Polynomial Stability of Universal Taylor Series

Definitions and mathematical statement

Put $D=\{z\in{\mathbb{C}}:|z|<1\}$. For a holomorphic function $f$ on $D$, write its Taylor partial sums at zero as \[ S_n(f,0)(z)=\sum_{j=0}^{n}\frac{f^{(j)}(0)}{j!}z^j. \] A standard convention for the class $U(D,0)$ is: for every compact $K\subset{\mathbb{C}}\setminus D$ with connected complement and every function $g$ continuous on $K$ and holomorphic in its interior, there is a strictly increasing sequence $n_r\to\infty$ such that \[ \sup_{z\in K}|S_{n_r}(f,0)(z)-g(z)|\longrightarrow0. \] The sequence may depend on $K$ and $g$. This conventional definition is made explicit for orientation; the supplied record does not state its compact-set or boundary convention, which must be checked, in particular for sets meeting $\partial D$.

The four source questions are:

1. For every $f\in U(D,0)$ and every $a\in{\mathbb{C}}$ with $|a|\geq1$, does $(z-a)f(z)$ belong to $U(D,0)$? The boundary case $|a|=1$ is included.
2. For every $f\in U(D,0)$ and every nonconstant polynomial $p\in{\mathbb{C}}[z]$, does $pf$ belong to $U(D,0)$? No restriction on the zeros of $p$ is stated.
3. For every $f\in U(D,0)$, does $f'$ belong to $U(D,0)$?
4. Does there exist an inner function $f$ belonging to $U_1(D,0)$?

Here, conventionally, an inner function is a bounded holomorphic function on $D$ whose radial boundary values have modulus one almost everywhere with respect to arc length on $\partial D$. The indexed class $U_1(D,0)$ is not defined anywhere in the supplied record. The fourth question therefore retains that symbol as an unresolved source-dependent class; it is not replaced by $U(D,0)$ or by a guessed variation of the compact-set condition.

The source labels the statement corrected and verified, with all four questions preserved. The supplied literature triage checked 2026-08-21 labels the questions open and says that later work on universal composition operators does not settle the multiplication and differentiation questions for $U(D,0)$. This is a dated source assessment, not new verification.

Short English statement

Ask whether multiplication by an exterior-root linear factor, multiplication by any nonconstant polynomial, and differentiation preserve universal Taylor series. Also ask whether an inner function lies in the separately named class $U_1(D,0)$, whose definition is missing from the record.

Sources

• Local UnsolvedMath registry, permanent ID 30000920, source code catalogue entry; supplied complete record, including its four-question corrected-verified note and dated assessment.
• Mini-Workshop: Complex Approximation and Universality (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/06, pp. 297--346: https://doi.org/10.4171/owr/2008/06.