Planar convex bodies, Fourier transform, lattice points, and irregularities of distribution
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- by L. Brandolini, A. Iosevich and G. Travaglini PDF
- Trans. Amer. Math. Soc. 355 (2003), 3513-3535 Request permission
Abstract:
Let $B$ be a convex body in the plane. The purpose of this paper is a systematic study of the geometric properties of the boundary of $B$, and the consequences of these properties for the distribution of lattice points in rotated and translated copies of $\rho B$ ($\rho$ being a large positive number), irregularities of distribution, and the spherical average decay of the Fourier transform of the characteristic function of $B$. The analysis makes use of two notions of “dimension” of a convex set. The first notion is defined in terms of the number of sides required to approximate a convex set by a polygon up to a certain degree of accuracy. The second is the fractal dimension of the image of the Gauss map of $B$. The results stated in terms of these quantities are essentially sharp and lead to a nearly complete description of the problems in question.References
- József Beck, Irregularities of distribution. I, Acta Math. 159 (1987), no. 1-2, 1–49. MR 906524, DOI 10.1007/BF02392553
- József Beck and William W. L. Chen, Irregularities of distribution, Cambridge Tracts in Mathematics, vol. 89, Cambridge University Press, Cambridge, 1987. MR 903025, DOI 10.1017/CBO9780511565984
- J. Beck and W. W. L. Chen, Note on irregularities of distribution. II, Proc. London Math. Soc. (3) 61 (1990), no. 2, 251–272. MR 1063047, DOI 10.1112/plms/s3-61.2.251
- J. Beck and W. W. L. Chen, Irregularities of point distribution relative to convex polygons. II, Mathematika 40 (1993), no. 1, 127–136. MR 1239135, DOI 10.1112/S0025579300013759
- Luca Brandolini and Leonardo Colzani, Localization and convergence of eigenfunction expansions, J. Fourier Anal. Appl. 5 (1999), no. 5, 431–447. MR 1755098, DOI 10.1007/BF01261637
- Luca Brandolini and Leonardo Colzani, Decay of Fourier transforms and summability of eigenfunction expansions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29 (2000), no. 3, 611–638. MR 1817712
- Luca Brandolini, Leonardo Colzani, and Giancarlo Travaglini, Average decay of Fourier transforms and integer points in polyhedra, Ark. Mat. 35 (1997), no. 2, 253–275. MR 1478780, DOI 10.1007/BF02559969
- Luca Brandolini, Marco Rigoli, and Giancarlo Travaglini, Average decay of Fourier transforms and geometry of convex sets, Rev. Mat. Iberoamericana 14 (1998), no. 3, 519–560. MR 1681584, DOI 10.4171/RMI/244
- Luca Brandolini and Giancarlo Travaglini, Pointwise convergence of Fejer type means, Tohoku Math. J. (2) 49 (1997), no. 3, 323–336. MR 1464180, DOI 10.2748/tmj/1178225106
- Joaquim Bruna, Alexander Nagel, and Stephen Wainger, Convex hypersurfaces and Fourier transforms, Ann. of Math. (2) 127 (1988), no. 2, 333–365. MR 932301, DOI 10.2307/2007057
- W. W. L. Chen, On irregularities of distribution. III, J. Austral. Math. Soc. Ser. A 60 (1996), no. 2, 228–244. MR 1375588, DOI 10.1017/S1446788700037629
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
- Leonardo Colzani, Fourier transform of distributions in Hardy spaces, Boll. Un. Mat. Ital. A (6) 1 (1982), no. 3, 403–410 (English, with Italian summary). MR 678657
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Ekkehard Krätzel, Lattice points, Mathematics and its Applications (East European Series), vol. 33, Kluwer Academic Publishers Group, Dordrecht, 1988. MR 998378
- Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543, DOI 10.1090/cbms/084
- F. L. Nazarov and A. N. Podkorytov, The behavior of the Lebesgue constants of two-dimensional Fourier sums over polygons, Algebra i Analiz 7 (1995), no. 4, 214–238 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 4, 663–680. MR 1356537
- A. N. Podkorytov, On the asymptotics of the Fourier transform on a convex curve, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 2 (1991), 50–57, 125 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 24 (1991), no. 2, 57–65. MR 1166380
- Burton Randol, On the Fourier transform of the indicator function of a planar set, Trans. Amer. Math. Soc. 139 (1969), 271–278. MR 251449, DOI 10.1090/S0002-9947-1969-0251449-9
- Fulvio Ricci and Giancarlo Travaglini, Convex curves, Radon transforms and convolution operators defined by singular measures, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1739–1744. MR 1814105, DOI 10.1090/S0002-9939-00-05751-8
- Carsten Schütt, The convex floating body and polyhedral approximation, Israel J. Math. 73 (1991), no. 1, 65–77. MR 1119928, DOI 10.1007/BF02773425
- Andreas Seeger and Sarah Ziesler, Riesz means associated with convex domains in the plane, Math. Z. 236 (2001), no. 4, 643–676. MR 1827499, DOI 10.1007/PL00004846
- W. T. Sledd and D. A. Stegenga, An $H^{1}$ multiplier theorem, Ark. Mat. 19 (1981), no. 2, 265–270. MR 650500, DOI 10.1007/BF02384484
- Marysia Tarnopolska-Weiss, On the number of lattice points in a compact $n$-dimensional polyhedron, Proc. Amer. Math. Soc. 74 (1979), no. 1, 124–127. MR 521885, DOI 10.1090/S0002-9939-1979-0521885-3
- A. A. Yudin and V. A. Yudin, Polygonal Dirichlet kernels and growth of Lebesgue constants, Mat. Zametki 37 (1985), no. 2, 220–236, 301 (Russian). MR 784367
- V. A. Judin, Lower bound of the Lebesgue constants, Mat. Zametki 25 (1979), no. 1, 119–122, 159 (Russian). MR 527005
Additional Information
- L. Brandolini
- Affiliation: Dipartimento di Ingegneria, Università di Bergamo, Viale G. Marconi 5, 24044 Dalmine (BG), Italy
- MR Author ID: 294667
- ORCID: 0000-0002-9670-9051
- Email: brandolini@unibg.it
- A. Iosevich
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri
- MR Author ID: 356191
- Email: iosevich@math.missouri.edu
- G. Travaglini
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy
- MR Author ID: 199040
- ORCID: 0000-0002-7405-0233
- Email: travaglini@matapp.unimib.it
- Received by editor(s): February 11, 2002
- Published electronically: April 25, 2003
- Additional Notes: The first and third authors are supported by MURST. The second author is supported by NSF grant DMS00-87339
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 3513-3535
- MSC (2000): Primary 42B10; Secondary 52A10
- DOI: https://doi.org/10.1090/S0002-9947-03-03240-9
- MathSciNet review: 1990161