Non-spectral problem for a class of planar self-affine measures
Non-spectral problem for a class of planar self-affine measures
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DOI：
10.1016/j.jfa.2008.04.001
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发表时间：
2008-12
影响因子：
1.7
通讯作者：
Jian-Lin Li
中科院分区：
文献类型：
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作者：
Jian-Lin Li
对应于扩张矩阵M∈Mn（R）和有限子集D ∈ R的自仿射测度μM，D在迭代函数系的吸引子（或不变集）上是可支撑的。近年来，μM，D上的谱问题和非谱问题，包括其中隐含的谱拼接问题，受到了广泛的关注。关于μM，D的非谱问题之一是估计L2（μM，D）中正交指数的个数并求出它们。本文证明了：如果a，B，c∈Z，|一|>1，|C|>1且ac∈Z <$（3 Z），则L2（μM，D）中至多存在3个相互正交的指数，且数目为3是最好的.这是几个已知结论的延伸。这一结果的证明依赖于傅里叶变换μ ∈ M，D的零点集的特征，并提供了一种处理非谱问题的方法。
The self-affine measure μM,Dcorresponding to an expanding matrix M∈Mn(R) and a finite subset D⊂Rnis supported on the attractor (or invariant set) of the iterated function system [Formula: see text] . The spectral and non-spectral problems on μM,D, including the spectrum-tiling problem implied in them, have received much attention in recent years. One of the non-spectral problem on μM,Dis to estimate the number of orthogonal exponentials in L2(μM,D) and to find them. In the present paper we show that if a,b,c∈Z, |a|>1, |c|>1 and ac∈Z∖(3Z), then there exist at most 3 mutually orthogonal exponentials in L2(μM,D), and the number 3 is the best. This extends several known conclusions. The proof of such result depends on the characterization of the zero set of the Fourier transform μˆM,D, and provides a way of dealing with the non-spectral problem.