Abstract
Let \(C,D\subset \mathbb{N}\) be disjoint sets, and \(\mathcal{C}=\{1/2^{c}\colon c\in C\}, \mathcal{D}=\{1/2^{d}\colon d\in D\}\). We consider the associate bases of dyadic, axis-parallel rectangles \(\mathcal{R}_{\mathcal{C}}\) and \(\mathcal{R}_{\mathcal{D}}\). We give necessary and sufficient conditions on the sets \(\mathcal{C} and \mathcal{D}\) such that there is a positive function \(f\in L^{1}([0,1)^{2})\) so that the integral averages are convergent with respect to \(\mathcal{R}_{\mathcal{C}}\) and divergent for \(\mathcal{R}_{\mathcal{D}}\). We next apply our results to the two-dimensional Fourier--Haar series and characterize convergent and divergent sub-indices. The proof is based on some constructions from the theory of low-discrepancy sequences such as the van der Corput sequence and an associated tiling of the unit square.
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Acknowledgements
The authors thank Shigeki Akiyama and Tomas Persson for helpful discussions. We thank Håkan Hedenmalm for useful suggestions. We would also like to thank the referees of this paper for their valuable comments, which substantially helped to improve the descriptions.
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The second author was supported by the Knut and Alice Wallenberg Foundation of Sweden (KAW).
The first author was partially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 19K03558.
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Hirayama, M., Karagulyan, D. On the coexistence of convergence and divergence phenomena for integral averages and an application to the Fourier–Haar series. Anal Math 50, 149–187 (2024). https://doi.org/10.1007/s10476-024-00010-3
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DOI: https://doi.org/10.1007/s10476-024-00010-3
Keywords and phrases
- differentiation of integrals
- differentiation bases
- low-discrepancy sequence
- rectangular partial sums of the Fourier–Haar series