On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem
On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem
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关于谱 Cantor-Moran 测度和布尔干正弦和问题的变体
DOI：
10.1016/j.aim.2019.04.014
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发表时间：
2019-06-20
影响因子：
1.7
通讯作者：
Lai, Chun-Kit
中科院分区：
文献类型：
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作者：
An, Lixiang;Fu, Xiaoye;Lai, Chun-Kit
本文证明了：如果有一个Hadamard三元组序列{（N-n，B-n，L-n）}，其B-n子集为{0，1，...，N-n- 1}，其中n = 1，2，...，除了极端情况之外，则相关联的Cantor Moran测度μ = μ（N-n，B-n）=delta 1/N1 B1 * delta 1/N1 N2 B2 * delta 1/N（1）N（22）N（3）B3 *.=在[0，1]内具有支撑的mu（n）* mu> n 0总是允许指数正交基E（λ）= {e（2 pi i λ x）：λ是λ的元素}：A E Al对于L-2（mu），其中λ是通过适当地修改L-n而获得的。这里，μ（n）是前71个Dirac测度的卷积，μ>n表示尾项，我们证明了E（λ）的完备性一般依赖于Cantor-Moran测度v>n（.）= μ.n（（N-1. N-n）（-1）（.））。这种等正性可以用{v>n}的弱极限的整周期零点集来分析。这一结果对指数函数的完备性提供了一个新的概念性理解，并显著改进了最近研究中的许多部分结果，这些研究的重点是#B-n
In this paper, we show that if we have a sequence of Hadamard triples {(N-n, B-n, L-n)} with B-n subset of {0, 1,.., N-n- 1} for n = 1, 2,..., except an extreme case, then the associated Cantor Moran measuremu = mu(N-n, B-n) =delta 1/N1B1 * delta 1/N1N2B2 * delta 1/N(1)N(22)N(3)B3 * ...=mu(n) * mu> n 0 with support inside [0, 1] always admits an exponential orthonormal basis E(Lambda) = {e(2 pi i lambda x) : lambda is an element of Lambda} : A E Al for L-2(mu), where Lambda is obtained from suitably modifying L-n. Here, mu(n) is the convolution of the first 71 Dirac measures and mu>n, denotes the tail-term.We show that the completeness of E(Lambda) in general depends on the "equi-positivity" of the sequence of the pull-backed tail of the Cantor-Moran measure v>n(.) = mu.n ((N-1 ... N-n)(-1) (.)).Such equi-positivity can be analyzed by the integral periodic zero set of the weak limit of {v>n}. This result offers a new conceptual understanding of the completeness of exponential functions and it improves significantly many partial results studied by recent research, whose focus has been specifically on #B-n