Polynomial Smoothing Splines

  • Amir Z. Averbuch
  • Pekka Neittaanmaki
  • Valery A. Zheludev


Interpolating splines is a perfect tool for approximation of a continuous-time signal \(f(t)\) in the case when samples \(x[k]=f(k),\;k\in \mathbb {Z}\) are available. However, frequently, the samples are corrupted by random noise. In such case, the so-called smoothing splines provide better approximation. In this chapter we describe periodic smoothing splines in one and two dimensions. The SHA technique provides explicit expression of such splines and enables us to derive optimal values of the regularization parameters.


  1. 1.
    E.T. Whittaker, On a new method of graduation. Proc. Edinb. Math. Soc. 41, 63–75 (1922)Google Scholar
  2. 2.
    I.J. Schoenberg, Spline functions and the problem of graduation. Proc. Nat. Acad. Sci. U.S.A. 52(4), 947–950 (1964)Google Scholar
  3. 3.
    C.H. Reinsch, Smoothing by spline functions. Numer. Math. 10, 177–183 (1967)Google Scholar
  4. 4.
    J.H. Ahlberg, E.N. Nilson, J.L. Walsh, The theory of splines and their applications (Academic Press, New York, 1987)Google Scholar
  5. 5.
    J.C. Holladay, A smoothest curve approximation. Math. Tables Aids Comput. 11, 233–243 (1957)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Amir Z. Averbuch
    • 1
  • Pekka Neittaanmaki
    • 2
  • Valery A. Zheludev
    • 1
  1. 1.School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  2. 2.Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland

Personalised recommendations