Indicators of linear combinations

Definitions and mathematical statement

Fix $\rho>0$ and $n\geq1$. Let $A\subset\mathbb C^n$ be a set such that every choice of $n$ distinct vectors is linearly independent. For each $a\in A$, prescribe a continuous $2\pi$-periodic real function $h_a$ satisfying $h_a''+\rho^2h_a\geq0$ in the sense of distributions. This is the meaning of $\rho$-trigonometric convexity.

An entire function $f$ has order $\rho$ and normal type when, with $M(r,f)=\max_{|z|=r}|f(z)|$, $0<\limsup_{r\to\infty}r^{-\rho}\log M(r,f)<\infty$. Its indicator at this order is \[ h_f(\theta)=\limsup_{r\to\infty} r^{-\rho}\log|f(re^{i\theta})|. \]

Characterize the prescribed data for which there are entire functions $f_1,\ldots,f_n$ of order $\rho$ and normal type such that, for every $a=(a_1,\ldots,a_n)\in A$, the nonzero combination $g_a=\sum_ja_jf_j$ has indicator $h_a$. All combinations use the same finite collection of functions; separately realizing each indicator is insufficient.

Also characterize realization with completely regular growth. For a function $g$ with indicator $h_g$, this means \[ t^{-\rho}\log|g(tz)|\longrightarrow |z|^\rho h_g(\arg z)\quad(t\to\infty) \] in the distribution space $\mathcal D'(\mathbb C)$. In this version require that the realizing functions and the specified combinations have this regular growth. The target is necessary and sufficient compatibility conditions, for arbitrary admissible vector sets and indicator assignments.

Short English statement

When can the indicators of many linear combinations of one finite family of entire functions be prescribed simultaneously?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700012 (AMR-036-0012), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Indicators, Problem 1. Original source indexed by AMR-036-0012. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Indicators of linear combinations, Problem 1. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/indi2.pdf.