Fourier Techniques in the Theory of Irregularities of Point Distribution

  • W. W. L. Chen
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


By the use of two examples, we discuss the techniques of Fourier analysis in the study of problems in irregularities of point distribution. Such techniques include classical Fourier series and transforms as well as Fourier-Walsh analysis and wavelet analysis. We also show that often the Fourier analysis can be combined with ideas and techniques in number theory, geometry, probability theory, group theory, characters and duality.


Characteristic Function Point Distribution Diophantine Approximation Walsh Function Fourier Technique 
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  1. 1.
    Beck, J., Irregularities of distribution, Acta Math. 159 (1987), 1–49.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Beck, J. and Chen, W.W.L., Note on irregularities of distribution II, Proc. London Math. Soc. 61 (1990), 251–272.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beck, J. and Chen, W.W.L., Irregularities of point distribution relative to convex polygons III, J. London Math. Soc. 56 (1997), 222–230.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Brandolini, L., Colzani, L., and Travaglini, G., Average decay of Fourier transforms and integer points in polyhedra, Ark. Mat. 35 (1997), 253–275.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brandolini, L., Iosevich, Á., and Travaglini, G., Planar convex bodies, Fourier transform, lattice points and irregularities of distribution, Trans. Amer. Math. Soc. 355 (2003), 3513–3535.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Brandolini, L., Rigoli, M., and Travaglini, G., Average decay of Fourier transforms and geometry of convex sets, Rev. Mat. Iberoam. 14 (1998), 519–560.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Chen, W.W.L., On irregularities of distribution II, Quart. J. Math. Oxford 34 (1983), 257–279.MATHCrossRefGoogle Scholar
  8. 8.
    Chen, W.W.L. and Skriganov, M.M., Davenport’s theorem in the theory of irregularities of point distribution, Zapiski Nauch. Sem. POMI 269 (2000), 339–353.Google Scholar
  9. 9.
    Chen, W.W.L. and Skriganov, M.M., Explicit constructions in the classical mean squares problem in irregularities of point distribution, J. Reine Angew. Math. 545 (2002), 67–95.MathSciNetMATHGoogle Scholar
  10. 10.
    Chen, W.W.L. and Skriganov, M.M., Orthogonality and digit shifts in the classical mean squares problem in irregularities of point distribution (preprint).Google Scholar
  11. 11.
    Daubechies, I., Ten Lectures on Wavelets, SIAM, 1992.Google Scholar
  12. 12.
    Davenport, H., Note on irregularities of distribution, Mathematika 3 (1956), 131–135.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Davenport, H., A note on diophantine approximation II, Mathematika 11 (1964), 50–58.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Fine, N.J., On the Walsh functions, Trans. American Math. Soc. 65 (1949), 373–414.CrossRefGoogle Scholar
  15. 15.
    Halton, J.H. and Zaremba, S.K., The extreme and L 2 discrepancies of some plane sets, Monats. Math. 73 (1969), 316–328.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Lidl, R. and Niederreiter, H., Finite Fields, Addison-Wesley, 1983.Google Scholar
  17. 17.
    Meyer, Y., Wavelets and Operators, Cambridge University Press, 1992.Google Scholar
  18. 18.
    Pollington, A.D., Haar wavelets and irregularities of distribution (manuscript).Google Scholar
  19. 19.
    Price, J.J., Certain groups of orthonormal step functions, Canadian J. Math. 9 (1957), 413–425.MATHCrossRefGoogle Scholar
  20. 20.
    Roth, K.F., On irregularities of distribution, Mathematika 1 (1954), 73–79.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Roth, K.F., On irregularities of distribution III, Acta Arith. 35 (1979), 373–384.MathSciNetMATHGoogle Scholar
  22. 22.
    Roth, K.F., On irregularities of distribution IV, Acta Arith. 37 (1980), 67–75.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • W. W. L. Chen
    • 1
  1. 1.Department of MathematicsMacquarie UniversitySydneyAustralia

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