Interpolating sequences in the polydisc

Abstract

ABSTRACT. Let H°°(Dn) denote the set of all bounded analytic functions defined on the polydisc D" of Cn. In this note, we give a sufficient condition for sequences of points in £>" to be interpolating sequences for H°°(Dn). We also discuss some conditions for interpolation of general domains.Let G(X) be the set of all bounded continuous functions on a compact set X and let A C C(X) be a uniform algebra equipped with the sup norm. We say that a sequence {a0} of points in X is an interpolating sequence for A if A \ {aj} = l°°. In other words, {aj} is an interpolating sequence for A if whenever given a bounded sequence {aj}, there is a function / G A such that f(aj) = aj. In the case when X = D = the unit disc in C and A -H°° (D) = all the bounded analytic functions on D, Carleson proved the following famous theorem (see [3]). THEOREM (CARLESON). The following are equivalent:(a) There is a constant 6 > 0 such that TT Qfc ~ aj >6>0 for all j. If to each point aj we associate an arc Ij in dD centered at a3/\aj\ with length 2(1 -\aj\), then the second part of condition (c) indicates that there is a constant G such that for all arcs I in 3D. (Here \E\ denotes the Lebesgue measure of the set E.)The purpose of this note is to study the analogue of these conditions on the polydisc Dn of Cn. On the same topic, the readers are referred to earlier works of Kronstadt [7] and E. Amar [1]. Our approach to the problem is quite different from theirs. First we recall some definitions:

Interpolation problem for l1 and a uniform algebra

“…'-interpolating sequence and under some mild condition it is an £'-interpolating sequence. In some sense, this type of theorem for an ^-interpolating sequence was conjectured in [1]. In Section 5, we apply the results from the previous sections to concrete uniform algebras.…”

Spectra of composition operators on algebras of analytic functions on Banach spaces

“…In fact, the explicit form of the metric ρ(z, w) given in (3.2) shows that any sequence {z k } satisfying (5.2) satisfies (1 − z k ) < ∞. An interpolating sequence for H ∞ (B H ) that does not satisfy this condition is given by {z k = 1 2 w k }, where {w k } is any orthonormal subset of H. It is not known whether the condition (5.2) in the context of an arbitrary uniform algebra guarantees that the sequence is interpolating (see [3]). …”

“…Furthermore, the interpolation constant of the sequence depends only on δ. Berndtsson was able to extend to several variables a construction for the interpolating functions due to P. Jones and show that the condition (5.2), where ρ is now the pseudo-hyperbolic metric of the open unit ball B n of C n , is a sufficient condition for a sequence of points in B n to be an interpolating sequence for H ∞ (B n ). As pointed out in [3], Berndtsson's interpolation constant depends on δ and not on the dimension n. By interpolating on finite subsets of the sequence with uniform bounds and applying a normal families argument, we can pass to a limit as n → ∞. We then have the following version of Berndtsson's theorem, where ρ is now the pseudo-hyperbolic metric on B H .…”

The corona problem with two pieces of data

“…Some partial results on the corona problem in several complex variables appear, for instance, in [15], [16], [17], and [2].…”