<p>It is known to all that the Moran measure <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu _{\rho ,\{D_n\}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>ρ</mi> <mo>,</mo> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> can be expressed as <Equation ID="Equ17"> <EquationSource Format="TEX">$\begin{aligned} \mu _{\rho ,\{D_n\}}=\delta _{\rho D_1}\delta _{\rho ^2 D_2}\delta _{\rho ^3 D_3}*\cdots \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>μ</mi> <mrow> <mi>ρ</mi> <mo>,</mo> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow> <mi>ρ</mi> <msub> <mi>D</mi> <mn>1</mn> </msub> </mrow> </msub> <mrow /> <mo>∗</mo> <msub> <mi>δ</mi> <mrow> <msup> <mi>ρ</mi> <mn>2</mn> </msup> <msub> <mi>D</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow /> <mo>∗</mo> <msub> <mi>δ</mi> <mrow> <msup> <mi>ρ</mi> <mn>3</mn> </msup> <msub> <mi>D</mi> <mn>3</mn> </msub> </mrow> </msub> <mrow /> <mo>∗</mo> <mo>⋯</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in the sense of weak convergence. We prove that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\rho ^{-1} \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>ρ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _{\rho ,\{D_n\}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>ρ</mi> <mo>,</mo> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> is a spectral measure under the conditions <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {Z}(\widehat{\delta }_{D_n}) \subseteq \alpha _n \mathbb {Z}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">Z</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> <msub> <mi>D</mi> <mi>n</mi> </msub> </msub> <mo stretchy="false">)</mo> </mrow> <mo>⊆</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mi mathvariant="double-struck">Z</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n \in \mathbb {N}^{+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. And we provided an example to explain this phenomenon.</p>

Spectrality of Some Moran Measures in $\mathbb {R}$

期刊论文
发表日期：

2026年3月28日

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Xin Yang

摘要

It is known to all that the Moran measure $\mu _{\rho ,\{D_n\}}$ $μ_{ρ, {D_{n}}}$ can be expressed as $\begin{aligned} \mu _{\rho ,\{D_n\}}=\delta _{\rho D_1}*\delta _{\rho ^2 D_2}*\delta _{\rho ^3 D_3}*\cdots \end{aligned}$ $\begin{matrix} μ_{ρ, {D_{n}}} = δ_{ρ D_{1}} * δ_{ρ^{2} D_{2}} * δ_{ρ^{3} D_{3}} * \dots \end{matrix}$ in the sense of weak convergence. We prove that $\rho ^{-1} \in \mathbb {N}$ $ρ^{- 1} \in N$ when $\mu _{\rho ,\{D_n\}}$ $μ_{ρ, {D_{n}}}$ is a spectral measure under the conditions $\mathcal {Z}(\widehat{\delta }_{D_n}) \subseteq \alpha _n \mathbb {Z}$ $Z ({\hat{δ}}_{D_{n}}) \subseteq α_{n} Z$ for all $n \in \mathbb {N}^{+}$ $n \in N^{+}$ . And we provided an example to explain this phenomenon.

Spectrality of Some Moran Measures in \(\mathbb {R}\)

摘要