Multiple interpolation and extremal functions in the Bergman spaces

Transcription

1 Multiple interpolation and extremal functions in the Bergman spaces Mark Krosky and Alexander P. Schuster Abstract. Multiple interpolation sequences for the Bergman space are characterized. In addition, relationships between interpolation sequences and extremal functions are explored.. Introduction. For 0 < p < +, the Bergman space A p is the set of functions analytic in the unit disk D with f p p = f(z) p da(z) <, π D where da denotes Lebesgue area measure. The space obtained for p = 2, A 2, is a Hilbert space with inner product f, g = f(z)g(z)da(z) π and reproducing kernel D k ζ (z) = ( ζ z) 2, corresponding to the point evaluation functional for ζ D. That is, we have f(a) = f, k a for all f A 2. It actually turns out that this holds for all f A. A sequence Γ = {z k } k is said to be an interpolation sequence for A p if for every sequence {w k } satisfying ( z k 2 ) 2 w k p < +, k= there is an f A p such that f(z k ) = w k for all k. These sequences were characterized completely by Seip [6], using a density condition to be described shortly. The above definition of interpolation sequences makes sense only when Γ consists of distinct points. A natural question thus concerns sequences containing points of 99 Mathematics Subject Classification. Primary 30H05, 46E5. Key words and phrases. Bergman space, interpolation, zero sequence, canonical divisor. Typeset by AMS-TEX

2 2 MARK KROSKY AND ALEXANDER P. SCHUSTER higher multiplicity. In analogy with the work of Øyma [3], who considered multiple interpolation in the Hardy space, we then define Γ to be a multiple interpolation sequence for A p if for every sequence of n-tuples {(wk 0, w k,..., wn k )} k satisfying (.) n l=0 k= ( z k 2 ) 2+lp wk l p < +, there is an f A p such that f (l) (z k ) = wk l for all k and 0 l n. In this paper we will henceforth assume that n is fixed. We mention in passing that in [2], Brekke and Seip state and solve a multiple interpolation problem for the Bargmann-Fock space. We say that Γ (with or without multiple points) is an A p zero sequence if there is a non-trivial function in A p which vanishes precisely on Γ. By a theorem of Horowitz [8], any subsequence of an A p zero sequence is again an A p zero sequence, so to prove that a sequence Γ is a zero sequence it actually suffices to find a non-identically vanishing function in A p that vanishes along Γ. An interpolation sequence (multiple if you like) for A p is also a zero sequence, as we see by interpolating the values 0 at all points but one, where we take the value. Then we multiply the interpolating function by a suitable first degree polynomial to have the resulting function vanish along the entire sequence. Associated to each A p zero sequence is a canonical divisor, which is the analogue of the Blaschke product in the Hardy space. If Γ = {z j } j is an A p zero sequence which avoids the origin, we consider the extremal problem (.2) sup{ g(0) : g(z j ) = 0 j, g p }. (If 0 occurs in Γ with multiplicity n, we consider instead the problem of maximizing g (n) (0), subject to the same constraints as in (.2).) This problem has a solution G Γ for 0 < p < + (which is unique if we stipulate that G Γ (0) > 0) and as was found by Hedenmalm [6] for p = 2 and later generalized by Duren, Khavinson, Shapiro and Sundberg [4, 5] to all other values of p plays a role in A p similar to that of the Blaschke product in H p. In particular, if Γ is an A p zero sequence, then G Γ vanishes precisely on Γ and is a contractive divisor in A p, that is, f/g Γ p f p for all f A p that vanish along Γ. It is shown in [5] that interpolation sequences can be described in terms of the canonical divisors of Möbius transformations of zero sequences. In the present article, we obtain further relationships between interpolation sequences, canonical divisors and kernel functions. We also characterize the multiple interpolation sequences for A p, both in terms of densities and in terms of canonical divisors.

3 MULTIPLE INTERPOLATION 3 2. The Description of Interpolation Sequences. The pseudo-hyperbolic metric ρ is defined on D by ρ(z, ζ) = φ ζ (z), where φ ζ (z) = ζ z, z, ζ D. ζz A sequence Γ = {z k } k is said to be uniformly discrete if there is a δ > 0 such that ρ(z i, z j ) δ for all i j. For Γ uniformly discrete and /2 < r <, let z D(Γ, r) = k <r ( z k ) log. r The upper uniform density is defined to be D + (Γ) = lim sup r sup D(φ ζ (Γ), r). The following result is proved in [6] for p = 2 using methods which may be extended to A p (0 < p < + ). We refer to [4] for a proof of this general case. Theorem A. A sequence Γ of distinct points in the disk is an interpolation sequence for A p if and only if Γ is uniformly discrete and D + (Γ) < p. In [5] the following theorem is proved: Theorem B. A sequence Γ of distinct points in the disk is an interpolation sequence for A p if and only if there is a δ > 0 such that ζ D (2.) G φzk (Γ\{z k })(0) δ for all k. 3. Interpolation and Extremal Functions. We say that a sequence Γ is a Hardy interpolation sequence if for every sequence {w k } satisfying ( z k 2 ) w k p <, k= there is an f H p such that f(z k ) = w k for all k. It is a well-known theorem of Shapiro and Shields [8] for p < +, and Kabaǐla [9] for 0 < p <, that Γ is a Hardy interpolation sequence if and only if it is uniformly separated, that is, there exists a δ > 0 such that z j z k z j z k δ j:j k for all k. Let B Γ be the Blaschke product associated with the sequence Γ, that is B Γ (z) = k= z k z k z k z z k z.

4 4 MARK KROSKY AND ALEXANDER P. SCHUSTER Note that B Γ is the normalized solution to the extremal problem sup{ f(0) : f(z j ) = 0 j, f H p }. It is easy to see that Γ is uniformly separated if and only if there is a δ > 0 such that δ B φzk (Γ\{z k })(0) for all k. Thus Theorem B is the precise analogue of the Shapiro-Shields result. However, it is also true that uniform separation can be expressed by the condition (3.) δ B Γ\{zk }(z k ) for all k. Our goal is to determine an analogue of this for the Bergman space. We first find the canonical divisor of the Möbius transformation of a sequence. Lemma 3.. Let Γ be an A p zero sequence not containing the origin. If ζ / Γ, then ( φ ζ (z)k GΓ (3.2) G φζ (Γ)(z) = γ G Γ (φ ζ (z)) p(φ ζ(z), ζ) ) 2 p (, k GΓ p(ζ, ζ)) p where γ is a constant of modulus, chosen so that the expression on the right is positive when evaluated at 0. Here k ω (, ζ) is the reproducing kernel with respect to the point ζ for the Hilbert space A 2 (ω), where A 2 (ω) is the closure of the analytic polynomials in the space L 2 (π ωda). We first note that (3.2) is a well-defined analytic function, since k GΓ p does not vanish on D 2. This is a consequence of Theorem of [7], which states that 2 k G Γ p(ζ, ζ)( ζ 2 ) 2 k ζz 2 GΓ p(z, ζ) 2 ζz 2 for all z, ζ D. We remark that this result actually holds for a more general class of weights ω, specifically those which are logarithmically subharmonic and are reproducing for the origin. (See [7] for definitions.) Proof of Lemma 3.. Define ( φ ζ (z)k GΓ f Γ,ζ (z) = γ G Γ (φ ζ (z)) p(φ ζ(z), ζ) ) 2 p (. k GΓ p(ζ, ζ)) p We first prove the lemma in the case that Γ is finite. By Theorem 4 of [4], G φζ (Γ)(z) = B φζ (Γ)(z) (k 2 B φζ (Γ) p(z, 0)) p (k Bφζ (Γ) p(0, 0)). p

5 A standard argument shows that MULTIPLE INTERPOLATION 5 (3.3) k Bφζ (Γ) p(z, a) = φ ζ (a)k B Γ p(φ ζ(z), φ ζ (a))φ ζ(z) for all a, z D. Since B Γ (φ ζ (z)) = αb φζ (Γ)(z), where α is a complex number of modulus, we see by (3.3) that G φζ (Γ)(z) = αb Γ (φ ζ (z)) (φ ζ (0)k B Γ p(φ ζ(z), ζ)φ ζ (z)) 2 p which, again by Theorem 4 of [4], equals ( (3.4) α φ ζ(0) 2 k B Γ k BΓ p(ζ, ζ) p(0, 0) (φ ζ (0)k B Γ p(ζ, ζ)φ ζ (0)) p ) ( p G Γ (φ ζ (z)) φ ζ (0)k B Γ p(φ ) 2 p ζ(z), ζ) k BΓ p(φ ζ(z), 0) φ ζ(z). The above mentioned result of [4] may also be used to prove that (3.5) k GΓ p(z, ζ) = k B Γ p(z, ζ)k B Γ p(0, 0) k p(z, BΓ 0)k B Γ p(0, ζ). (Note that a peaking point argument of Hedenmalm [6] shows the existence of a positive constant η such that k BΓ p(z, 0) η for all z D.) Equation (3.5) implies that (3.6) k GΓ p(ζ, ζ) = k B p(ζ, Γ ζ)k B Γ p(0, 0) k BΓ p(ζ, 0) 2 and (3.7) k p(φ GΓ ζ(z), ζ) = k B Γ p(φ ζ(z), ζ)k BΓ p(0, 0) k BΓ p(φ ζ(z), 0)k BΓ p(0, ζ). We then combine (3.4), (3.6) and (3.7) to obtain the equation where γ = α G φζ (Γ)(z) = f Γ,ζ (z), ( ) 2 φ p ζ (0)k BΓ p(0, ζ) φ ζ (0)k. B Γ p(0, ζ) It is now straightforward to generalize the result to infinite sequences. To this end, let Γ n = {z k } n k=, where Γ = {z k} k=. By Lemma 7 of [4], G Γ n (z) G Γ (z) and G φζ (Γ n )(z) G φζ (Γ)(z), both in the norm of A p. We have shown above that G φζ (Γ n )(z) = f Γn,ζ(z), so to complete the proof of the lemma, we need only demonstrate that f Γn,ζ(z) f Γ,ζ (z). This follows immediately from the fact that k GΓ p(z, w) k n G Γ p(z, w). Note that the restrictions on ζ and Γ in Lemma 3. are artificial and only there for convenience. If 0 occurs in Γ with multiplicity n, one simply maximizes g (n) (0) instead of g(0) in the extremal problem (.2). The lemma holds when the canonical divisor is defined in this way. Lemma 3. may be used to prove the following result:,

6 6 MARK KROSKY AND ALEXANDER P. SCHUSTER Theorem 3.2. Suppose Γ = {z k } is a sequence of distinct points in D. Then the following are equivalent: (3.8) Γ is an interpolation sequence for A p. (3.9) G φzk (Γ\{z k })(0) = G Γ\{zk }(z k ) ( z k 2 ) 2 p (k GΓ\{zk } p(z k, z k )) p δ all k. (3.0) G φζ (Γ)(0) = G Γ (ζ) ( ζ 2 ) 2 p (k GΓ p(ζ, ζ)) p δρ(ζ, Γ) for all ζ D. (3.) G φ zk (Γ) (0) = G Γ(z k ) ( z k 2 ) + 2 p (k GΓ p(z k, z k )) p δ for all k. Here ρ(ζ, Γ) = inf k ρ(ζ, z k ) and δ is a generic positive constant. Proof. (3.8) (3.9). The equality in (3.9) follows immediately from an application of Lemma 3. to Γ \ {z k } and φ zk. The equivalence of (3.8) and (3.9) is thus given by Theorem B. (3.9) (3.0). The equality in (3.0) follows, as above, from an application of Lemma 3.. If (3.0) does not hold, there is a sequence {ζ n } and an ɛ > 0 such that G φζn (Γ)(0) 0 and ρ(ζ n, Γ) ɛ. Since (3.9) implies Γ is an interpolation sequence, work in [6] shows that there is a constant C, depending on Γ and ɛ, such that M (Γ {ζ n }) C for all n. (M (Γ) is defined in 4.) An argument from [5] shows that if Λ is an interpolation sequence, then for all λ k Λ. Therefore a contradiction. G φλk (Λ\{λ k })(0) M (Λ) p G φζ n (Γ) (0) = G φζ n (Γ ζn\{ζn}) (0) (M (Γ {ζ n })) p C p, (3.0) (3.). By Lemma 3., G G φzk φ zk (Γ)(0) = lim (Γ) (φ zk (ζ)) ζ z k ρ(ζ, z k ) G Γ (ζ) φ 2 z = lim k (φ zk (ζ))k GΓ p(ζ, z k ) p ζ z k ρ(ζ, z k )(k p(z GΓ k, z k )) p G Γ (ζ) = lim ζ z k ζ z k giving us the equality in (3.). z k ζ φ z lim k (φ zk (ζ))k GΓ p(ζ, z k ) ζ z k (k GΓ p(z k, z k )) p = G Γ(z k ) ( z k 2 ) φ z k (0) 2 p (k GΓ p(z k, z k )) p = G Γ(z k ) ( z k 2 ) + 2 p (k GΓ p(z k, z k )) p, 2 p

7 Again by Lemma 3., we obtain MULTIPLE INTERPOLATION 7 G G φzk φ zk (Γ)(0) = lim (Γ) (φ zk (ζ)) G φzk = lim (Γ) (φ zk (ζ)) ζ z k ρ(ζ, z k ) ζ z k G φζ (Γ)(0) G φζ (Γ)(0) ρ(ζ, z k ) G Γ (ζ) φ 2 z δ lim k (φ zk (ζ)) p (k GΓ p(ζ, z k)) 2 p (k GΓ p(z k, z k )) p ζ z k G Γ (ζ) φ ζ (0) 2 p (k GΓ p(ζ, ζ)) p = δ lim ζ z k ( ρ(ζ, z k ) 2 ) 2 p (k GΓ p(ζ, z k)) 2 p (k GΓ p(z k, z k )k GΓ p(ζ, ζ)) p = δ. (3.) (3.8). An application of Jensen s formula in the disk of radius r < to G φzk (Γ)(z)/z yields w φ zk (Γ), w <r log r w = 2π 2π 0 p log ( 2π log G φzk (Γ)(re iθ ) dθ log G φ zk (Γ)(0) log r 2π 0 ) G φzk (Γ)(re iθ ) p dθ + C. Here we have used (3.) and the arithmetic-geometric mean inequality. This is the estimate given by (7) on p. 228 of [5], where it is used to prove that Γ is an interpolation sequence. The rest of the proof is then identical to that in [5]. The equivalence of (3.8) and (3.9) gives us the desired analogue of (3.). Since H 2 A 2 ( G Λ p ) A 2 holds for each A p zero set Λ with contractive imbeddings (cf. [5], [3], [4]), we have the estimates ζ 2 k G Λ p(ζ, ζ) ( ζ 2 ) 2. It follows that the requirement that for some δ > 0, G Γ\{zk }(z k ) δ, is necessary, and ( z k 2 ) p GΓ\{zk }(z k ) δ is sufficient for interpolation in A p. The following argument allows us to describe interpolation sequences in the case p = 2 in another way. An A 2 function g is said to be A 2 -inner if it has unit norm and g(z) 2 z n da(z) = 0 for n =, 2,.... D

8 8 MARK KROSKY AND ALEXANDER P. SCHUSTER It is easy to see that for any A 2 -inner function g, the mapping given by multiplication by g is an isometry from A 2 ( g 2 ) to [g], the closure in A 2 of polynomial multiples of g. This leads to the relationship k [g] (z, ζ) = g(z)g(ζ)k g 2(z, ζ), where k [g] is the reproducing kernel of [g]. In particular, since extremal functions are known to be A 2 -inner, k [GΓ\{zk }](z k, z k ) = G Γ\{zk }(z k ) 2 k GΓ\{zk } 2(z k, z k ). Therefore, by Theorem 3.2, Γ is an interpolation sequence for A 2 if and only if Similarly, since k [GΓ\{zk }](z k, z k ) δ 2 ( z k 2 ) 2. k [GΓ](ζ, ζ) = G Γ (ζ) 2 k GΓ 2(ζ, ζ), we may again apply Theorem 3.2 to obtain the result that Γ is an interpolation sequence for A 2 if and only if k [GΓ](ζ, ζ) δ2 (ρ(ζ, Γ)) 2 ( ζ 2 ) 2 for all ζ D. Taking square roots, we see that this statement is equivalent to (3.2) k [GΓ ](, ζ) 2 δρ(ζ, Γ) k(, ζ) 2 for all ζ D. A theorem of Aleman, Richter and Sundberg [] states that [G Γ ] is the subspace of A 2 consisting of functions vanishing on Γ. Thus, (3.2) may be viewed as a statement about norms of reproducing kernels of a certain subspace of A 2, rather than about extremal functions. A simple calculation shows that a sequence Γ is uniformly separated if and only if there is a δ > 0 such that B Γ(z k ) δ z k 2 for all k. The equivalence of (3.8) and (3.) is the Bergman space analogue of this statement. 4. Multiple Interpolation. One characterization of multiple interpolation sequences is given by the following theorem. Recall that each point in Γ has multiplicity n.

9 MULTIPLE INTERPOLATION 9 Theorem 4.. The sequence Γ is a multiple interpolation sequence for A p if and only if Γ is uniformly discrete and D + (Γ) < np. Proof. First we give the proof of the necessity. A standard argument based on the closed graph theorem yields that if Γ is a multiple interpolation sequence for A p, then there is a constant M n (Γ) such that if (.) holds, then the multiple interpolation problem can be solved by a function satisfying n f p p M n (Γ) ( z k 2 ) 2+lp wk l p. l=0 k= Given k, let wj,k 0 = δ jk( z k 2 ) 2 p and let w l j,k = 0 for l n and all j. By hypothesis, there exists an f k A p such that f (l) k (z j) = wj,k l for all j and 0 l n. Also, f k p p M n (Γ) for all k. Defining g k (z) = f k (φ zk (z))(φ z k (z)) 2 p, we see that gk (0) =, g k has zeroes of order n on φ zk (Γ \ {z k }) and g k p p = f k p p M n (Γ) for all k. By the previous paragraph then, (M n (Γ)) p problem defining G (p) φ zk (Γ\{z k. Hence }) (n) g k is a candidate for the extremal (4.) G (p) φ zk (Γ\{z k }) (n) (0) g k(0) (M n (Γ)) p =. (M n (Γ)) p Lemma 8 of [4] states that (G (p) φ zk (Γ\{z k }) (n) ) n = G (np) φ zk (Γ\{z k }). Strictly speaking, the lemma asserts this only for finite sets, but it is clear that the argument given there will work for infinite zero sets as well. Therefore (4.) implies that G (np) φ zk (Γ\{z k })(0) = G(p) φ zk (Γ\{z k (0) }) (n) n (Mn (Γ)) np. This implies by Theorem B that Γ is an interpolation sequence for A np and then by Theorem A that Γ is uniformly discrete and D + (Γ) < np. The proof of the sufficiency is by induction on the multiplicity n. The case n = is Theorem A. Suppose that we are given a sequence {wk l } satisfying (.). By the induction hypothesis, we find a function F in A p such that F (l) (z k ) = wk l for all k and 0 l n 2. We need the following result, which is a direct consequence of Lemma 2. of []. Lemma 4.. If Γ is uniformly discrete, then there exists a constant C = C(n, p, Γ) such that ( z k 2 ) 2+np f (n) (z k ) p C f p p k=

10 0 MARK KROSKY AND ALEXANDER P. SCHUSTER for all f analytic in D. Applying this lemma to the function F yields (4.2) ( z k 2 ) 2+(n )p F (n ) (z k ) p C F p p <. k= Let u k = w n k F (n ) (z k ). Then by (.) and (4.2) we obtain ( z k 2 ) 2+(n )p u k p <. k= If there is a function f in A p such that for all k, f (l) (z k ) = 0 for 0 l n 2 and f (n ) (z k ) = u k, we can let H = F + f and see that H (l) (z k ) = wk l for 0 l n. Our problem has therefore been reduced to finding such a function f, the construction of which we proceed with now. We will use an argument similar to that given in [6] and [4]. Let ɛ = 2 ( np D+ (Γ)). For each k, let Γ k = {φ zk (Γ)} {a < z < } = φ zk (Γ \ {z k }). The second equality holds if we choose a < inf j k ρ(z j, z k ). At this point, we require the following result of Seip [6], which is a strengthening of a theorem of Korenblum [0]. We remark that Seip [7] has achieved a further sharpening of the result. Lemma 4.2. If Λ is a sequence of points contained in the annulus {a < z < } and if D + (Λ) < α, then there exists a function g analytic in the disk such that g(λ) = 0, g(0) =, and g(z) C( z 2 ) α, where C depends only on a and D + (Λ). Applying Lemma 4.2 to Γ k and α = np ɛ yields a function g k such that g k (φ zk (z j )) = 0 j k, g k (0) =, g(z) C( z 2 ) ɛ np. Then Let now h k (z) = (g k(φ zk (z))) n (φ zk (z)) n ( φ z k (z)) p nɛ. (n )! h (l) k (z j) = 0 0 l n, j k, h (l) k (z k) = 0 0 l n 2, h (n ) k (z k ) = ( z k 2 ) nɛ p (n ), h k (z) C( z 2 ) nɛ p.

11 Let f(z) = k= where s > max{nɛ +, 2 p }. MULTIPLE INTERPOLATION u k ( z k 2 ) p +(n ) nɛ h k (z) ( z k 2 ) s ( z k z) s, It is routine to verify that f (l) (z k ) = 0 for 0 l n 2 and that f (n ) (z k ) = u k. To show that f is actually in A p, we use the same argument employed by Seip in [6] for p = 2. See also [4], where this is carried out for general p. We leave the details to the reader. This completes the proof of Theorem 4.. We remark that during the course of the proof, we have proven the following analogue of Theorem B for multiple interpolation: Theorem 4.5. A sequence Γ is a multiple interpolation sequence for A p if and only if there is a δ > 0 such that G φzk (Γ\{z k }) (n)(0) δ for all k. (For a sequence of distinct points Λ, we let Λ (n) be the sequence having n copies of each point of Λ.) 5. Uniformly Minimal Families. The following definitions can be found in [2]. Let X be a Banach space and suppose {x n } is a family of vectors in X. The family {x n } is said to be minimal in X if x n / X (x k : k n), where X denotes the closed linear span in X. The family {x n } is said to be uniformly minimal in X if there is a δ > 0 such that dist X ( xn / x n, X (x k : k n) ) δ for all n. The family {x n} of continuous linear functionals on X is biorthogonal to {x n } if x n, x k = δ nk for all n and k. The following result is taken directly from [2]. Lemma 5.. Suppose {x n } is minimal in X and X (x n) = X. Then {x n } admits a unique biorthogonal family {x n} and dist X ( xn / x n, X (x k : k n) ) = x n x n for all n. The Bloch space B consists of functions f analytic in D with f B = f(0) + sup( z 2 ) f (z) <, z D

12 2 MARK KROSKY AND ALEXANDER P. SCHUSTER and the little Bloch space B 0 is the closure of the polynomials in the Bloch norm. It is well known that with respect to the pairing f, g = lim f(z)g(z)da(z), r π rd the dual of A p ( < p < ) can be identified with A q, where p + q =. Furthermore, the dual of B 0 can be identified with A and the dual of A can be identified with B. The Hardy space H 2 is the set of functions f analytic in D satisfying f 2 H 2 = sup 0<r< 2π 2π 0 f(re iθ ) 2 dθ <. We say that an analytic function f is in the Dirichlet space D if f 2 D = f 2 H + f (z) 2 da(z) <. 2 π The dual of D can be identified with A 2 with respect to the Cauchy pairing f, g # = lim r 2π D 2π 0 f(re iθ )g(re iθ )dθ. We will refer to the Szegö kernel as K a (z) = ( az). It has the property that f, K a # = f(a) for all f H 2 and a D. It is shown in [2] that Γ = {z n } is an interpolation sequence for H 2 if and only if {K zn } is a uniformly minimal family in H 2 (K zn ). We obtain the following analogue for the Bergman space. Theorem 5.2. Suppose < p < and Γ = {z n } is a sequence of distinct points in D. Then (5.) Γ is an interpolation sequence for A p if and only if {k zn } is uniformly minimal in A q (k zn ). (5.2) Γ is an interpolation sequence for A if and only if {k zn } is uniformly minimal in B (k z n ). (5.3) Γ is an interpolation sequence for A 2 if and only if {K zn } is uniformly minimal in D (K z n ). Proof. (5.). Let x n = k zn and define x n(z) = G φz n (Γ\{z n})(φ zn (z))( φ z n (z)) 2 p ( zn 2 ) 2 p /Gφz n (Γ\{z n})(0), where G is the extremal function in A p, so that {x n} is biorthogonal to {x n }.

13 A calculation shows that MULTIPLE INTERPOLATION 3 x n q ( z n 2 ) 2 p, where A(n) B(n) means the existence of positive constants bounding the ratio of A(n) and B(n) from above and below. Since x n p = ( z n 2 ) 2 p G φz n (Γ\{zn}) (0), we see that x n p x n q G φz n (Γ\{zn}) (0), and so we apply Theorem 3.2 and Lemma 5. to obtain the desired result. (5.2). Here we let x n and x n be as above, making the substitution p =. We see again that {x n} is biorthogonal to {x n }. A straightforward estimate shows that x n B ( z n 2 ) 2. Therefore x n B x n G φzn (Γ\{z n })(0) and so, as above, we arrive at the desired conclusion. (5.3). In this case we let x n be the Szegö kernel K zn and x n as above, substituting in the value p = 2. Then x n D ( z n 2 ), so This completes the proof of the theorem. x n D x n 2 G φzn (Γ\{z n })(0). For the case p = 2, Theorem 5.2 may be interpreted in another way. Let us return for a moment to a more general setting. Suppose that we have a reproducing kernel Hilbert space H of analytic functions on a domain Ω with inner product,. We will denote by k ζ the kernel that reproduces function values at the point ζ Ω, that is, f, k ζ = f(ζ) for all f in H. In this case, Γ is an interpolation sequence for H if for every {a j } satisfying a j 2 k zj 2 <, there is an f in H with j f(z j ) = a j. We say that Γ = {z n } is a weak interpolation sequence for H if there is a constant C such that for every n there is a function f n H satisfying (5.4) f n (z m ) = δ nm and (5.5) f n C/ k zn. It is clear that for an interpolation sequence, we can obtain functions satisfying (5.4), and an application of the closed graph theorem yields (5.5). It is thus immediate that every interpolation sequence is a weak interpolation sequence. The following proposition, combined with (5.) of Theorem 5.2, shows that like the Hardy space, the Bergman space A 2 has the property that every weak interpolation sequence is actually an interpolation sequence.

14 4 MARK KROSKY AND ALEXANDER P. SCHUSTER Proposition 5.3. The family {k zn } is uniformly minimal in H (k z n ) if and only if Γ is a weak interpolation sequence for H. Proof. Suppose first that {k zn } is uniformly minimal and let {f n } be a family that is biorthogonal to {k zn }, that is f n (z m ) = δ nm. Since {k zn } is uniformly minimal, we may apply Lemma 5. to find a constant C, such that f n k zn C, which implies (5.5). If Γ is a weak interpolation sequence, then the family {f n } is biorthogonal to {k zn }, and so (5.5), combined with Lemma 5., implies uniform minimality. Acknowledgement. We would like to extend our thanks to Haakan Hedenmalm, who gave us significant help with the paper, especially with 3. Many of the ideas were conceived during our numerous discussions with him. Note. After we finished this article, we received a preprint of N. Marco entitled Interpolation in weighted Bergman spaces and finite unions of interpolating sequences, in which he considers a more general version of multiple interpolation. Our Theorem 4. seems to follow from his main result, but we remark that the methods of proof are completely different. We would like to thank S.M. Shimorin for pointing out to us the version of Carleson s theorem stated in terms of uniformly minimal families, which led to the work in 5. References. A. Aleman, S. Richter and C. Sundberg, Beurling s theorem for the Bergman space, Acta Math. 77 (996), S. Brekke and K. Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space III, Math. Scand. 73 (993), P. L. Duren, D. Khavinson and H.S. Shapiro, Extremal functions in invariant subspaces of Bergman spaces, Illinois J. Math. 40 (996), P. L. Duren, D. Khavinson, H.S. Shapiro and C. Sundberg, Contractive zero-divisors in Bergman spaces, Pacific J. Math. 57 (993), P. L. Duren, D. Khavinson, H. S. Shapiro, C. Sundberg, Invariant subspaces in Bergman spaces and the biharmonic equation, Michigan Math. J. 4 (994), H. Hedenmalm, A factorization theorem for square area-integrable analytic functions, J. Reine Angew. Math. 422 (99), H. Hedenmalm, An off-diagonal estimate of Bergman kernels, J. Math. Pures Appl. (to appear). 8. C. Horowitz, Zeros of functions in the Bergman spaces, Duke Math. J. 4 (974), V. Kabaǐla, Interpolation sequences for the H p classes in the case p <, (in Russian), Litovsk. Mat. Sb. 3 (963), B. Korenblum, An extension of the Nevanlinna theory, Acta. Math. 35 (975), D. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 07 (985), N.K. Nikolskii, Treatise on the shift operator, Springer-Verlag, Berlin, K. Øyma, Interpolation in H p spaces, Proc. Amer. Math. Soc. 76 (979), A. Schuster, Ph.D. thesis, University of Michigan, Ann Arbor, A. Schuster and K. Seip, A Carleson-type condition for interpolation in Bergman spaces, J. Reine Angew. Math. 497 (998), K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 3 (994), K. Seip, On Korenblum s density condition for the zero sequences of A α, J. Anal. Math. 67 (995),

15 MULTIPLE INTERPOLATION 5 8. H.S. Shapiro and A.L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (96), S.M. Shimorin, Approximate spectral synthesis in the Bergman space, Duke Math. J. (to appear). 20. K. Zhu, Operator theory in function spaces, Marcel Dekker, New York, 990. Department of Mathematics, University of Michigan, Ann Arbor, Michigan address: mkrosky@math.lsa.umich.edu Department of Mathematics, San Francisco State University, San Francisco, California address: schuster@sfsu.edu