Research Problems in Function Theory — Problem 5.62

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 $\mathbb T=\partial\mathbb D$. The disc algebra $A(\overline{\mathbb D})$ consists of the functions continuous on $\overline{\mathbb D}$ and analytic in $\mathbb D$. For $u\in C(\mathbb T;\mathbb R)$, the extension condition is \[\exists f\in A(\overline{\mathbb D})\quad\forall\zeta\in\mathbb T\quad\operatorname{Re}f(\zeta)=u(\zeta).\]

Statement recorded in the supplied source.

Find a necessary and sufficient condition, expressed in terms of $u$, for this extension condition. [S2]

Short English statement

Characterize exactly the continuous real boundary functions that are real parts of functions in the disc algebra.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2305062 (AMR-022-5062). 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 5.62, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.