On non-uniqueness sets for spaces of holomorphic functions

Problems of description of zero subsequences for weight spaces of holomorphic functions are reduced, according to a general scheme, to solving certain problems in weight classes of subharmonic functions. Let be a domain in the complex plane C. We associate with every at most countable sequence Λ = { 𝑘}𝑘=1,2,... ⊂ 𝐷, without accumulation points in 𝐷, the counting measure 𝑛Λ(𝑆) := Σ︀ 𝑘∈𝑆 1. We denote by Hol(𝐷) the vector space of all holomorphic functions in 𝐷. For 0 ̸= ∈ Hol(𝐷), denote by Zero𝑓 zero sequence of with account of multiplicities. A sequence Λ ⊂ is called the non-uniqueness sequence for a subspace ⊂ Hol(𝐷), if there exists a nonzero function ∈ such that Λ ⊂ Zero𝑓, i. e. 𝑛Λ( ) ≤ 𝑛Zero𝑓 ( ) for all ∈ 𝐷. We denote by sbh(𝐷) the convex cone of all subharmonic functions in ⊂ C. For -∞ ̸≡ ∈ sbh(𝐷) we denote by the Riesz measure of 𝑠. A Borel positive measure is called the submeasure for a subset ⊂ sbh(𝐷), if there exists a function ∈ 𝑆, 𝑠 ̸≡ -∞, with the Riesz measure ≥ on 𝐷. For a (weight) function 𝑀: → [-∞,+∞] we define the weight classes sbh(𝐷;𝑀] := {𝑠 ∈ sbh(𝐷) : ≤ + const на 𝐷} and Hol(𝐷; exp𝑀] := {𝑓 ∈ ∈ Hol(𝐷) : |𝑓| ≤ const · exp𝑀 на 𝐷}, where const is a constant. Let be a subset of the extended complex plane C∞ := C∪{∞}. Denote by clos and bd the closure and the boundary of in C∞ resp. Let dist(·, ·) be the Euclidean distance between two objects (points or subsets) in C. Let 𝑑: → (0, 1] be a continuous function such that 0 0 weight functions 𝑁,𝑀 on C (see Section 2, Theorem 2).