Research Problems in Function Theory — Problem 8.13

Definitions and mathematical statement

Under the usual circle-measure interpretation, let $\mathbb T={\mathbb{R}}/(2\pi{\mathbb{Z}})$, with Lebesgue measure $m$. For a finite Borel measure $\mu$, its Fourier--Stieltjes coefficients are \[ \widehat\mu(k)=\int_{\mathbb T}e^{-ik\theta}\,{\,\mathrm{d}}\mu(\theta),\qquad k\in{\mathbb{Z}}. \] For a positive measure, singularity means that a Borel set $E$ satisfies $m(E)=0$ and $\mu(\mathbb T\setminus E)=0$. Holland's question, in this conventional setting, is whether there exists a nonzero finite singular measure $\mu$ belonging to Zygmund's class $A^*$ such that \[ \widehat\mu(k)\in[0,\infty)\quad\text{for every }k\in{\mathbb{Z}}. \] Nonnegativity requires real coefficients, not merely nonnegative real parts. Nonzero is needed to avoid the zero-measure triviality; for a nonzero positive measure normalization to total mass one does not change the properties.

The record does not define $A^*$, the underlying space, or whether measures must be positive. For orientation, a common Zygmund-type convention controls second differences of a distribution primitive $F$ by $|F(x+h)-2F(x)+F(x-h)|\le C|h|$; an associated measure convention compares equal adjacent intervals. This is not asserted to be the exact $A^*$ definition intended by the source. Until that definition and the measure category are supplied, the class-membership condition remains an unresolved essential datum. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no definitive resolution located.

Short English statement

Is there a nontrivial singular measure in Zygmund's $A^*$ class with every Fourier--Stieltjes coefficient real and nonnegative? The class definition, measure category, and underlying space are absent.

Sources

• Local UnsolvedMath registry, permanent ID $2308013$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 8.13."
• Hayman, Research Problems in Function Theory (2018), Problem 8.13, attributed to F. Holland; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• Zygmund's class is named without a defining reference or formula. The supplied OPEN-TRIAGE summary provides no separate authoritative current-status citation.