Research Problems in Function Theory — Problem 5.25

Definitions and mathematical statement

The analytic Wiener algebra is \[ W^+=\left\{f(z)=\sum_{n\ge0}a_nz^n:\sum_{n\ge0}|a_n|<\infty\right\}, \qquad\|f\|_{W^+}=\sum_{n\ge0}|a_n|. \] Each member is holomorphic in $\mathbb D$ and continuous on its closure, with multiplication given by pointwise products of functions. A function $f\in W^+$ generates this unital Banach algebra if \[ \overline{\{P\circ f:P\in\mathbb C[z]\}}^{\|\cdot\|_{W^+}}=W^+. \] Characterize these generators. In particular, is injectivity of $f$ on the entire closed disc $\overline{\mathbb D}$ sufficient? Injectivity there is necessary because evaluations at two points with the same $f$-value agree on every polynomial in $f$. Density is required in the sum-of-absolute-coefficients norm. Uniform approximation on the closed disc is weaker and does not give the required conclusion. The source records sufficient hypotheses supplementing univalence, including $f'\in H^1$, where $H^1$ means bounded circular integral means of $|f'|$. The main question removes such extra smoothness assumptions and asks whether closed-disc injectivity alone already forces generation.

Short English statement

Does every function in the analytic Wiener algebra that is injective on the closed disc generate the algebra by polynomial approximation in coefficient norm?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2305025 (AMR-022-5025), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Hayman - Research Problems in Function Theory (2018), Problem 5.25. Original source indexed by AMR-022-5025. https://arxiv.org/abs/1809.07200.