Spectrality of a measure consisting of two line segments

2025/01/20 by Mihail N. Kolountzakis, Kolountzakis, Mihail N., Sha Sha Wu +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2501.11367

Abstract

Take an interval [t, t+1] on the x-axis together with the same interval on the y-axis and let ρ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lai, Liu and Prince (2021) as well as Ai, Lu and Zhou (2023) we examine the spectrality of this measure for all different values of t (being spectral means that there is an orthonormal basis for L²(ρ) consisting of exponentials e^{2πi (λ₁ x + λ₂ y)}). We almost complete the study showing that for -\frac12

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