Abstract
We study the concentration problem on compact two-point homogeneous spaces for finite expansions of eigenfunctions of the Laplace–Beltrami operator using large sieve methods. We derive upper bounds for concentration in terms of the maximum Nyquist density. Our proof uses estimates of the spherical harmonics basis coefficients of certain zonal filters and an ordering result for Jacobi polynomials for arguments close to one.
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Notes
We could as well consider geodesic balls \(B(y,r)=\{x\in {\mathbb {M}}:\ {\mathrm {d}}(x,y)<r\}\). Note that \({\mathcal {C}}_\delta (y)=B(y,\gamma ^{-1}\arccos \delta )\) or equivalently \(B(x,r)={\mathcal {C}}_{\cos \gamma r}(x)\). It turns out that caps are more convenient here.
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Acknowledgements
M.S. was supported by the Austrian Science Fund (FWF) through an Erwin-Schrödinger Fellowship (J-4254). The authors wish to thank the referees for their careful reading of the manuscript.
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Communicated by Ilya Krishtal.
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Jaming, P., Speckbacher, M. Concentration estimates for finite expansions of spherical harmonics on two-point homogeneous spaces via the large sieve principle. Sampl. Theory Signal Process. Data Anal. 19, 9 (2021). https://doi.org/10.1007/s43670-021-00008-0
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DOI: https://doi.org/10.1007/s43670-021-00008-0
Keywords
- Large sieve inequalities
- Concentration estimates
- Two-point homogeneous spaces
- Eigenfunctions of Laplace–Beltrami operator
- Jacobi polynomials