Spectrality of Weighted Measures on Two Line Segments

Abstract

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.