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[in Journal: International Journal of Mathematics] AND [Keyword: Zeros] (1)	26 Mar 2025	Run
[in Journal: Modern Physics Letters A] AND [Keyword: Heat] (1)	26 Mar 2025	Run
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articleNo Access
Infinite orthogonal exponentials for a class of Moran measures
- Sha Wu and
- Jing-Cheng Liu
International Journal of Mathematics11 Oct 2023
Preview Abstract
For a sequence ${M_{n}}_{n}^{\infty}$ ${M_{n}}_{n = 1}^{\infty}$ of expanding matrices with $M_{n} \in M_{d} (ℤ)$ and a sequence ${D_{n}}_{n = 1}^{\infty}$ of finite digit sets with $D_{n} \subset ℤ^{d}$ , the Moran measure $μ_{{M_{n}}, {D_{n}}}$ is defined by the infinite convolution
$μ_{{M_{n}}, {D_{n}}} = δ_{M_{1}^{- 1} D_{1}} * δ_{M_{1}^{- 1} M_{2}^{- 1} D_{2}} * δ_{M_{1}^{- 1} M_{2}^{- 1} M_{3}^{- 1} D_{3}} * \dots$
and the convergence is in the weak sense. Under some additional assumptions, we show that $L^{2} (μ_{{M_{n}}, {D_{n}}})$ contains an infinite orthogonal set of exponential functions if and only if there exists an infinite subsequence ${n_{k}}_{k = 1}^{\infty}$ of ${n}_{n = 1}^{\infty}$ such that
$(M_{n_{k} + 1}^{*} M_{n_{k} + 2}^{*} \dots M_{n_{k + 1}}^{*} Z_{n_{k + 1}}) \cap ℤ^{d} \neq \emptyset$
for any $k \in ℕ^{+}$ , where $Z_{n_{k + 1}} : = {x : \sum_{α \in D_{n_{k + 1}}} e^{2 π i 〈 α, x 〉} = 0} \cap [0, 1)^{d}$ . This extends the results of [J. L. Li, A necessary and sufficient condition for the finite $μ_{M, D}$ -orthogonality, Sci. China Math. 58 (2015) 2541–2548].

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