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Spectrality of infinite convolutions with three-element digit sets. (English) Zbl 1371.28018

Let \(0 < \rho <1\) and let \(\{a_n,b_n\}_{n=1}^\infty\) be a sequence of integers with finite upper and lower bounds. Associated with \(\rho\) and \(\{a_n,b_n\}_{n=1}^\infty\) there exists a unique Borel probability measure \(\mu_{\rho,\{0,a_n,b_n\}}\) on \(\mathbb{R}^n\) generated by the infinite convolution product \[ \mu_{\rho,\{0,a_n,b_n\}} = \underset{n = 1}{\overset{\infty}{\ast}} \delta_{\rho^n, \{0,a_n,b_n\}} \] in the sense of weak convergence. Here \(\delta_E := \frac{1}{\#E}\sum\limits_{e\in E} \delta_e\) and \(\gcd(a_n,b_n) = 1\), \(\forall\,n\in \mathbb{N}\). The authors show that \(L^2 (\mu_{\rho,\{0,a_n,b_n\}})\) admits an orthonormal basis consisting of exponentials iff \(\rho^{-1}\in 3\mathbb{N}\) and \(\{a_n,b_n\} \equiv \{1,2\}\, (\!\!\!\!\mod 3)\), \(\forall\,n\in \mathbb{N}\).

MSC:

28A80 Fractals
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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