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Lattice Point Problem and Questions of Estimation and Detection of Smooth Multivariate Functions

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Let N d (m) be the number of points of the integer lattice that belong to a d-dimensional ball of radius m (in the l 1- and l 2-norms). The aim of the paper is to study the asymptotic behavior of N d (m) as d → ∞, m → ∞. It is shown that if d tends to infinity much faster than m, then the asymptotic is different from the asymptotic volume of a d-dimensional ball of radius m. Bibliography: 6 titles.

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References

  1. F. Gӧtze, “Lattice point problem and the Central Limit Theorem in Euclidean Spaces,” Documenta Math. Extra Volume ICM, III, 245–255 (1998).

  2. I. A. Ibragimov and R. Z. Khasminskiǐ, “Asymptotic properties of some nonparametric estimators in a Gaussian white noise,” in: Proc. 3rd Summer School on Probab. Theory and Math. Statist., Varna 1978, Sofia (1980), pp. 31–64.

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Authors and Affiliations

  1. Itmo University, St.Petersburg, Russia

    I. A. Suslina

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  1. I. A. Suslina
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Correspondence to I. A. Suslina.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 431, 2014, pp. 198–208.

Translated by the author.

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Suslina, I.A. Lattice Point Problem and Questions of Estimation and Detection of Smooth Multivariate Functions. J Math Sci 214, 554–561 (2016). https://doi.org/10.1007/s10958-016-2798-x

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  • DOI: https://doi.org/10.1007/s10958-016-2798-x

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