Research Problems in Function Theory — Problem 9.11

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $\mathbb D=\{z\in\mathbb C:|z|<1\}$, and let $H^{\infty}(D)$ denote the algebra of bounded analytic functions on a planar domain $D$, with supremum norm. Let $(z_j)\subset D$ and suppose there is one finite $C>0$ such that for every $j$ there exists $F_j\in H^{\infty}(D)$ with \[F_j(z_j)=1,\quad F_j(z_k)=0\ (k\ne j),\quad\|F_j\|_\infty\le C.\] The norm bound applies to these functions $F_j$, not to an unspecified additional function.

Statement recorded in the supplied source.

Does this imply that for every bounded complex sequence $(a_j)\in\ell^{\infty}$ there exists $F\in H^{\infty}(D)$ with $F(z_j)=a_j$ for every $j$? [S2]

Short English statement

Do uniformly bounded single-node interpolants imply interpolation of arbitrary bounded data on every planar domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309011 (AMR-022-9011). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200. Problem 9.11, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.