Abstract
Let \(I=(a,b)\times (c,d)\subset {\mathbb R}_{+}^2\) be an index set and let \(\{G_{\alpha }(x) \}_{\alpha \in I}\) be a collection of Gaussian functions, i.e. \(G_{\alpha }(x) = \exp (-\alpha _1 x_1^2 - \alpha _2 x_2^2)\), where \(\alpha = (\alpha _1, \alpha _2) \in I, \, x = (x_1, x_2) \in {\mathbb R}^2\). We present a complete description of the uniformly discrete sets \(\Lambda \subset {\mathbb R}^2\) such that every bandlimited signal f admits a stable reconstruction from the samples \(\{f *G_{\alpha } (\lambda )\}_{\lambda \in \Lambda }\).
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Notes
Sometimes, the term uniformly separated is used.
References
Aldroubi, A., Baskakov, A., Krishtal, I.: Slanted matrices, Banach frames, and sampling. J. Funct. Anal. 255(7), 1667–1691 (2008). https://doi.org/10.1016/j.jfa.2008.06.024
Aldroubi, A., Cabrelli, C., Çakmak, A.F., Molter, U., Petrosyan, A.: Iterative actions of normal operators. J. Funct. Anal. 272(3), 1121–1146 (2017). https://doi.org/10.1016/j.jfa.2016.10.027
Aldroubi, A., Cabrelli, C., Molter, U., Tang, S.: Dynamical sampling. Appl. Comput. Harmon. Anal. 42(3), 378–401 (2017). https://doi.org/10.1016/j.acha.2015.08.014
Aldroubi, A., Gröchenig, K., Huang, L., Jaming, P., Kristal, I., Romero, J.L.: Sampling the flow of a bandlimited function. J. Geom. Anal. 31, 9241–9275 (2021). https://doi.org/10.1007/s12220-021-00617-0
Beurling, A.: Balayage of Fourier-Stieltjes Transforms, The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhäuser, Boston (1989)
Beurling, A.: Local Harmonic Analysis with Some Applications to Differential Operators, The collected Works of Arne Beurling. Harmonic Analysis, vol. 2. Birkhäuser, Boston (1989)
Dunkl, C.F., Xu, Y.: Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 155, 2nd edn. Cambridge University Press, Cambridge (2014)
Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), 105–132 (2004). https://doi.org/10.1007/s00041-004-8007-1
Gröchenig, K., Romero, J.L., Stöckler, J.: Sharp results on sampling with derivatives in shift-invariant spaces and multi-window Gabor frames. Constr. Approx. 51(1), 1–25 (2020). https://doi.org/10.1007/s00365-019-09456-3
Jaming, P., Negreira, F., Romero, J.L.: The Nyquist sampling rate for spiraling curves. Appl. Comput. Harmon. Anal. 52, 198–230 (2021). https://doi.org/10.1016/j.acha.2020.01.005
Landau, H.J.: Necessary density conditions for sampling and interpolation of certain entire functions. Acta Math. 117, 37–52 (1967). https://doi.org/10.1007/BF02395039
Ya. Levin, B.: Lectures on Entire Functions, AMS Transl. of Math. Monographs, vol. 150, Amer. Math. Soc., Providence, RI (1996)
Nitzan, S., Olevskii, A.: Revisiting Landau’s density theorems for Paley-Wiener spaces. C. R. Math. Acad. Sci. Paris 350(9–10), 509–512 (2012). https://doi.org/10.1016/j.crma.2012.05.003
Olevskii, A., Ulanovskii, A.: Functions with Disconnected Spectrum: Sampling, Interpolation, Translates, AMS, University Lecture Series, 65, (2016)
Olevskii, A., Ulanovskii, A.: On multi-dimensional sampling and interpolation. Anal. Math. Phys. 2(2), 149–170 (2012). https://doi.org/10.1007/s13324-012-0027-4
Ortega-Cerdà, J., Seip, K.: Fourier frames. Ann. Math. (2) 155(3), 789–806 (2002). https://doi.org/10.2307/3062132
Rashkovskii, A., Ulanovskii, A., Zlotnikov, I.: On 2-dimensional mobile sampling, preprint. arXiv:2005.11193
Seip, K.: Interpolation and Sampling in Spaces of Analytic Functions. University Lecture Series, vol. 33. AMS, Providence (2004)
Ulanovskii, A., Zlotnikov, I.: Reconstruction of bandlimited functions from space-time samples. J. Funct. Anal. 280(9), 108962 (2021). https://doi.org/10.1016/j.jfa.2021.108962
Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (2001)
Acknowledgements
I am grateful to A. Ulanovskii for stimulating discussions and to D. Stolyarov for the proof of Lemma 8. I also thank the anonymous referees for the suggestions and constructive remarks.
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Communicated by Akram Aldroubi.
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This research was supported by the Russian Science Foundation (Grant No. 18-11-00053), https://rscf.ru/project/18-11-00053/
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Zlotnikov, I. On Planar Sampling with Gaussian Kernel in Spaces of Bandlimited Functions. J Fourier Anal Appl 28, 55 (2022). https://doi.org/10.1007/s00041-022-09948-0
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DOI: https://doi.org/10.1007/s00041-022-09948-0