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Spectrality of Moran Sierpinski-type measures on ${\mathbb R}^2$

Part of: Harmonic analysis in one variable Classical measure theory Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  18 January 2021

Min-Min Zhang
Min-Min Zhang*
Affiliation:
Department of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P.R.China.
*
e-mail: zhangminmin0907@163.com
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Abstract

Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$, where $a_n, b_n\in \{-1,1\}$. The associated Borel probability measure

$$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$
is called a Moran Sierpinski-type measure. In this paper, we show that $\mu _{M, \{D_n\}}$ is a spectral measure if and only if $3\mid \rho _i$ for each $i=1, 2$. The special case is the Sierpinski-type measure with $a_n=b_n=1$ for all $n\in {\mathbb N}$, which is proved by Dai et al. [Appl. Comput. Harmon. Anal. (2020), https://doi.org/10.1016/j.acha.2019.12.001].

Keywords

Moran measuresFourier transformspectral measureconvolutionsorthogonal

MSC classification

Primary: 28A80: Fractals
Secondary: 42A85: Convolution, factorization 42C05: Orthogonal functions and polynomials, general theory

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Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 1024 - 1040
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Copyright
© Canadian Mathematical Bulletin 2021

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