<p>In this paper, we investigate the spectrality of a class of Moran measures <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu _{\{R_{n}\},\{B_{n}\}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> generated by a sequence of integers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{R_{n}\}_{n=1}^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> and a sequence of product-form digit sets <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{B_{n}\}_{n=1}^{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R_{n}=N^{q_{n}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo>=</mo> <msup> <mi>N</mi> <msub> <mi>q</mi> <mi>n</mi> </msub> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$B_{n}=\{0,1,\ldots ,N-1\}\oplus N^{p_{n}}\{0,1,\ldots ,N-1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> <mo>⊕</mo> <msup> <mi>N</mi> <msub> <mi>p</mi> <mi>n</mi> </msub> </msup> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>N</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$N\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$q_{n},~p_{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mi>n</mi> </msub> <mo>,</mo> <mspace width="3.33333pt" /> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> are positive integers for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We give some sufficient conditions for <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mu _{\{R_{n}\},\{B_{n}\}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> to be a spectral measure, i.e., there exists a countable set <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\{e^{2\pi i\lambda \cdot x}:\lambda \in \Lambda \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <msup> <mi>e</mi> <mrow> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>λ</mi> <mo>·</mo> <mi>x</mi> </mrow> </msup> <mo>:</mo> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is an orthonormal basis in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$L^2(\mu _{\{R_{n}\},\{B_{n}\}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>R</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>B</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our results partially extend the work of Liu et al. [<CitationRef CitationID="CR19">19</CitationRef>], where <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$p_n=p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo>=</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$q_n=q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mi>n</mi> </msub> <mo>=</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we show that our results can be extended to higher-dimensional settings.</p>

Spectrality of a class of Moran measures with product-form digit sets

期刊论文
发表日期：

2026年5月15日

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Kailing Lai

摘要

In this paper, we investigate the spectrality of a class of Moran measures $\mu _{\{R_{n}\},\{B_{n}\}}$ $μ_{{R_{n}}, {B_{n}}}$ generated by a sequence of integers $\{R_{n}\}_{n=1}^{\infty }$ ${R_{n}}_{n = 1}^{\infty}$ and a sequence of product-form digit sets $\{B_{n}\}_{n=1}^{\infty }$ ${B_{n}}_{n = 1}^{\infty}$ , where $R_{n}=N^{q_{n}}$ $R_{n} = N^{q_{n}}$ , $B_{n}=\{0,1,\ldots ,N-1\}\oplus N^{p_{n}}\{0,1,\ldots ,N-1\}$ $B_{n} = {0, 1, \dots, N - 1} \oplus N^{p_{n}} {0, 1, \dots, N - 1}$ with $N\ge 2$ $N \geq 2$ and $q_{n},~p_{n}$ $q_{n}, p_{n}$ are positive integers for $n\ge 1$ $n \geq 1$ . We give some sufficient conditions for $\mu _{\{R_{n}\},\{B_{n}\}}$ $μ_{{R_{n}}, {B_{n}}}$ to be a spectral measure, i.e., there exists a countable set $\Lambda $ $Λ$ such that $\{e^{2\pi i\lambda \cdot x}:\lambda \in \Lambda \}$ ${e^{2 π i λ \cdot x} : λ \in Λ}$ is an orthonormal basis in $L^2(\mu _{\{R_{n}\},\{B_{n}\}})$ $L^{2} (μ_{{R_{n}}, {B_{n}}})$ . Our results partially extend the work of Liu et al. [19], where $p_n=p$ $p_{n} = p$ and $q_n=q$ $q_{n} = q$ for all $n\ge 1$ $n \geq 1$ . Furthermore, we show that our results can be extended to higher-dimensional settings.