Uniqueness sets with angular density for spaces of entire functions, II: $ρ=1$

TL;DR

The work addresses determining the critical uniqueness type for sets with angular density in spaces of entire functions of exponential type, removing the Lindelöf restriction. It combines a minimal regularization of a non-regular set by adding a ray and a zero-density sequence to form a regularized angular measure $\Delta^*$, and then applies Levin-Pfluger theory to relate the zero-set type to the circumradius $R_{\Delta^*}$. The main result is the precise equality $\sigma_U(\Lambda) = R_{\Delta^*}$ where $\Delta^*=\Delta + A_0\delta_{\alpha_0}$ with $A_0e^{i\alpha_0}=-\int_0^{2\pi} e^{it}\,d\Delta(t)$ and $A_0\ge 0$, with a constructive description of the minimal regularization. The proof combines two directions: an upper bound via regularization and Levin-Pfluger theory, and a matching lower bound via Grishin-Sodin with carefully chosen trigonometrically convex test functions, covering both finite and general angular densities and highlighting the geometric role of the circumradius in uniqueness questions for exponential-type spaces.

Abstract

In this note, which is the second part of a three-part series, we focus on uniqueness sets specifically in the case of spaces of entire functions of exponential type. As in the first part, we consider sets with angular density; however, now we abandon the Lindelöf condition restriction. This part is essentially self-contained and can be read independently of the first one.

Figures (3)

• Figure 1: Triangles $\mathcal{T}_M$ and $\mathcal{T}_N$.
• Figure 2: $S(\mathcal{T}_N)=S(N_1N_2M_2)+S(N_1N_3M_2)$, where $S(N_1N_2M_2)=\frac{1}{2}|N_1N_2|\cdot|M_2P_2|$, and similarly for the second triangle.
• Figure 3: Triangles $\mathcal{T}_M$ and $\mathcal{T}_N.$

Theorems & Definitions (1)

• Theorem 1.1