两条线段上加权测度的谱性

中文摘要 AI 辅助

我们研究了$\nmathbb R^d$中两条线段上具有正可积密度的测度的谱性。我们证明，当两条线段不重叠时，谱性迫使每条线段上的密度几乎处处为常数。当两条线段重叠时，谱性迫使并集上的总密度几乎处处为常数。然后，我们根据线段的几何形状研究具有正常数密度的结果测度。对于两条不共面的线段，每个这样的测度都允许一个包含在直线中的谱。对于位于不同平行线上的两条线段，当且仅当两个密度相等时，该测度才是谱测度。对于非平行的共面线段，一个合适的可逆线性变换将该测度转换为无权的弧长测度。当这些线段在其仿射平面中观察时，谱测度的每个谱都包含在一条直线中。最后，我们给出例子说明密度如何决定线谱的方向，并为加权测度构造显式谱。

英文摘要

We study the spectrality of measures with positive integrable densities supported on two line segments in $\mathbb R^d$. We prove that, when the two segments are non-overlapping, spectrality forces the density on each segment to be constant almost everywhere. When the two segments are overlapping, spectrality forces the total density to be constant almost everywhere on their union. We then study the resulting measures with positive constant densities according to the geometric of the segments. For two non-coplanar segments, every such measure admits a spectrum contained in a straight line. For two segments lying on distinct parallel lines, the measure is spectral if and only if the two densities are equal. For non-parallel coplanar segments, a suitable invertible linear transformation converts the measure into an unweighted arc-length measure. When these segments are viewed in their affine plane, every spectrum of a spectral measure is contained in a straight line. Finally, we give examples showing how the densities determine the directions of line spectra and construct explicit spectra for weighted measures.

发表机构

• School of Mathematics, Guangxi University（广西大学数学学院）

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