Recursive interpolating sequences

Abstract: This paper is devoted to pose several interpolation problems on the open unit diskD of the complex plane in a recursive and linear way. We look for interpolating sequences (zn) in D so that given a bounded sequence (an) and a suitable sequence (wn), there is a bounded analytic function f onD such that f(z1) = w1 and f(zn+1) = an f(zn) + wn+1. We add a recursion for the derivative of the type: f ′(z1) = w1 and f ′(zn+1) = an [(1−∣zn ∣)/(1−∣zn+1∣)] f (zn)+wn+1, where (an) is bounded and (wn) is an appropriate sequence, andwe also look for zero-sequences verifying the recursion for f ′. The conditions on these interpolating sequences involve the Blaschke product with zeros at their points, one of them being the uniform separation condition.


Introduction
Interpolation problems on the unit disk D are a classical branch of complex analysis. Several types of interpolating sequences for di erent classes of analytic functions have been addressed since the middle of the last century, beginning with the celebrated works of W.K. Hayman [1], D.J. Newman [2] and L. Carleson [3] about the so-called "universal" interpolation problem, which consists in characterizing the sequences (z n ) in D verifying that for any bounded sequence (w n ), there is a bounded analytic function f on D such that f (z n ) = w n .
On the other hand, recursion appears in many areas of mathematics: formulas, algorithms, optimization..., providing alternative de nition procedures. Since there exists a speci c theory for recursive numerical sequences and interpolating sequences have not been studied from a recursive perspective, we think it is interesting to pose recursive-type interpolation problems.
We want to emphasize that our approach converts the universal interpolation problem and other problems related to it into trivial cases of those that we introduce. Furthermore, most conditions involved are new and depend not only on the separation of the points of the sequence in D, but also on the sequences that we employ to de ne recursion.
We begin with the necessary notation. Let H ∞ be the space of all analytic functions f on D such that f ∞ = sup z∈D f (z) < ∞ and let l ∞ be the Banach space of all sequences of complex numbers (w n ) such that (w n ) ∞ = sup n w n < ∞. We put Z = (z n ) for any sequence of di erent points in D verifying the Blaschke condition ∑ n ( − z n ) < ∞, which characterizes the zero-sequences of functions in H ∞ . For two points z and w in D, we write ψ(z, w) = z − w −zw , so that ρ = ψ is their pseudo-hyperbolic distance. Let B be the Blaschke product with zeros at Z, that is, If E is a subsequence of Z, we put B E for the Blaschke product with zeros at E and for a xed m ∈ N, we denote B Z∖{z m } by B m . We write c for strictly positive constants that may change from one occurrence to the next one. First, we recall that Z is interpolating if given any (w n ) ∈ l ∞ , there exists f ∈ H ∞ such that f (z n ) = w n . Interpolating sequences are characterized by the well-known Carleson's theorem: (1) Sequences satisfying (1) are called uniformly separated (u.s.). From the Schwarz lemma, it follows that and thus, it is said that Z is interpolating in di erences if given These sequences are the union of two u.s. [4], characterized as follows.

Lemma 1.2 ([5]
). For a sequence Z, the following are equivalent (a) Z is the union of two u.s. sequences.
It is proved in [6] that these sequences are also the u.s. ones.
Next we consider the following three quantities for the terms of a sequence T = (t n ) ∈ l ∞ : We need them to take suitable target spaces in our interpolation problems and they also appear in the results we get (Section 2).
for some constants c T,Z , c ′ T,Z > and for all m ∈ N. From now on A = (a n ) and A ′ = (a ′ n ) will denote sequences in l ∞ . Our purpose is to examine the following distinguished sequences of D:

De nition 1.3. We say that Z is A-interpolating if given
and w n+ ≤ c Γ (a n ), Clearly if A ∞ < , then the sum in (3) is bounded by (3) and

De nition 1.4. We say that Z is A-interpolating in di erences if given (w n ) satisfying
there is f ∈ H ∞ verifying recursion (5). (3) and (4) and and there exists f ∈ H ∞ satisfying recursion (5) and De nition 1.6. We say that Z is zero and A ′ -interpolating if given (w ′ n ) verifying (7), and there is f ∈ H ∞ vanishing on Z and satisfying recursion (9).
and recursion (9) is equivalent to f ′ (z n ) = µ ′ n , with Thus, we must have (3) and (7) to state that sequences (µ n ) and (µ ′ n ( − z n )) are bounded. We impose that data sequences (w n ) and (w ′ n ) verify (4), (6), (8) and (11), because they are intrinsic to recursions (a technical reason justi es (10)). In e ect, (4) and (6) are obtained taking into account (2) in the inequalities respectively. On the other hand, and See [7] for (14), [8] for (15) and [9] for (16). Thus, (8) is obtained using (14) in the inequality If on the right of this last inequality, we put then (11) is obtained using (16) for the rst summand and (15) for the second one. We introduce these interpolating sequences because they provide a generalization of the usual interpolation problems, in the sense that ( )-interpolating sequences, ( )-interpolating in di erences and (( ), ( ))interpolating are interpolating, interpolating in di erences and double interpolating, respectively.
Extending recursion to an arbitrary order or increasing the degree of derivability are projects certainly cumbersome, so that we con ne ourselves to order one and the rst derivative. Nevertheless, we think it would be interesting to consider these types of sequences for other spaces of analytic functions, such as the Lipschitz class and the Bloch space, for which the pseudo-hyperbolic distance in (2) is replaced by the Euclidean and hyperbolic distance, respectively (interpolating sequences for these spaces are characterized in [10] and [11]).
While the proofs of results turn out to be rather standard (Carleson's theorem is used repeatedly), we appreciate the following separation conditions, which are consistent with the problems posed and appear in a natural way.

De nition 1.7. We say that (Z, A) satis es condition (S) if
We say that (Z, A ′ ) satis es condition (M) if We name (S), (D) and (M) to the above conditions because these are the initials of simple, di erences and mixed, respectively, and we will see in the next section that condition (S) is related to interpolating sequences in a simple sense; (D), in a di erences sense, and (M), in a mixed sense (zero and interpolating).

Statement of results
Our results are the following ones.

Proposition 2.2. (i) If (Z, A) veri es (S), then Z is A-interpolating.
(ii) If (Z, A) veri es (D) and A is such that a n+ ≤ r a n (17) for some r ∈ ( , ), then Z is A-interpolating in di erences. (iii) If Z is u.s., then Z is (A, A ′ )-interpolating for any sequences A and A ′ .
then Z is zero and A ′ -interpolating.

Proof of results
Proof of Proposition 2.1. (i) For a xed m ∈ N, let (w n ) be de ned by w m+ = Γ (a m ), w m+ = −a m+ w m+ and w n = if n ≠ m + , m + . Since Γ (a n ) = O(Γ (a n+ )), it follows that and w m+ also veri es (4). Since the operator (3) and (4) and (S) holds.
(ii) For a xed m ∈ N, let (w n ) be de ned by w m+ = Π(a m ) and the other terms as in (i). This sequence satis es (6), because taking into account that Π(a n ) ∼ Π(a n+ ), Condition (D) is obtained proceeding exactly as in the proof of (i).
(iii) Let (w n ) be de ned as in (i) and let (w ′ n ) be de ned by We have that w ′ m+ also satis es (8), because Proceeding as in (i), there exists g m ∈ H ∞ such that f m = g m B m+ . Since ρ(z m , z m+ ) ≤ Γ (a m ) (see de nition of Γ ) and B m+ (z m+ ) < ρ(z m , z m+ ), we have Thus, it follows that Z ∖ {z } is u.s. and so is Z.
Proof of Proposition 2.2. Let (µ n ) be as in (12) and (µ ′ n ) as in (13). (i) We situate ourselves in (c) of Lemma 1.2. If Z is u.s., then Carleson's theorem provides f performing the interpolation f (z n ) = µ n . Otherwise, Z is the union of E = (z m ) and F = (z m+ ), both u.s. Let g ∈ H ∞ such that g(z m ) = µ m . We look for a function h ∈ H ∞ such that h(z m+ ) = λ m , where Taking into account that µ m+ = a m µ m + w m+ (20) and using (3) and (4), .
Thus, (λ n ) ∈ l ∞ and Carleson's theorem provides h. Putting it follows that f is in H ∞ and performs the above interpolation on Z.
Thus, (λ n ) ∈ l ∞ and the proof continues as in (i).
(iii) Since Z is u.s., then Z is double interpolating and there is f in H ∞ performing f (z n ) = µ n and f ′ (z n ) = µ ′ n . (iv) In case that Z = E ∪ F, where E = (z m ) and F = (z m+ ) are u.s., we know that there is g ∈ H ∞ vanishing on E and satisfying g ′ (z m ) = µ ′ m . We look for a function h ∈ H ∞ such that h (z m+ ) = λ m , where .