谱间隙函数的离散唯一集
Discrete Uniqueness Sets for Functions with Spectral Gaps
摘要：
众所周知，其谱属于固定有界集$S$的整函数允许实一致离散唯一集$Λ$。我们表明，同样的情况下，更广泛的空间的连续函数。特别地，当$S$是具有“周期间隙”的无限测度集时，索伯列夫空间具有该性质。周期性条件至关重要。对于具有随机分布间隙的集合$S$，我们证明了均匀离散集合$Λ$满足一个强非唯一性：每个离散函数$c(λ)\in l^2(Λ)$都可以被$S$中的一个带谱的解析$L^2$ -函数插值.
英文摘要：
It is well-known that entire functions whose spectrum belongs to a fixed bounded set $S$ admit real uniformly discrete uniqueness sets $Λ$. We show that the same is true for much wider spaces of continuous functions. In particular, Sobolev spaces have this property whenever $S$ is a set of infinite measure having "periodic gaps". The periodicity condition is crucial. For sets $S$ with randomly distributed gaps, we show that the uniformly discrete sets $Λ$ satisfy a strong non-uniqueness property: Every discrete function $c(λ)\in l^2(Λ)$ can be interpolated by an analytic $L^2$-function with spectrum in $S$.
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