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Fourier orthonormal bases of two dimensional Moran measures with four-element digits. (English) Zbl 1487.42061

Summary: Let \(\mu_{\{M_n\},\{D_n\}}\) be a Moran measure generated by \(\{(M_n,D_n)\}_{n=1}^\infty\), where \(M_n=\begin{pmatrix} p_n & 0\\ 0 & q_n\end{pmatrix}\in M_2(\mathbb{Z})\) is an expanding matrix and \(D_n=\left\{ \begin{pmatrix} 0 \\ 0 \end{pmatrix},\;\begin{pmatrix} a_n\\ 0 \end{pmatrix}, \;\begin{pmatrix} 0\\ b_n \end{pmatrix},\;\begin{pmatrix} -a_n\\ -b_n \end{pmatrix}\right\} \subseteq\mathbb{Z}^2\) is a finite digit set. In the present paper we will study the problem of how to determine the Hilbert space \(L^2(\mu_{\{M_n\},\{D_n\}})\) has a Fourier basis. We first obtain a sufficient condition for this aim and give some examples to explain the theory. Moreover, we completely settle the corresponding problem if \(a_n= b_n= 1\) for all \(n\ge 1\).

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A65 Completeness of sets of functions in one variable harmonic analysis
28A80 Fractals
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