Mathematical Notes

, Volume 96, Issue 1–2, pp 228–238 | Cite as

On the Riesz constants for systems of integer translates

  • E. A. Kiselev
  • L. A. Minin
  • I. Ya. Novikov
  • S. M. Sitnik


In this paper, one-parameter families of integer translates of the Gaussian and Lorentz functions are studied. For a Lorentz function, we obtain formulas for the coefficients of the series defining node functions and show that the limit value of node functions is given by a sample function. For systems of translates generated by the Gaussian and Lorentz functions as well as by the node functions related to them, we obtain explicit expressions for the Riesz constants and study the parameter-dependent behavior of these constants. While proving some of the results of this paper, we establish the monotonicity of a special ratio of two Jacobi theta functions, a fact which is of interest in itself.


Riesz constant Gaussian function Lorentz function system of integer translates node function Jacobi theta function Riesz system 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. C. Kashin and A. A. Saakyan, Orthogonal Series (Izd. AFTs, Moscow, 1999) [in Russian].zbMATHGoogle Scholar
  2. 2.
    I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory (Fizmatlit, Moscow, 2005) [in Russian].zbMATHGoogle Scholar
  3. 3.
    N. S. Bakhvalov, N. P. Zhidkov, and G. M. Kobel’kov, Numerical Methods (Nauka, Moscow, 1987) [in Russian].zbMATHGoogle Scholar
  4. 4.
    D. Jankov and T. K. Pogány, “Integral representation of Schlömilch series,” J. Classical Anal. 1(1), 75–84 (2012).Google Scholar
  5. 5.
    Ch. K. Chui, An Introduction to Wavelets, Wavelet Analysis and Its Applications (Academic Press, Boston, MA, 1992; Mir, Moscow, 2001).zbMATHGoogle Scholar
  6. 6.
    O. Christensen, An Introduction to Frames and Riesz Bases, in Appl. Numer. Harmon. Anal. (Birkhäuser Boston, Boston, 2003).Google Scholar
  7. 7.
    A. I. Drobyshev, Foundations of Atomic Spectral Analysis (Izd. St. Petersburg Univ., St. Petersburg, 1997) [in Russian].Google Scholar
  8. 8.
    M. V. Zhuravlev, E. A. Kiselev, L. A. Minin, and S. M. Sitnik, “Jacobi theta-functions and systems of integer shifted Gaussian functions,” J. Math. Sci. (N. Y.) 173(2), 231–241 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 3rd ed. (Cambridge Univ. Press, Cambridge, 1927; GIFML, Moscow, 1963).zbMATHGoogle Scholar
  10. 10.
    V. Maz’ya and G. Schmidt, Approximate Approximations, in Math. Surveys Monogr. (Amer. Math. Soc., Providence, RI, 2007), Vol. 141.Google Scholar
  11. 11.
    M. V. Zhuravlev, “On the Riesz constants for systems of integer translates of the Gaussian function,” Nauchn. Vedom. BelGU Mat. Fiz. 22(5), 39–46 (2011).Google Scholar
  12. 12.
    Th. Schlumprecht and N. Sivakumar, “On the sampling and recovery of bandlimited functions via scattered translates of the Gaussian,” J. Approx. Theory 159(1), 128–153 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    NIST Handbook of Mathematical Functions (Cambridge Univ. Press, Cambridge, 2010).Google Scholar
  14. 14.
    D. F. Lawden, Elliptic Functions and Applications, in Appl. Math. Sci. (Springer-Verlag, New York, 1989), Vol. 80.Google Scholar
  15. 15.
    M. V. Zhuravlev, L. A. Minin, and S. M. Sitnik, “On the computational singularities of interpolation using integer translates of Gaussian functions,” Nauchn. Vedom. BelGU Mat. Fiz. 17/2(13 (68)), 89–99 (2009).Google Scholar
  16. 16.
    A. Yu. Solynin, “Harmonic measure of radial line segments and symmetrization,” Mat. Sb. 189(11), 121–138 (1998) [Sb. Math. 189 (11), 1701–1718 (1998)].CrossRefMathSciNetGoogle Scholar
  17. 17.
    A. Dixit, A. Roy, and A. Zaharescu, “Convexity of quotients of theta functions,” J. Math. Anal. Appl. 386(1), 319–331 (2012).CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    K. Schiefermayr, Some New Properties of Jacobi’s Theta Functions, arXiv: math. CV/1306.6220v1 (2013).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • E. A. Kiselev
    • 1
  • L. A. Minin
    • 1
  • I. Ya. Novikov
    • 1
  • S. M. Sitnik
    • 2
  1. 1.Voronezh State UniversityVoronezhRussia
  2. 2.Voronezh Institute of the Ministry of Internal Affairs of RussiaVoronezhRussia

Personalised recommendations