Crystalline measures in two dimensions
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Yves Meyer
Université Paris-Saclay
Some crystalline measures supported by a Delone set Λ ⊂ R2 are constructed in this note. This gives a new proof of a remarkable theorem by Pavel Kurasov and Peter Sarnak.
Paraules clau
Fourier transform, crystalline measure
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Meyer, Yves. «Crystalline measures in two dimensions». Publicacions Matemàtiques, 2023, vol.VOL 67, núm. 1, p. 469-80, https://raco.cat/index.php/PublicacionsMatematiques/article/view/412720.
Referències
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P. Kurasov and P. Sarnak, Stable polynomials and crystalline measures, J. Math. Phys. 61(8) (2020), 083501, 13 pp. DOI: 10.1063/5.0012286
N. Lev and A. Olevskii, Quasicrystals and Poisson’s summation formula, Invent. Math. 200(2) (2015), 585–606. DOI: 10.1007/s00222-014-0542-z
Y. Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, Afr. Diaspora J. Math. 13(1), Special issue (2012), 1–45.
Y. Meyer, Measures with locally finite support and spectrum, Rev. Mat. Iberoam. 33(3) (2017), 1025–1036. DOI: 10.4171/RMI/962
A. Olevskii and A. Ulanovskii, A simple crystalline measure, Preprint (2020). arXiv:2006.12037v2
A. Olevskii and A. Ulanovskii, Fourier quasicrystals with unit masses, C. R Math. Acad. Sci. Paris 358(11-12) (2020), 1207–1211. DOI: 10.5802/crmath.142
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J.-P. Kahane, “Lectures on Mean Periodic Functions”, Tata Institute of Fundamental Research, Bombay 1959.
J.-P. Kahane and S. Mandelbrojt, Sur l’´equation fonctionnelle de Riemann et la formule sommatoire de Poisson, Ann. Sci. Ecole Norm. Sup. (3) 75(1) (1958), 57–80. DOI: 10.24033/asens.1066
P. Kurasov and P. Sarnak, Stable polynomials and crystalline measures, J. Math. Phys. 61(8) (2020), 083501, 13 pp. DOI: 10.1063/5.0012286
N. Lev and A. Olevskii, Quasicrystals and Poisson’s summation formula, Invent. Math. 200(2) (2015), 585–606. DOI: 10.1007/s00222-014-0542-z
Y. Meyer, Quasicrystals, almost periodic patterns, mean-periodic functions and irregular sampling, Afr. Diaspora J. Math. 13(1), Special issue (2012), 1–45.
Y. Meyer, Measures with locally finite support and spectrum, Rev. Mat. Iberoam. 33(3) (2017), 1025–1036. DOI: 10.4171/RMI/962
A. Olevskii and A. Ulanovskii, A simple crystalline measure, Preprint (2020). arXiv:2006.12037v2
A. Olevskii and A. Ulanovskii, Fourier quasicrystals with unit masses, C. R Math. Acad. Sci. Paris 358(11-12) (2020), 1207–1211. DOI: 10.5802/crmath.142
L. Schwartz, “Théorie des distributions”, Tome I, Publ. Inst. Math. Univ. Strasbourg 9, Actualit´es Scientifiques et Industrielles [Current Scientific and Industrial Topics] 1091, Hermann & Cie, Paris, 1950
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