Abstract
We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : ℝd → ℂ. In particular, we prove that such functions have a special structure that simplifies the construction of partitions of unity with specific properties. We also prove that it is possible to modify the function g in such a way that it becomes symmetric with respect to a given symmetry group on ℤd. The results are illustrated with constructions of dual pairs of Gabor frames for L2(ℝd). In addition, we obtain general approaches to construct dual Gabor frames whose window functions are symmetric with respect to an arbitrary symmetry group. Through sampling and periodization, these dual Gabor frames for L2(ℝd) lead to dual pairs of discrete Gabor frames for ℓ2(ℤd) and finite Gabor frames for periodic sequences on ℤd.
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Acknowledgments
The authors would like to thank the reviewers for useful suggestions, in particular for the suggestion to add the section on discretization. Ole Christensen would also like to thank the National University of Singapore for its warm hospitality during visits in 2018 and 2019.
Funding
This research was supported in part by the National University of Singapore funding C-141-000-058-001. It was also supported in part by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2016R1D1A1B02009954), as well as by the 2017 Yeungnam University Research Grant.
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Communicated by: Felix Krahmer
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Christensen, O., Goh, S.S., Kim, H.O. et al. Translation partitions of unity, symmetry properties, and Gabor frames. Adv Comput Math 47, 49 (2021). https://doi.org/10.1007/s10444-021-09851-0
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DOI: https://doi.org/10.1007/s10444-021-09851-0