solve the Interpolation problem locally. We then find functions fj that are holomorphic in Δ,· = Δ (ζ,·, ε 0 ) satisfying fj(Zj) = c j and in Aj. Then put where g is the same cut-off function s in section 2. gain, we need to solve a S-equation, but this time we want to estimate the solution in uniform norm. Let tp(z)= sup |ζ-ζ|<«ο/ We apply the following theorem from [Be3]: Theorem F. Lei ψ be a plurisubharmonic function in C" such that id^ip > δ > 0. Then iff is a Ή-closed (0, \)-form in C n and u is t he solution to?)u = fwhich is of minimal norm in L 2 (C",<r v ), usatisfies This theorem has s a consequence a more precise Statement that we will need. Let h be any harmonic function in C and write h = 9l/f where H is entire. Then v = ue~H I2 is the canonical solution to 'S v = e~H /2 fm L 2 (e~t p + h ). Applying Theorem F to this Situation instead we see that ψ (z) can be replaced by where h is any harmonic function in C. It is even enough to assume that h is harmonic in Δ ζ =:Δ(ζ,ε 0 /2), since any such h can be approximated by functions that are globally defined. Choosing h to be the harmonic extension oft/; from <9Δ 2 to the interior of Δ ζ , we see that φ (z) can be replaced by ψ (z) -G (z) where G is the Green's potential of Δ ψ over Δ ζ . In particular, if the laplacian of ψ is uniformly bounded, the Green's potential will be bounded, so Theorem F holds with ψ replaced by φ.
In our case where ψ -φ H-υ one sees in a similar way that we can replace ψ by φ in (*).
Let us now for a moment suppose that our sequence Γ is finite. Then 3/ lies in L 2 so Theorem F applies and we see that the canonical solution to Ή u = ^/satisfies Moreover the canonical solution also satisfies the L 2 estimate from section 2, so it follows that u(Zj) = 0. Letting h =/-u we get an interpolating function in ΒΡ φ . Since the norm does not depend on the number of points in the sequence, we can also permit infinite sequences by a normal family argument.
We can also treat the disk case in a similar manner. Let φ be subharmonic in the disk, and suppose that the invariant laplacian of φ is uniformly bounded. Put ΒΡ φ (0) = {/etf(D); sup |/(z)| 2 e"* ^ C} . zeO A subsequence Γ of the disk is interpolating for ΒΡ φ if for any sequence {Cj} such that there is a function / in ΒΡ φ such that f(Zj) = Cj. We then have Theorem 6. Let Γ be a sequence in the disk which satisfies the hypothesis of Theorem 4 (a). Then Γ is interpolating for ΒΡ φ (Ώ).
The proof again follows the same pattern s in the L 2 case; the only difference being that we need to replace the L 2~e stimates for Ή by uniform estimates. In [Be3] there is also a theorem analogous to Theorem F for the case of the disk (or ball in C") but it requires that Αφ > 4. In one dimension one can avoid this restriction by instead appealing to the following theorem from [Be2].

Theorem G. Lei ψ be a subharmonic function and
Lei f be a function in D such that sup -e~* l2 < + 00. In our construction from section 3 it holds (after subtraction of a harmonic function like in the case of entire space), that ψ -φ is uniformly bounded, and moreover the corresponding η will be of size 1/(1 -|z| 2 ). Therefore Theorem 6 follows in the same way s Theorem 5.
Finally, it may be worth remarking that since we use the canonical solution of the 3equation, we can interpolate between L 2 and L 00 and get similar results in L p for p between 2 and infinity.

Appendix. A proof of Theorem E
The one dimensional case of H rmander's theorem that we used in section 2 says that if φ is subharmonic in a domain in C we can solve the equation 5w = g with the estimate (8) in Ω.
That Ή u = g in the sense of distributions means that if α is any function in C c°°( ) it holds that Taking the supremum of the norm of the right band side over all g such that J --e φ ^ l, we get from (8) da (9) |Δφ|α| 2 £ φ^ fdz for all α e C c°°( ). Conversely (9) implies H rmander's theorem, and the usu l proof of (8) consists in establishing (9) using Integration by parts.
A Standard functional analysis argument then shows that for a given g we may find u such that forallyeC c°° ( B), and This completes the proof of Theorem E.