Science in China Series A: Mathematics

, Volume 43, Issue 10, pp 1075–1082 | Cite as

Riesz basis, Paley-Wiener class and tempered splines

  • Gensun Fang
  • Jingfan Long


The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ℒ2m (T, ℝ) ∩ L2 (ℝ) which is the space of polynomial splines with irregularly distributed nodesT = {t j } j ∈ℤ, where {t j }j∈ℤ is a real sequence such that {e it ξ} j }j ∈ℤ constitutes a Riesz basis for L2([ −π,π]). From these results, the asymptotic relation
is proved, where B π,2 denotes the set of all functions from L2( R) which can be continued to entire functions of exponential type ⪯ ϕ, i.e. the classical Paley-Wiener class.


Riesz basis entire functions tempered splines asymptotic relation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akhieser, N. I., Theory of Approximation, New York:Ungar, 1956.Google Scholar
  2. 2.
    Lyubarskii, Y., Madych, W. R., The recovery of irregular sampled band limited functions via tempered splines, J. of Functional Analysis, 1994, 125:201.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Seip, K., On the connection between exponential bases and certain related sequence in L2 - ϕ, ϕ), J. of Functional Analysis, 1995, 130:131.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    DeVore, R. A., Lorentz, G. G., Constructive Approximation, Berlin:Springer-Verlag, 1993.MATHGoogle Scholar
  5. 5.
    Yong, R., An Introduction to Nonhannonic Fourier Series, New York:Academic Press, 1980.Google Scholar
  6. 6.
    Zygmund, A., Trigonometric Series, 2nd ed., London:Cambridge Univ. Press, 1959.MATHGoogle Scholar
  7. 7.
    Richards, F. B., Schoenberg, J. J., Notes on spline functions IV: A cardinal spline analogue of the theorem of brothers Markov, Israel J. Math., 1973, 16:94.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Fang Gensun, Approximation properties of entire functions of exponential type, J. Math. Anal. Appl., 1996, 201:642.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fang Gensun, On an asymptotic connection between approximation by the cardinal splines and entire functions of exponential type, Chinese Science Bulletin, 1996, 41(4):265.MATHMathSciNetGoogle Scholar
  10. 10.
    Korneichuk, N., Exact Constants in Approximation Theory, Cambridge:Cambridge Univ. Press, 1991.MATHGoogle Scholar
  11. 11.
    Timan, A. F., Theory of Approximation of Functions of a Real Variable, New York:MacMillan, 1963.MATHGoogle Scholar
  12. 12.
    Tikhomirov, V. M., Some Question of Approximation Theory, Moscow:Izdat Moscow 6s. Univ., 1976.Google Scholar

Copyright information

© Science in China Press 2000

Authors and Affiliations

  1. 1.Department of MathematicsBeijing Normal UniversityBeijingChina

Personalised recommendations