Advertisement

On Some Characteristics of Uniformity of Distribution and Their Applications

  • Igor E. Shparlinski
Part of the Mathematics and Its Applications book series (MAIA, volume 325)

Abstract

We consider some relatively new characteristics of uniformity of distribution of sequences that are not widely known, and show their connections to several classical measures such as discrepancy and exponential sums. They are connected to several problems from quite different areas such as choosing parameters of linear iteration processes for solving system of linear equations, choosing knots for polynomial interpolation, estimating the size of Varshamov codes correcting asymmetrical errors in binary channels.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Beck, The modulus of polynomials with zeros on the unit circle: A problem of Erdös, Ann. Math. 134 (1991), 609–651.CrossRefGoogle Scholar
  2. [2]
    J. Beck and W. W. L. Chen, Irregularities of distribution, Cambridge: Cambridge Univ. Press, 1987.zbMATHCrossRefGoogle Scholar
  3. [3]
    P. Borwein and C. Ingalls, The Prouchet—Tarry—Esscott problem revisited, L’Enseign. Math. 40 (1994), 3–27.zbMATHGoogle Scholar
  4. [4]
    W. W. L. Chen, Upper bounds in irregularities of distribution, Macquarie University Report 92–103 (1992), 1–74.Google Scholar
  5. [5]
    P. Erdös and G. Szekeres, On the product ru:_, (1 — Zak), Acad. Serbe Sci., Publ. Inst. Math. 13 (1959), 29–34.MathSciNetzbMATHGoogle Scholar
  6. [6]
    P. Erdös and R. Turan, On the uniformity dense distribution of certain sequences of points, Ann. Math. 41 (1940), 162–173.CrossRefGoogle Scholar
  7. [7]
    B. Fischer and L. Riesler, A stable Richardson iteration method for complex linear systems, Numer. Math. (1988) 54, 225–242.zbMATHGoogle Scholar
  8. [8]
    B. Fischer and L. Riesler, Newton interpolation in Fejer and Chebyshev Points, Math. of Comp. 53 (1989), 265–278.zbMATHGoogle Scholar
  9. [9]
    M. N. Kolountzakis, On nonnegative cosine polynomials with nonnegative, integral coefficients, Proc. Amer. Math. Soc. 120 (1994), 157–163.MathSciNetCrossRefGoogle Scholar
  10. [10]
    S. V. Konyagin, On estimates of Gaussian sums and Waring problem modulo a prime, Proc. Math. Inst. Russian Acad. Sci. 198 (1992), 111–124 (in Russian).Google Scholar
  11. [11]
    L. L. Kosachevskaja, V. V. Romanovcev and I. Shparlinski, On some iteration processes in the numerical solution of systems of linear algebraic equations, Zhurnal Vychisl. Matem. i Matem. Fiziki (J. Comp. Math. and Math. Phys.) 22 (1982), 1504–1508 (in Russian).Google Scholar
  12. [12]
    L. L. Kosachevskaja, V. V. Romanovcev and I. Shparlinski, On some sequences of the iteration parameters, Zhurnal Vychisl. Matem. i Matem. Fiziki (J. Comp. Math. and Math. Phys.) 25 (1985), 136–140 (in Russian).Google Scholar
  13. L. L. Kosachevskaja, V. V. Romanovcev and I. Shparlinski, On the choice of parameters in the alternating direction method,in: Proc. All-Union Workshop `Problems of Optimization of Computation’ Simferopol, 1987, pp. 112–113 (in Russian).Google Scholar
  14. [14]
    L. L. Kosachevskaja, V. V. Romanovcev, I. Shparlinski and A. N.Vystavkin, A new improved algorithm for the iterative solution of a system of linear algebraic equations, Computer Physics Commun. 27 (1982), 87–89.CrossRefGoogle Scholar
  15. [15]
    L. L. Kosachevskaja and I. Shparlinski, On the rate of convergence of some interpolation processes, Zhurnal Vychisl. Matem. i Matem. Fiziki (J. Comp. Math. and Math. Phys.) 24 (1984), 458–461 (in Russian).Google Scholar
  16. [16]
    L. Kuipers and H. Niederreiter, Uniform distribution of sequences New York: Wiley, 1974.Google Scholar
  17. [17]
    V. I. Lebedev and F. I. Finogenov, A solution of the problem of ordering of parameters in Chebyshev’s iteration processes, Zhurnal Vychisl. Matem. i Matem. Fiziki (J. Comp. Math. and Math. Phys.) 13 (1973), 18–33 (in Russian).zbMATHGoogle Scholar
  18. [18]
    V. I. Lebedev and F. I. Finogenov, On using ordering Chebyshev’s parameters in iteration processes, Zhurnal Vychisl. Matem. i Matem. Fiziki (J. Comp. Math. and Math. Phys.) 16 (1976), 895–907 (in Russian).zbMATHGoogle Scholar
  19. [19]
    C. N. Linden, The modulus of polynomials with zeros on the unit circle, Bull. Lond. Math. Soc. 9 (1977), 65–69.zbMATHCrossRefGoogle Scholar
  20. [20]
    L. Mazur, On some codes correcting asymmetrical errors, Problemy Peredachi Informacii (Probi. of Inform. Transmission) 10 (1974), 40–46 (in Russian).zbMATHGoogle Scholar
  21. [21]
    E. S. Nikolaev and A. A. Samarski, The choice of iterative parameters in the Richardson method, Zhurnal Vychisl. Matem. i Matem. Fiziki (J. Comp. Math. and Math. Phys.) 12 (1972), 960–973 (in Russian).MathSciNetzbMATHGoogle Scholar
  22. [22]
    A. M. Odlyzko, Minima of cosine sums and maxima of polynomials on the unit circle, J. London Math. Soc. 26 (1982), 412–420.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    A. M. Odlyzko and R. P. Stanley, Enumeration of power sums modulo a prime, J. Number Theory 10 (1978), 263–272.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    L. Riesler, The ordering of tridiagonal matrices in the cyclic reduction method for Poisson’s equation, Numer. Math. 56 (1989), 215–227.zbMATHGoogle Scholar
  25. [25]
    I. Shparlinski, On a property of a sequence of real numbers and the rate of convergence of some interpolation processes, in: Proc. 2 All-Union Conf. on Methods of Comp. Math. Krasnojarsk: Preprint of Comp. Centre of Siberian Depart. of Acad. Sci. of USSR, 1982, pp. 47–48 (in Russian).Google Scholar
  26. [26]
    I. Shparlinski, On the rate of convergency of the Newton interpolation process and the size of some codes, Uspechi Matem. Nauk (Advances in Math.) 39 (1984), 205–206 (in Russian).zbMATHGoogle Scholar
  27. [27]
    I. Shparlinski, On some generalizations of Chebyshev polynomials, Sibirskiy Matem. Zhurnal (Siberian Math. J. 31 (1990), 217–218 (in Russian).zbMATHGoogle Scholar
  28. [28]
    I. Shparlinski, On bounds of Gaussian sums, Matem. Zametki (Math. Notes) 50 (1991), 122–130 (in Russian).zbMATHGoogle Scholar
  29. [29]
    C. L. Siegel, The trace of totally positive and real algebraic integers, Ann. of Math. 46 (1945), 302–312.MathSciNetCrossRefGoogle Scholar
  30. [30]
    R. R. Varshamov, A class of codes for asymmetric channels and a problem from the additive theory of numbers, IEEE Trans. Inform. Theory 19 (1972), 92–95.Google Scholar
  31. [31]
    Yu. V. Vorob’ev, A random iteration process in the method of alternating directions, Zhurnal Vychisl. Matem. i Matem. Fiziki (J. Comp. Math. and Math. Phys.) 8 (1968), 458–461 (in Russian).Google Scholar
  32. [32]
    E. Wachspress, Optimum alternating direction implicit iteration parameters for a model problem, J. SIAM 10 (1962), 339–350.zbMATHGoogle Scholar
  33. [33]
    G. Wagner, On a problem of Erdös in Diophantine approximation, Bull. Lond. Math. Soc. 12 (1980), 81–88.CrossRefGoogle Scholar
  34. [34]
    G. Wagner, Erdös-Turan inequalities for distance function on spheres, Michigan Math. J. 39 (1992), 17–34.Google Scholar
  35. [35]
    A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. USA 34 (1948), 204–207.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Igor E. Shparlinski
    • 1
  1. 1.School of Mathematics, Physics, Computing and ElectronicsMacquarie UniversitySydneyAustralia

Personalised recommendations