Fourier frames for singular measures and pure type phenomena
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Abstract:
Let $\mu$ be a positive measure on $\mathbb {R}^d$. It is known that if the space $L^2(\mu )$ has a frame of exponentials, then the measure $\mu$ must be of “pure type”: it is either discrete, absolutely continuous or singular continuous. It has been conjectured that a similar phenomenon should be true within the class of singular continuous measures, in the sense that $\mu$ cannot admit an exponential frame if it has components of different dimensions. We prove that this is not the case by showing that the sum of an arc length measure and a surface measure can have a frame of exponentials. On the other hand we prove that a measure of this form cannot have a frame of exponentials if the surface has a point of non-zero Gaussian curvature. This is in spite of the fact that each “pure” component of the measure separately may admit such a frame.References
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Additional Information
- Nir Lev
- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel
- MR Author ID: 760715
- Email: levnir@math.biu.ac.il
- Received by editor(s): November 29, 2016
- Received by editor(s) in revised form: June 10, 2017
- Published electronically: March 30, 2018
- Additional Notes: This research was supported by ISF grant No. 225/13 and ERC Starting Grant No. 713927.
- Communicated by: Alexander Iosevich
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2883-2896
- MSC (2010): Primary 42C15, 42B10
- DOI: https://doi.org/10.1090/proc/13849
- MathSciNet review: 3787351