Kybernetika 49 no. 1, 60-72, 2013

Reference points based recursive approximation

Martina Révayová and Csaba Török


The paper studies polynomial approximation models with a new type of constraints that enable to get estimates with significant properties. Recently we enhanced a representation of polynomials based on three reference points. Here we propose a two-part cubic smoothing scheme that leverages this representation. The presence of these points in the model has several consequences. The most important one is the fact that by appropriate location of the reference points the resulting approximant of two successively assessed neighboring approximants will be smooth. We also show that the considered models provide estimates with appropriate statistical properties such as consistency and asymptotic normality.


asymptotic normality, consistency, approximation model


41A10, 62-07, 62F10, 62J05, 62L12, 65D05, 65D07, 65D10


  1. N. D. Dikoussar: Adaptive projective filters for track finding. Comput. Phys. Commun. 79 (1994), 39-51.   CrossRef
  2. N. D. Dikoussar: Kusochno-kubicheskoje priblizhenije I sglazhivanije krivich v rezhime adaptacii. Comm. JINR, P10-99-168, Dubna 1999.   CrossRef
  3. N. D. Dikoussar and Cs. Török: Automatic knot finding for piecewise-cubic approximation. Mat. Model. T-17 (2006), 3.   CrossRef
  4. N. D. Dikoussar and Cs. Török: Approximation with DPT. Comput. Math. Appl. 38 (1999), 211-220.   CrossRef
  5. S. Haykin: Adaptive Filter Theory. Prentice Hall, 2002   CrossRef
  6. E. A. Nadaraya: On estimating regression. Theory Probab. Appl. 9 (1964 ), 141-142.   CrossRef
  7. Ch. H. Reinsch: Smoothing by spline functions. Numer. Math. 10 (1967), 177-183.   CrossRef
  8. M. Révayová and Cs. Török: Piecewise approximation and neural networks. Kybernetika 43 (2007), 4, 547-559.   CrossRef
  9. B. D. Ripley: Pattern Recognision and Neural Networks. Cambridge University Press, 1996.   CrossRef
  10. Cs. Török: 4-point transforms and approximation. Comput. Phys. Commun. 125 (2000), 154-166.   CrossRef
  11. M. T. Wasan: Stochastic Approximation. Cambridge University Press, 2004.   CrossRef