Research Problems in Function Theory — Problem 8.7

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 $\mathbb T=\partial\mathbb D$. For $p>0$, the Bergman space is $A^p(\mathbb D)=\{f\text{ analytic}:\|f\|_{A^p}=(\int_{\mathbb D}|f|^p\,dA)^{1/p}<\infty\}$, where $dA=dx\,dy$ is ordinary area measure. Let $(z_n)_{n\ge1}\subset\mathbb D$ and set $k_n(z)=(1-z_nz)^{-2}$. Suppose $M=\overline{\operatorname{span}\{k_n:n\ge1\}}^{A^2}$ is a proper subspace of $A^2$. Let $f_j$ be finite complex linear combinations of the $k_n$, with $\|f_j-f\|_{A^2}\to0$ for some $f\in A^2$. Each finite combination is a rational function; removable poles are canceled when evaluating it outside the disk. For the analogous simple-pole construction the kernels are $h_n(z)=(1-z_nz)^{-1}$, with the same finite-linear-combination and proper-closed-span conditions in a specified normed space of analytic functions. The Hardy space mentioned by the source is $H^2(\mathbb D)=\{g\text{ analytic}:\sup_{0<r<1}(2\pi)^{-1}\int_0^{2\pi}|g(re^{it})|^2\,dt<\infty\}$.

Statement recorded in the supplied source.

If there exists an open disk $\Delta\subset\{|z|>1\}$ on which these rational functions are defined and $\sup_{z\in\Delta}|f_j(z)|\to0$, must $f\equiv0$? [S2] Hayman notes that similar questions can be formulated with the simple-pole kernels $h_n$ and with analytic function spaces other than $A^2$, and reports that the analogous $H^2$ result is known. [S2]

Short English statement

Does vanishing of finite kernel combinations on an exterior disk force their Bergman-norm limit to vanish when the kernels do not span the whole space? Hayman also notes analogous questions for other kernels and spaces.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2308007 (AMR-022-8007). 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 8.7, complete original question, all following updates, and the relevant chapter definitions. https://arxiv.org/abs/1809.07200.