Research Problems in Function Theory — Problem 9.14

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 $K\subset\mathbb C$ be compact. Write $\mathcal H(K)$ for the uniform closure in $C(K)$ of restrictions of functions analytic in some open neighborhood of $K$ (the neighborhood may depend on the function). Equivalently, $u\in\mathcal H(K)$ means there are analytic-near-$K$ functions $u_j$ with $\sup_K|u_j-u|\to0$.

Statement recorded in the supplied source.

For every $f\in C(K)$, does $f^2\in\mathcal H(K)$ imply $f\in\mathcal H(K)$? [S2]

Short English statement

Does uniform approximation by functions analytic near a compact set pass from the square of a continuous function to the function itself?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2309014 (AMR-022-9014). 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 9.14, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.