Take an interval <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$[t, t+1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> on the x-axis together with the same interval on the y-axis and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lev (2018), Lai, Liu and Prince (2021) as well as Ai, Lu and Zhou (2023) we examine the spectrality of this measure for all different values of t (being spectral means that there is an orthonormal basis for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^2(\rho )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> consisting of exponentials <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$e^{2\pi i (\lambda _1 x + \lambda _2 y)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>). We almost complete the study showing that for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-\frac{1}{2}<t<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo><</mo> <mi>t</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$t \notin {\mathbb Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∉</mo> <mi mathvariant="double-struck">Q</mi> </mrow> </math></EquationSource> </InlineEquation> the measure <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> is not spectral. The only remaining undecided case is <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$t=-\frac{1}{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> (Plus space). We also observe that, in all known cases of spectral instances of this measure, the spectrum is contained in a line, and we give an easy necessary and sufficient condition for such measures to have a line spectrum.

Spectrality of a Measure Consisting of Two Line Segments

期刊论文
发表日期：

2025年12月5日

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Mihail N. Kolountzakis,
Sha Wu

摘要

Take an interval $[t, t+1]$ $[t, t + 1]$ on the x-axis together with the same interval on the y-axis and let $\rho $ $ρ$ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lev (2018), 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 $L^2(\rho )$ $L^{2} (ρ)$ consisting of exponentials $e^{2\pi i (\lambda _1 x + \lambda _2 y)}$ $e^{2 π i (λ_{1} x + λ_{2} y)}$ ). We almost complete the study showing that for $-\frac{1}{2}<t<0$ $- \frac{1}{2} < t < 0$ and for all $t \notin {\mathbb Q}$ $t \notin Q$ the measure $\rho $ $ρ$ is not spectral. The only remaining undecided case is $t=-\frac{1}{2}$ $t = - \frac{1}{2}$ (Plus space). We also observe that, in all known cases of spectral instances of this measure, the spectrum is contained in a line, and we give an easy necessary and sufficient condition for such measures to have a line spectrum.