Journal of Statistical Theory and Practice

, Volume 2, Issue 3, pp 465–474 | Cite as

Asymptotic Optimality of Periodic Spline Interpolation in Non-parametric Regression

  • Jaerin ChoEmail author
  • Boris Levit


A class of interpolation type estimates based on the so-called periodic Lagrange splines is considered. Asymptotic rate optimality of these estimates is established for periodic Sobolev classes. Moreover, it is shown that these estimates are asymptotically optimal to the constant for certain classes of periodic analytic functions. An additional advantage of these estimates is a non-asymptotic upper risk bound which can be used, in principle, with any number of observations.

AMS Subject Classification

Primary 62G08 secondary 65D07 


Non-parametric estimation periodic Lagrange spline Sobolev classes analytic functional classes 


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  1. Arcangéli, R., de Silanes, M., Torrens, J., 2007. An extension of a bound for functions in Sobolev spaces, with application to (m,s)-spline interpolation and smoothing. Numer. Math., 107, 181–211.MathSciNetCrossRefGoogle Scholar
  2. de Boor, C., 1976. On the cardinal spline interpolant to e iut. SIAM J. Math. Anal, 7, 930–941.MathSciNetCrossRefGoogle Scholar
  3. Cai, T.T., 2003. Rate of convergence and adaptation over Besov spaces under pointwise risk. Statistica Sinica, 13, 881–902.MathSciNetzbMATHGoogle Scholar
  4. Cho, J., Levit, B., 2008. Cardinal splines in nonparametric regression. Math. Methods Statist., 17, 19–34.MathSciNetCrossRefGoogle Scholar
  5. Korneichuk, N.P., 1991. Exact Constants in Approximation Theory. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  6. Karlin, S., Studden, W.J., 1966. Tchebycheff Systems: With Applications in Analysis and Statistics. Wiley, NY.zbMATHGoogle Scholar
  7. Lee, D., 1986. A simple approach to cardinal lagrange and periodic lagrange splines. J. Approx. Theory, 47, 93–100.MathSciNetCrossRefGoogle Scholar
  8. Lee, D., 1992. On a minimal property of cardinal and periodic lagrange splines. J. Approx. Theory, 70, 335–338.MathSciNetCrossRefGoogle Scholar
  9. Levit, B., Stepanova, N., 2004. Efficient estimation of multivariate analytic functions in cube-like domains. Math. Methods Statist., 13, 253–281.MathSciNetzbMATHGoogle Scholar
  10. Schoenberg, I.J., 1973. Cardinal Spline Interpolation. New York: SIAM.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2008

Authors and Affiliations

  1. 1.Department of System Management EngineeringSungkyunkwan UniversitySuwonKorea
  2. 2.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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