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Spectrality of self-affine measures and generalized compatible pairs

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Abstract

Let \(\mu _{M,D}\) be a self-affine measure associated with an expanding matrix \(M\in M_{n}(\mathbb {Z})\) and a finite digit set \(D\subset \mathbb {Z}^{n}\). In this paper, we study the spectrality of \(\mu _{M,D}\) by introducing the generalized compatible pairs via Hadamard matrices. This is motivated by the problem of looking for conditions for \(\mu _{M,D}-\)orthogonal exponential system to be infinite. Based on the properties of Hadamard matrices, we first present some elementary properties concerning the generalized compatible pair. We then provide a method of getting \(\mu _{M,D}\)-orthogonal exponentials. Certain relationships between the generalized compatible pair and the spectrality of self-affine measure are established, which extend the known results in the appropriate manner. The research here is closely related to the spectral problem of self-affine measures.

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Acknowledgements

The author thanks Prof. Deguang Han for a helpful discussion on the subject during his visit to Xi’an in the autumn of 2014. The author also thanks the anonymous referees for their valuable suggestions. This work is supported by the National Natural Science Foundation of China (No.11571214).

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  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, People’s Republic of China

    Jian-Lin Li

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Communicated by A. Constantin.

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Li, JL. Spectrality of self-affine measures and generalized compatible pairs. Monatsh Math 184, 611–625 (2017). https://doi.org/10.1007/s00605-017-1096-0

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