Finite unions of interpolating sequences for Hardy spaces

A sequence which is a finite union of interpolating sequences for $H^infty$ have turned out to be especially important in the study of Bergman spaces. The Blaschke products $B(z)$ with such zero sequences have been shown to be exactly those such that the multiplication $f mapsto fB$ defines an operator with closed range on the Bergman space. Similarly, they are exactly those Blaschke products that boundedly divide functions in the Bergman space which vanish on their zero sequence. There are several characterizations of these sequences, and here we add two more to those already known. We also provide a particularly simple new proof of one of the known characterizations. One of the new characterizations is that they are interpolating sequences for a more general interpolation problem.

Random Interpolating Sequences in the Polydisc and the Unit Ball

We study almost sure separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0-1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov-Sobolev spaces $B_{2}^{sigma}left(mathbb{B}_{d}right)$, in the range $0 < sigmaleq1 / 2$. For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $mathrm{H}^2(mathbb{D}^d)$ and its multiplier algebra $mathrm{H}^infty(mathbb{D}^d)$: in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $mathrm{H}^infty(mathbb{D}^d)$-interpolating almost surely. Those two conditions do not coincide, due to the fact that the deterministic starting point is less descriptive of interpolating sequences than its counterpart for the unit ball. On the other hand, we give the $0-1$ law for random interpolating sequences for $mathrm{H}^2(mathbb{D}^d)$.

Complete interpolating sequences, the discrete Muckenhoupt condition, and conformal mapping

We extend the parameterization of sine-type functions in terms of conformal mappings onto slit domains given by Eremenko and Sodin to the more general case of generating functions of real complete interpolating sequences. It turns out that the cuts have to fulfill the discrete Muckenhoupt condition studied earlier by Lyubarskii and Seip.

Interpolating sequences for the Bergman space and the $barpartial$-equation in weighted $L^p$

The author showed that a sequence in the unit disk is a zero sequence for the Bergman space $A^p$ if and only if a certain weighted space $L^p(W}$ contains a nontrivial analytic function. In this paper it is shown that the sequence is an interpolating sequence for $A^p$ if and only if it is separated in the hyperbolic metric and the $barpartial$-equation $(1 - |z|^2)barpartial u = f$ has a solution $u$ belonging to $L^p(W)$ for every $f$ in $L^p(W)$.

Random harmonic functions in growth spaces and Bloch-type spaces

Let $h^infty_v(mathbf D)$ and $h^infty_v(mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v $ that fulfill a doubling condition. In the two-dimensional case let $u (re^{ita},xi) = sum_{j=0}^infty (a_{j0} xi_{j0} r^j cos jtheta +a_{j1} xi_{j1} r^j sin jtheta)$ where $xi ={xi_{ji}}%_{k=0}^infty $ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji} $ which imply that $u$ is in $h^infty_v(mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu.