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SPECTRALITY AND NON-SPECTRALITY OF PLANAR SELF-SIMILAR MEASURES WITH FOUR-ELEMENT DIGIT SETS

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. China

and

Key Laboratory of High Performance Computing and, Stochastic Information Processing (HPCSIP), (Ministry of Education of China), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China

E-mail Address: tmw33@163.com

Corresponding author.

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https://doi.org/10.1142/S0218348X20501303Cited by:9 (Source: Crossref)

Abstract

Let $I$ $I$ be the unit matrix and $D = \{(\begin{matrix} 0 \\ 0 \end{matrix}), (\begin{matrix} 1 \\ 0 \end{matrix}), (\begin{matrix} 0 \\ 1 \end{matrix}), (\begin{matrix} - 1 \\ - 1 \end{matrix}) \}$ . In this paper, we consider the self-similar measure $μ_{ρ I, D}$ on $ℝ^{2}$ generated by the iterated function system ${ϕ_{d} (x) = ρ I (x + d)}_{d \in D}$ where $0 < | ρ | < 1$ . We prove that there exists $Λ$ such that $E_{Λ} = \{e^{2 π i < λ, x >} : λ \in Λ\}$ is an orthonormal basis for $L^{2} (μ_{ρ I, D})$ if and only if $| ρ | = 1 / (2 q)$ for some integer $q > 0$ .

Keywords:

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