Asymptotics of variance of the lattice point count
Tóm tắt
The variance of the number of lattice points inside the dilated bounded set rD with random position in ℝ
d
has asymptotics ∼ r
d−1 if the rotational average of the squared modulus of the Fourier transform of the set is O(ϰ
−d−1). The asymptotics follow from Wiener’s Tauberian theorem.
Tài liệu tham khảo
S. Bochner and K. Chandrasekharan: Fourier transforms. Princeton University Press, 1949.
L. Brandolini, S. Hofmann and A. Iosevich: Sharp rate of average decay of Fourier transform of a bounded set. Geom. Func. Anal. 13 (2003), 671–680.
J. Janáček: Variance of periodic measure of bounded set with random position. Comment. Math. Univ. Carolinae 47 (2006), 473–482.
D. G. Kendall: On the number of lattice points inside a random oval. Quarterly J. Math. 19 (1948), 1–26.
D. G. Kendall and R. A. Rankin: On the number of points of a given lattice in a random hypersphere. Quarterly J. Math., 2nd Ser. 4 (1953), 178–189.
B. Matérn: Precision of area estimation: a numerical study. J. Microsc. 153 (1989), 269–283.
G. Matheron: Les variables regionalisés et leur estimation. Masson et CIE, Paris, 1965.
R. C. Rao: Linear statistical inference and its applications. 2nd ed., John Wiley & Sons, New York, 1973.
J. Rataj: On set covariance and three-point test sets. Czech. Math. J. 54 (2004), 205–214.
G. N. Watson: A treatise on the theory of Bessel functions. 2nd edition, Cambridge University Press, 1922.
W. Rudin: Functional Analysis. McGraw-Hill Book Company, 1973.
A. Varchenko: Number of lattice points in families of homothetic domains in ℝn. Func. Anal. Appl. 17 (1983), 79–83.
N. Wiener: The Fourier integral and certain of its applications. Dover Publications Inc., New York, 1933.