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Abstract We discuss random interpolating sequences in weighted Dirichlet spaces $${{\mathcal{D}}}_\alpha $$, $$0\leq \alpha \leq 1$$, when the radii of the sequence points are fixed a priori and the arguments are uniformly distributed. Although conditions for deterministic interpolation in these spaces depend on capacities, which are very hard to estimate in general, we show that random interpolation is driven by surprisingly simple distribution conditions. As a consequence, we obtain a breakpoint at $$\alpha =1/2$$ in the behavior of these random interpolating sequences showing more precisely that almost sure interpolating sequences for $${{\mathcal{D}}}_\alpha $$ are exactly the almost sure separated sequences when $$0\le \alpha <1/2$$ (which includes the Hardy space $$H^2={{\mathcal{D}}}_0$$), and they are exactly the almost sure zero sequences for $${{\mathcal{D}}}_\alpha $$ when $$1/2 \leq \alpha \le 1$$ (which includes the classical Dirichlet space $${{\mathcal{D}}}={{\mathcal{D}}}_1$$).
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Random Interpolating Sequences in the Polydisc and the Unit Ball
Abstract We study almost surely 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 $$B_{2}^{\sigma }\left( \mathbb {B}_{d}\right) $$ B 2 σ B d , in the range $$0 < \sigma \le 1 / 2$$ 0 < σ ≤ 1 / 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 $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) and its multiplier algebra $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( D d ) : in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $$\mathrm {H}^\infty (\mathbb {D}^d)$$ H ∞ ( 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 $$0-1$$ 0 - 1 law for random interpolating sequences for $$\mathrm {H}^2(\mathbb {D}^d)$$ H 2 ( D d ) .
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- Award ID(s):
- 1800057
- PAR ID:
- 10430220
- Date Published:
- Journal Name:
- Computational Methods and Function Theory
- Volume:
- 23
- Issue:
- 1
- ISSN:
- 1617-9447
- Page Range / eLocation ID:
- 165 to 198
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s40315-022-00448-2
