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Existence of Exponential Orthonormal Bases for Infinite Convolutions on \({{\mathbb {R}}}^n\)

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Abstract

In this paper we investigate the harmonic analysis of infinite convolutions generated by admissible pairs on Euclidean space \({{\mathbb {R}}}^n\). Our main results give several sufficient conditions so that the infinite convolution \(\mu \) to be a spectral measure, that is, its Hilbert space \(L^2(\mu )\) admits a family of orthonormal basis of exponentials. As a concrete application, we give a complete characterization on the spectral property for certain infinite convolution on the plane \({{\mathbb {R}}}^2\) in terms of admissible pairs.

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Notes

  1. Here, the notation “expanding” denotes that, for the sequence \(\{R_k\}\) with finitely many distinct matrices, there exists \(r > 1\) such that \(\Vert R_kx\Vert _2\ge r\Vert x\Vert _2\) for all k, where \(\Vert \cdot \Vert _2\) denotes the Euclidean 2-norm on \({{\mathbb {R}}}^n\), see [34, pp. 216, line 1–2]

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Acknowledgements

The authors would like to thank the referees for his/her many valuable comments and suggestions.

Funding

Yan-Song Fu is supported by the National Natural Science Foundation of China (Grant Nos. 12371090, 11801035) and the Fundamental Research Funds for the Central Universities (No. 2023ZKPYLX01). Min-Wei Tang is supported by National Natural Science Foundation of China (Grant No. 12201208) and the Hunan Provincial NSF (Grant Nos. 2023JJ40422, 2024JJ3023).

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Authors and Affiliations

  1. School of Science, China University of Mining and Technology, Beijing, Beijing, 100083, People’s Republic of China

    Yan-Song Fu

  2. Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Min-Wei Tang

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  1. Yan-Song Fu
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Correspondence to Min-Wei Tang.

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Communicated by Yurii Lyubarskii.

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Fu, YS., Tang, MW. Existence of Exponential Orthonormal Bases for Infinite Convolutions on \({{\mathbb {R}}}^n\). J Fourier Anal Appl 30, 31 (2024). https://doi.org/10.1007/s00041-024-10088-w

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