ON SEIP S DESCRIPTION OF SAMPLING SEQUENCES FOR BERGMAN SPACES. Alexander P. Schuster

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1 ON SEIP S DESCRIPTION OF SAMPLING SEQUENCES FOR BERGMAN SPACES Alexader P. Schuster Abstract. I his 993 paper [], Kristia Seip characterizes samplig ad iterpolatio sequeces for the space A ad gives a outlie of the proof of the correspodig theorem for the Bergma space A 2. Usig his techiques, we provide a complete proof of the result cocerig samplig sequeces ot oly for A 2, but also for A p, p <.. Itroductio For < p <, the Bergma space A p is the set of fuctios f aalytic i the uit disk D = {z : z < } with f p p = f(z) p da(z) <, π where da deotes Lebesgue area measure. The space obtaied whe p = 2, A 2, is a Hilbert space with ier product f, g = π D D f(z)g(z)da(z) ad reproducig kerel k ζ (z) = ( ζz) 2 at ζ D. That is, f, k ζ = f(ζ) for all f A 2. We say that a sequece Γ = {γ } of distict poits i the disk is a samplig sequece for A p if there exist positive costats K ad K 2 such that K f p p ( γ 2 ) 2 f(γ ) p K 2 f p p () for all f A p. We will deote by f Γ p the p-th root of the sum i the above equatio. The sequece Γ is a uiqueess sequece for A p if the oly elemet of A p vaishig alog Γ is the zero fuctio. Otherwise, we say that Γ is a o-uiqueess sequece. It is easy to see from the defiitio that every A p samplig sequece is a uiqueess sequece for A p. 99 Mathematics Subject Classificatio. 3H5, 46E5. Key words ad phrases. Bergma space, samplig. The cotet of this paper forms a part of the author s doctoral dissertatio at the Uiversity of Michiga, writte uder the directio of Professor Peter Dure, whose help is greatly appreciated. Typeset by AMS-TEX

2 I a geeral Hilbert space X, Parseval s equatio tells us that for a orthoormal basis {u α } of X, x 2 = x, u α 2 α for every x X. The family {v α } of liearly idepedet vectors i X is said to be a frame for X if there are positive costats A ad B such that A x 2 x, v α 2 B x 2 (2) α for all x X. I the Bergma space the most atural buildig blocks are the kerel fuctios, as they correspod to poit evaluatio fuctioals. Sice they are ot mutually orthogoal, they will ever form a orthoormal basis for A 2, but they ca make up a frame. Comparig () ad (2), we see that Γ is a samplig sequece for A 2 precisely whe the set of ormalized reproducig kerels {k γ / k γ 2 } is a frame for A 2. Because of the requiremet i () that two iequalities eed to be satisfied simultaeously, the mere existece of samplig sequeces is ot immediate. I fact, a easy argumet shows that the aalogous coditio i the Hardy space H 2, i.e. that the ormalized Szegö kerels form a frame for H 2, ca ever hold, so that samplig sequeces do ot exist i that cotext. They do, however, exist i the Bergma space ad are characterized by Seip [] for p = 2. More precisely, he describes samplig sequeces for the growth space A (see Sectio 6 for defiitios) ad provides a sketch of the proof for A 2. I this article we exted the techiques of [] to give a complete proof of the case p <. 2. Statemet of the theorem The sequece Γ = {γ } is said to be uiformly discrete if δ(γ) = if m ρ(γ, γ m ) >, where ρ(z, ζ) = (ζ z)/( ζz) is the equatio defiig the pseudohyperbolic metric. The disk of cetre ζ ad radius r i this metric will be deoted by (ζ, r). For < s <, let (Γ, ζ, s) be the umber of poits of Γ lyig i (ζ, s). Defie, for Γ uiformly discrete, r (Γ, ζ, s)ds E(Γ, ζ, r) = 2 r a( (ζ, s))ds, where a(ω) = π Ω ( z 2 ) 2 da(z) is the hyperbolic area of a measurable subset Ω of the disk. The lower uiform desity of Γ is defied to be D (Γ) = lim if r if E(Γ, ζ, r). ζ D The followig result was proved i [] for A 2 ad stated for A p, < p <, i [4]. We remark that the sufficiecy part has bee proved for p = 2 i [] usig differet methods. 2

3 Theorem A. A sequece Γ of distict poits i the disk is a samplig sequece for A p if ad oly if it is a fiite uio of uiformly discrete sequeces ad it cotais a uiformly discrete subsequece Γ for which D (Γ ) > p. The iterpolatio problem is related, ad i some sese dual, to the samplig questio. Here we cosider agai a sequece Γ of distict poits i the uit disk, ad we say that Γ is a iterpolatio sequece for A p if the iterpolatio problem f(γ ) = a has a solutio i A p wheever ( γ 2 ) 2 a p <. I [] Seip characterizes iterpolatio sequeces for A ad oce agai gives a sketch of the proof for the Hilbert space A 2 usig techiques which may be exteded to A p, < p <. Theorem B. A sequece Γ of distict poits i the disk is a iterpolatio sequece for A p if ad oly if it is uiformly discrete ad D + (Γ) = lim sup r sup E(Γ, ζ, r) < p. ζ D We oly state Theorem B here for the sake of compariso with Theorem A, but we refer the reader to the author s thesis [7] for a complete ad detailed treatmet. Jevtic, Massaeda ad Thomas [5] also provide some of the details of the proof of Theorem B. For alterate characterizatios of iterpolatio sequeces for the Bergma space, see [8] ad [9]. 3. Prelimiary results Samplig sequeces are ivariat uder Möbius trasformatios of the disk, ad this ivariace will play a crucial role i the proof of Theorem A. We ext list some basic properties of Möbius trasformatios, which will be used repeatedly throughout this article. We will deote by φ ζ the aalytic automorphism of the disk that iterchages ζ ad the origi, that is, φ ζ (z) = (ζ z)/( ζz). Simple algebraic maipulatios yield φ ζ (ζ) =, φ ζ (φ ζ (z)) = z ad From the idetity we obtai φ ζ (z) 2 = ( ζ 2 )( z 2 ) ζz 2. (3) ( φ ζ (z)φ ζ (w)) 2 φ ζ(w)φ ζ (z) = ( zw) 2, φ ζ(z) = φ ζ(z) 2 z 2. (4) The relatioship betwee ρ ad the hyperbolic metric h ca be expressed by the equatio h(z, ζ) = log ( ) + ρ(z, ζ), ρ(z, ζ) 3

4 ad so ρ satisfies the triagle iequality which implies that ρ(z, ζ) ρ(z, w) + ρ(w, ζ) + ρ(z, w)ρ(w, ζ), ( (z, ɛ), z + ɛ ) + ɛ z for z D ad ɛ >. Oe applicatio of this cotaimet relatio is the ext result, which shows that the coutig fuctio associated to a uiformly discrete sequece caot grow too quickly. Lemma 3.. Let Γ be a uiformly discrete sequece of poits i D. There is a costat C, depedig oly o δ(γ), such that for all r <. (Γ,, r) C r Proof. Lettig ɛ = ( δ/2, we ) ca, by (5), fid pairwise disjoit disks (γ, ɛ), whose uio is cotaied i, r+ɛ +rɛ. The (5) (Γ,, r) a( (, r+ɛ +rɛ )) a( (, ɛ)) ( + ɛ)2 ɛ 2 r. This lemma implies immediately that D (Γ) <. I combiatio with the ext result, it also allows us to express the lower uiform desity i a form more ameable to certai types of calculatios. Lemma 3.2. Let Γ be uiformly discrete ad a R. There is a costat C = C(p, a, δ(γ)), such that ( γ 2 ) a f(γ ) p C f(z) p ( z 2 ) a 2 da(z) γ Ω ρ(z,ω)<δ for all measurable subsets Ω of D ad all f aalytic i D. Proof. As before, we let ɛ = δ(γ)/2, so that the disks (γ, ɛ) are pairwise disjoit. By subharmoicity, there is a costat C such that f() p C f(z) p ( z 2 ) a 2 da(z). (,ɛ) Applyig this to f φ γ, chagig variables ad by (4), we obtai f(γ ) p C f(z) p ( φ γ (z) 2 ) a ( z 2 ) 2 da(z). (γ,ɛ) 4

5 We replace f(z) by f(z)( γ z) 2a/p ad use (3) to arrive at f(γ ) p ( γ 2 ) 2a C whece (γ,ɛ) = C( γ 2 ) a f(γ ) p ( γ 2 ) a C f(z) p γ z 2a ( φ γ (z) 2 ) a ( z 2 ) 2 da(z) (γ,ɛ) (γ,ɛ) Summig over γ i Ω, we obtai the desired result. f(z) p ( z 2 ) a 2 da(z), f(z) p ( z 2 ) a 2 da(z). (6) We remark that Lemma 3.2 implies i particular that every uiformly discrete sequece has the property that ( γ ) +ɛ < for ɛ >. (7) We ow examie the lower uiform desity defied i Sectio 2. A easy calculatio ivolvig (4) shows that r r 2 a( (ζ, s))ds = 2 a( (, s))ds = log + r r 2r. We the use itegratio by parts to show that r (Γ, ζ, s)ds = r (φ ζ (Γ),, s)ds = φ ζ (γ ) <r = (r )(φ ζ (Γ),, r) + φ ζ (γ ) <r (r φ ζ (γ ) ) ( φ ζ (γ ) ). If Γ is uiformly discrete, we are permitted to apply Lemma 3., the iequalities x log x x + ( x)2 for 2 x <, ad (7) to see that where D (Γ) = lim if r D(Γ, ζ, r) = if D(Γ, ζ, r), ζ D /2< φ ζ (γ ) <r log φ ζ (γ ) log( r ). Aother cosequece of Lemma 3.2 is the followig result, which gives a ecessary ad sufficiet coditio o Γ for the upper samplig iequality to hold. 5

6 Lemma 3.3. There is a costat C such that ( γ 2 ) 2 f(γ ) p C f p p (8) for all f A p if ad oly if Γ ca be expressed as a fiite uio of uiformly discrete sequeces. Proof. We ote first that (8) holds precisely whe the measure ( γ 2 ) 2 δ γ is a Carleso measure for A p. Sice Carleso measures are kow to be idepedet of p (see [3] or [6], for example), it suffices to prove the result for p = 2. Suppose ow that (8) is satisfied. If Γ is ot a fiite uio of uiformly discrete sequeces, the there is a sequece of poits {ζ m } i D such that (Γ, ζ m, /2) is ubouded as m becomes large. Lettig f m (z) = k ζm (z)/ k ζm 2 = ( ζ m 2 )( ζ m z) 2, we see by (3) that ( γ 2 ) 2 f m (γ ) 2 = γ (ζ m,/2) γ (ζ m,/2) ( ζ m 2 ) 2 ( γ 2 ) 2 ζ m γ 4 ( ρ(ζ m, γ ) 2 ) (Γ, ζ m, /2), which approaches ifiity, cotradictig (8) ad the fact that f m 2 =. The fact that (8) is satisfied whe Γ is a fiite uio of uiformly discrete sequeces follows immediately from Lemma 3.2. For a give sequece Γ, the largest umber K for which the lower iequality i () holds will be deoted by K (Γ). Lemma 3.3 thus shows that a uiformly discrete sequece Γ is a samplig sequece if K (Γ) >. A chage of variables argumet ad (4) ca be used to prove that the map f(z) f(φ ζ (z))(φ ζ (z))2/p is a isometry from A p to itself. It is ow straightforward to show that K (φ ζ (Γ)) = K (Γ). 4. Perturbed samplig sequeces We would like to kow that samplig sequeces retai their properties uder sufficietly small perturbatios. The atural topology o sequeces i D is defied by the Fréchet distace with respect to the pseudo-hyperbolic metric. If A is a closed set i D ad t, the A t = {z D : ρ(z, a) t for some a A}. For A ad B closed, the Fréchet distace betwee A ad B is [A, B] = if{t : A B t ad B A t }. The mai goal of this sectio is to prove the followig result: 6

7 Lemma 4.. Let Γ be uiformly discrete. There are costats δ ad C, depedig oly o δ(γ) ad p, such that for every sequece Λ with [Γ, Λ] < δ. K (Γ) p K (Λ) p C [Γ, Λ] This will follow from the uiform cotiuity with respect to the pseudo-hyperbolic metric ejoyed by Bergma space fuctios. Lemma 4.2. Let ɛ >. There are costats C ad δ, depedig oly o p ad ɛ, such that give z, ζ D with ρ(z, ζ) δ, f(z) ( z 2 ) 2 p for all f A p. Proof. Assume ρ(z, ζ) < ɛ/4. Let ( ) p f(ζ) ( ζ 2 ) 2 p C ρ(z, ζ) f(w) p da(w) (z,ɛ) S(w) = ( w 2 ) 2 p f(w) ad Sζ (w) = ( w 2 ) 2 p fζ (w), where f ζ (w) = f(φ ζ (w))(φ ζ (w)) 2 p. Usig (4), oe ca check without difficulty that S ζ (φ ζ (z)) = S(z) ad S ζ () = S(ζ). Therefore, S(z) S(ζ) = Sζ (φ ζ (z)) S ζ () Sζ (φ ζ (z)) S ζ () = ( φ ζ (z) 2 ) 2 p fζ (φ ζ (z)) f ζ () ( φ ζ (z) 2 ) 2 p fζ (φ ζ (z)) f ζ (φ ζ (z)) + f ζ (φ ζ (z)) f ζ () = ( ( φ ζ (z) 2 ) 2 p ) fζ (φ ζ (z)) + f ζ (φ ζ (z)) f ζ () = Q + Q 2. The, by (4) ad (6) with a = 2, Q C ( ( φ ζ(z) 2 ) 2 p ) ( φ ζ (z) 2 ) 2 p ( (z,ɛ) f(w) p da(w) We write f ζ (φ ζ (z)) f ζ () = φ ζ (z) f ζ (w)dw, apply Cauchy s itegral formula to f ζ (w) ad use stadard estimates to obtai where s < mi{ φ ζ (z), ɛ/4}). Q 2 φ ζ(z) s sup f ζ (w), w φ ζ (z) +s ) p. 7

8 We ext apply (6) to f ζ, ad sice (φ ζ (w), ɛ/4) (z, ɛ), arrive at f ζ (w) p C( w 2 ) 2 f(u) p da(u), which implies that ad so (z,ɛ) sup f ζ (w) C ( ( φ ζ (z) + s) 2) 2 p w φ ζ (z) +s ( φ ζ (z) Q 2 C s ( ( φ ζ (z) + s) 2) 2 p (z,ɛ) ( (z,ɛ) f(u) p da(u) f(u) p da(u) Sice we could have take ay value of s satisfyig < s < φ ζ (z) ad s < ɛ/4, ad addig the estimates for Q ad Q 2, we have ) p ( ) p S(z) S(ζ) Cu(ρ(z, ζ)) f(w) p da(w), (z,ɛ). ) p, where u(r) = mi <s< r <s<ɛ/4 r s( (r + s) 2 ) 2 p Sice u(r) = O(r) as r, we obtai the desired result. + ( r2 ) 2 p ( r 2 ) 2 p Proof of Lemma 4.. We may assume that the elemets of Λ = {λ } are ordered so that ρ(γ, λ ) [Γ, Λ] for each. Let ɛ = δ(γ)/2. By Mikowski s iequality ad Lemma 4.2, f Γ p f Λ p { ( γ 2 ) 2/p f(γ ) ( λ 2 ) 2/p f(λ ) { C [Γ, Λ] (γ,ɛ). } /p p f(z) p da(z)} /p C [Γ, Λ] f p. Let ow γ >. By the defiitio of K (Γ), there is a fuctio f i A p such that f Γ p < (K (Γ) p + γ) f p. We may assume, without loss of geerality, that f p =, so that f Γ p < K (Γ) p + γ ad K (Λ) p f Λ p. We combie these two iequalities to see that K (Λ) p K (Γ) p C [Γ, Λ] + γ. We iterchage the roles of Γ ad Λ to obtai the desired result. Oe applicatio of Lemma 4. is that it allows for the assumptio that samplig sequeces are uiformly discrete. Namely, if Γ is a samplig sequece for A p, the there is a uiformly discrete subsequece Λ which is also a samplig sequece for A p. 8

9 I the proof of the ecessity part of Theorem A, we will apply Lemma 4. to sequeces which are shifted radially outward. To that ed, we defie for η > ad z, g η (z) = z + η z + η z z. It is easy to see that [Γ, g η (Γ)] η. Moreover, there is a costat C = C(δ) such that ( ) D(g η (Γ),, r) ( η)d(γ,, r) + C log( r ) () for all r < ad η /4. Ideed, by Lemma 3., log( r )D(g η(γ),, r) = /2< γ <r /2< γ <r ( η) log log /2< γ <r from which () follows readily. /2< g η (γ ) <r log g η (γ ) + g η (γ ) /4 γ </2 log + log(4)(γ,, /2) g η (γ ) g η (γ ) log γ + C = ( η) log( )D(Γ,, r) + C, r 5. The proof of the ecessity part of Theorem A Suppose Γ is a samplig sequece for A p. By the remarks i the previous sectio, we may assume that Γ is uiformly discrete. Let α = D (Γ) ad suppose ɛ j. By the expressio for D (Γ), there exist poits w j i the disk such that D(φ wj (Γ),, r j ) = D(Γ, w j, r j ) α + ɛ j, () where r j > ɛ j. We assume, without loss of geerality, that oe of the w j s lie i Γ. Let Γ j = φ wj (Γ) ad choose η < mi{k (Γ) p /C, δ, /4}, where C ad δ are as i Lemma 4.. The K (g η (Γ j )) p K (Γ j ) p C [Γ j, g η (Γ j )] K (Γ) p C η >, so that g η (Γ j ) is a samplig sequece for A p. Writig g η (Γ j ) = {γ (j) }, we defie for each j, f j (z) = γ (j) <r j 9 γ (j) γ (j) z γ (j) z.

10 The, by Lemma 3. ad (), f j g η (Γ j ) p p = C w r j w r j w r j = C C w r j w r j ( w 2 ) 2 γ (j) <r j ( w 2 ) 2 γ (j) <r j ( w 2 ) 2 γ (j) p γ (j) p /2< γ (j) <r j γ (j) γ (j) γ (j) p w w ( w 2 ) 2 exp(p log( )D(g η (Γ j ),, r j )) r j ( w 2 ) 2 exp(p log( )( η)d(γ j,, r j )). r j Applyig Lemma 3.2 to p = a = 2, f ad Ω = {z : r j < z < }, yields Therefore, w r j ( w 2 ) 2 C( r j ). < (K (Γ) p C η) p K (g η (Γ j )) Sice ɛ j, we must have α > /p. ( w 2 ) 2 f j (w) p / f j p p ( w 2 ) 2 f j (w) p / f j () p = f j g η (Γ j ) p p C( r j ) exp(p log( )( η)d(γ j,, r j )) r j C( r j ) ( p( η)(α+ɛ j)). 6. Samplig i A The proof of the sufficiecy part of Theorem A will deped o the characterizatio of samplig sequeces for the closely related growth space A, which cosists of those aalytic fuctios f such that f = sup( z 2 ) f(z) <. z D p

11 The sequece Γ = {γ m } of distict poits i D is a samplig sequece for A if there is a positive costat L such that f L sup( γ m 2 ) f(γ m ) (2) m for all f A. As i A p, we let L(Γ) be the smallest costat L for which (2) holds. The map f(z) f(φ ζ (z))(φ ζ(z)) is a isometry i A for every ζ D, which allows us to show that L(φ ζ (Γ)) = L(Γ). I [], Seip proves the followig result: Theorem 6.. Γ is a samplig sequece for A if ad oly if there is a uiformly discrete subsequece Γ of Γ such that D (Γ ) >. The proof of the ecessity part of this theorem follows alog the lies of the proof of the ecessity part of Theorem A. For details, see []. We tur ow to the coverse, which will be used i the proof of the sufficiecy part of Theorem A ad is icluded here for completeess. Here we require the cocept of weak covergece itroduced by Beurlig [2]. We say that a sequece A of closed sets coverges weakly to A (A A) if [(A K) K, (A K) K] for every compact set K D. We deote by W (Γ) the collectio of sequeces Λ such that φ ζ (Γ) Λ for some sequece of Möbius trasformatios {φ ζ }. It is ot difficult to show that if Γ is uiformly discrete, the every sequece of elemets of W (Γ) has a subsequece that coverges to a member of W (Γ). The followig lemma is the aalogue of Theorem 3 of [2] (p. 345). Lemma 6.2. Let Γ be a uiformly discrete sequece i D. If every Λ W (Γ) is a uiqueess sequece for A, the Γ is a samplig sequece for A. Proof. Assume, without loss of geerality, that Γ does ot cotai the origi. If Γ is ot a samplig sequece, the there is a sequece {f k } of elemets of uit orm i A such that sup m ( γ m 2 ) f k (γ m ) becomes arbitrarily small as k approaches ifiity. We may therefore choose, for each k, a z k D such that ( z k 2 ) f k (z k ) = 2. Cosider Γ k = φ zk (Γ) ad assume that Γ k Λ. (Here Λ may be empty.) Defie g k (z) = f k (φ zk (z))(φ z k (z)). Sice g k = f k =, we use a ormal family argumet to show the existece of a g A which is the locally uiform limit of a subsequece of {g k }. Sice g k () = f k (z k ) ( z k 2 ) = /2, it follows that g is ot the zero elemet. To show that g vaishes o Λ, we observe that sup( γ m 2 ) g k (γ m ) sup( γ m 2 ) f k (γ m ), m m

12 which approaches as k becomes large. A applicatio of the triagle iequality ad the A versio of Lemma 4.2 (Lemma 2. i []) yields the desired result. Thus Λ W (Γ) is a o-uiqueess sequece for A, ad we have our cotradictio. We ow complete the proof of the sufficiecy part of Theorem 6.. Suppose that a uiformly discrete sequece Γ satisfies D (Γ) = α > ad that there is a Λ = {λ } W (Γ) which admits a o-trivial fuctio f i A that vaishes o Λ. Sice D (Λ) D (Γ), we have D(Λ,, r) + ɛ = α ɛ, (3) where ɛ = (α )/2 ad r is sufficietly close to. We assume, without loss of geerality, that f() = ad apply Jese s formula to obtai 2π 2π log f(re iθ ) dθ = λ <r log /2< λ <r r λ /2< λ <r log λ + C log r r log r = log D(Λ,, r) + C r r ( + ɛ) log r + C log r r. Here we have used Lemma 3. ad (3). This implies that sup log f(re iθ ) ( + ɛ) log θ r + C log r r, log + (Λ,, r) log r λ which cotradicts the fact that f A ad so, by Lemma 5.2, Γ must be samplig for A. 7. Proof of the sufficiecy part of Theorem A Suppose we have a uiformly discrete sequece Γ satisfyig D (Γ) = α > p. Theorem 6., Γ is a samplig sequece for A ( p +ɛ), where ɛ > is chose so small that α > /p + ɛ. Thus, Γ is also samplig for A ( p +ɛ), where A is the closure of the polyomials i A. Here samplig sequeces are defied by the same iequality (2), which ow eed hold oly for f A. We obtai the followig iterpolatio formula: Lemma 7.. There is a costat C, such that give ζ D, s R ad f A ( p +ɛ), there is a sequece {h (ζ)} such that By ( ζ 2 ) p +ɛ f(ζ) = ( ) ( γ 2 ) p ζ +ɛ 2 s f(γ )h (ζ), (4) ζγ 2

13 where ad h (ζ) C (5) h (ζ) C ( ζ 2 )( γ 2 ) γ ζ 2. (6) It suffices for our purposes to let s =, but we give the result i full geerality, sice it is of idepedet iterest. Proof. Defie T : A ( p +ɛ) c by f {( γ 2 ) p +ɛ f(γ )}, where c cosists of sequeces covergig to. It is clear that T is a bouded liear trasformatio, ad the fact that Γ is a samplig sequece for A ( p +ɛ) implies that T is bouded below. Thus a = T (A ( p +ɛ) ) is closed ad T maps a boudedly oto A ( p +ɛ). For ζ D, defie ψ ζ to be the liear fuctioal f ( ζ 2 ) p +ɛ f(ζ) for f A ( p +ɛ). It geerates a liear map ψ ζ which is defied by the equatio ψ ζ ({w }) = ψ ζ (T ({w })) for all {w } i a. The boudedess of ψ ζ the follows from the boudedess of ψ ζ ad T. I fact, oe shows that its orm is at most T, sice the orm of ψ ζ is less tha or equal to. Usig the Hah-Baach theorem, we ca exted ψ ζ to a bouded liear fuctioal o c ad so, by duality, there is a sequece {g (ζ)} such that g (ζ) T (7) ad ψ ζ ({w }) = w g (ζ) for all {w } a. I other words, ( ζ 2 ) p +ɛ f(ζ) = ( γ 2 ) p +ɛ f(γ )g (ζ). (8) Now apply this formula to the fuctio f(z) ( ( ζ 2 )/( ζz) ) s to obtai ( ζ 2 ) p +ɛ f(ζ) = ( ) ( γ 2 ) p ζ +ɛ 2 s f(γ )g (ζ). (9) ζγ 3

14 Defie for each ζ D, Λ ζ = { γ : ρ(γ, ζ) > 2, g (ζ) > ρ 2 (γ, ζ) }. Combiig (3) with (7) yields that Λ ζ is a Blaschke sequece, to which we the associate the Blaschke product B ζ (z) = γ γ z γ γ z. γ Λ ζ We apply (9) to the fuctio B ζ f to obtai (4) with h (ζ) = B ζ (γ )g (ζ)(b ζ (ζ)). Because of the iequality x e 2( x), which holds for /4 x, we see that B ζ (ζ) 2 = ρ 2 (γ, ζ) exp( 2( ρ 2 (γ, ζ))) γ Λ ζ γ Λ ζ = exp( 2 ρ 2 ) (γ, ζ) exp( 2 g (ζ) ), γ Λ ζ γ Λ ζ which is bouded below by (7). Therefore, (5) is a cosequece of (7). Oe ca likewise check that B ζ (γ )g (ζ) C ( ζ 2 )( γ 2 ) γ ζ 2 by cosiderig separately the cases γ Λ ζ ad γ / Λ ζ. The iequality (6) follows from this immediately. We are ow i a positio to complete the proof of the theorem. Cosider first the case p =. Here we require that ɛ < ad we let s =. By (4) ad(6), ( ζ 2 ) +ɛ f(ζ) C ( γ 2 ) 2+ɛ f(γ ) ( ζ 2 ) γ ζ 2. By Lemma of [], there is a costat C such that wheever < t < s, ( ζ 2 ) t 2 zζ s da(ζ) C( z 2 ) t s for all z D. (2) This implies that f(ζ) da(ζ) C D D C ( γ 2 ) 2+ɛ ( ζ 2 ) ɛ f(γ ) D γ ζ 2 da(ζ) ( γ 2 ) 2 f(γ ). For < p <, we eed ɛ < /p ad here it also suffices to take s =. By (4), ( ζ 2 ) p +ɛ f(ζ) ( γ 2 ) p +ɛ f(γ )h (ζ), 4

15 which implies, by Hölder s iequality, (5) ad (6), that ( ζ 2 ) +pɛ f(ζ) p ( ( γ 2 ) p +ɛ f(γ )h (ζ) ) p C ( γ 2 ) +pɛ f(γ ) p h (ζ) ( h (ζ) ) p q ( γ 2 ) +pɛ f(γ ) p ( ζ 2 )( γ 2 ) γ ζ 2. Therefore, a applicatio of (2) yields f(ζ) p da(ζ) C ( γ 2 ) 2+pɛ f(γ ) p D C ( γ 2 ) 2 f(γ ) p. D ( ζ 2 ) pɛ γ ζ 2 da(ζ) The samplig iequality thus holds for all f A ( p +ɛ). Sice A ( p +ɛ) A p is dese i A p, we see that Γ is a samplig sequece for A p. Ackowledgemet. The author would like to thak Kristia Seip for may helpful coversatios ad commets. Refereces. B. Berdtsso ad J. Ortega-Cerdà, O iterpolatio ad samplig i Hilbert spaces of aalytic fuctios, J. Reie Agew. Math. 464 (995), L. Carleso, P. Malliavi, J. Neuberger ad J. Wermer, The collected Works of Are Beurlig, vol. 2, Harmoic Aalysis, Birkhäuser, p , Bosto, W.W. Hastigs, A Carleso measure theorem for Bergma spaces, Proc. Amer. Math. Soc. 52 (975), H. Hedemalm, S. Richter ad K. Seip, Iterpolatig sequeces ad ivariat subspaces of give idex i the Bergma spaces, J. Reie Agew. Math. 477 (996), M. Jevtic, X. Massaeda ad P. Thomas, Iterpolatig sequeces for weighted Bergma spaces of the ball, Michiga Math. J. 43 (996), D. Lueckig, A techique for characterizig Carleso measures o Bergma spaces, Proc. Amer. Math. Soc. 87 (983), A. Schuster, Ph.D. thesis, Uiversity of Michiga, A Arbor, A. Schuster ad K. Seip, Weak coditios for iterpolatio i holomorphic spaces, Publ. Mat. (to appear). 9. A. Schuster ad K. Seip, A Carleso-type coditio for iterpolatio i Bergma spaces, J. Reie Agew. Math. 497 (998), K. Seip, Beurlig type desity theorems i the uit disk, Ivet. Math. 3 (993), K. Zhu, Operator theory i fuctio spaces, Marcel Dekker, New York, 99. Departmet of Mathematics, Sa Fracisco State Uiversity, Sa Fracisco, Califoria, 9432, USA address: schuster@sfsu.edu 5