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On minimax cardinal spline interpolation

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Abstract

It is shown that in the problem of cardinal interpolation, spline interpolants of various degrees are R-minimax, with respect to corresponding Sobolev and Hardy functional classes, under restrictions determined by the interference between their oscillating variance and bias. The results raise a natural question: what degrees of interpolating splines are more appropriate, for given Sobolev or Hardy classes? It turns out that the scales of such functional classes can be divided into “very smooth” and “not-so-smooth” subfamilies, whereby “very smooth” classes can benefit from higher degrees of cardinal splines, and vice versa.

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References

  • Bateman H, Erdélyi A (1955) Higher transcendental functions, vol 1. McGraw-Hill, New York

    MATH  Google Scholar 

  • Cho J (2007) Optimal design in regression and spline smoothing, Ph.D. Thesis, Queen’s University

  • de Boor Carl (1976) On the cardinal interpolant to \(e^{iut}\). SIAM J Math Anal 7:930–941

    Article  MathSciNet  Google Scholar 

  • Euler L (1949) Differential analysis. Gostechizdat, St. Petersburg

    Google Scholar 

  • Feller W (1971) An introduction to probability theory and its applications, vol 2. Wiley, New York

    MATH  Google Scholar 

  • Fichtengolts GM (1966) A course in analysis, vol 3. Nauka, Moscow (in Russian)

    Google Scholar 

  • Finch SR (2003) Mathematical constants. Cambridge University Press, Cambridge, pp 255–257

    MATH  Google Scholar 

  • Gradshteyn IS, Ryzhik IM (1994) Table of integrals, series, and products, 5th edn. Academic Press, New York

    MATH  Google Scholar 

  • He Tian-**ao (2012) Eulerian plolynomials and B-splines. J Comput Appl Math 236:3763–3773

    Article  MathSciNet  Google Scholar 

  • Korneichuk N (1991) Exact constants in approximation theory. Cambridge University Press, New York

    Book  Google Scholar 

  • Levit B (2013) Some new perspectives in best approximation and interpolation of random data. Math Methods Stat 22:165–192

    Article  MathSciNet  Google Scholar 

  • Levit B (2015) Variance-optimal data approximation using the Abel–Jacobi functions. Math Methods Stat 24:37–54

    Article  MathSciNet  Google Scholar 

  • Levit B (2016) Optimal methods of interpolation in nonparametric regression. Math Methods Stat 25:235–261

    Article  MathSciNet  Google Scholar 

  • Levit B (2018) On optimal cardinal interpolation. Math Methods Stat 27:245–267

    Article  MathSciNet  Google Scholar 

  • Markushevich AI (1985) Theory of functions of a complex variable. Chelsea Publisher, New York

    Google Scholar 

  • Micchelli CA (1976) Cardinal \({\cal{L}}\)-splines. In: Karlin A, Misshelli CA, Pinkkus A, Schoenberg IJ (eds) Studies in spline functions and approximation theory. Academic Press, Cambridge, pp 203–250

    Google Scholar 

  • Schoenberg IJ (1973a) Cardinal interpolation and spline functions IV. The exponential Euler splines. In: International Series Numerical Math., Birkhäuser. (Reprinted in: I.J. Schoenberg, Selected papers, vol 2, pp 88–110, Springer, Berlin, 1988)

  • Schoenberg IJ (1973b) Cardinal spline interpolation. SIAM, Philadelphia

    Book  Google Scholar 

  • Schoenberg IJ, Sharma A (1971) The interpolatory background of the Euler–Maclaurin formula. Bull Am Math Soc 77:1034–1038

    Article  Google Scholar 

  • Schumaker LL (1981) Spline functions: basic theory. Wiley, New York

    MATH  Google Scholar 

  • Vainikko G (2009) Error estimates for the cardinal spline interpolation. J Anal Appl 28:205–222

    MathSciNet  MATH  Google Scholar 

  • Volkovyskii LI, Lunts GL, Aramanovich IG (1991) A collection of problems on complex analysis. Dover, New York

    MATH  Google Scholar 

  • Zygmund A (2002) Trigonometic series, vol 1–2, 3rd edn. Cambridge University Press, Cambridge

    Google Scholar 

Download references

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  1. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, K7L 3N6, Canada

    B. Levit

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Levit, B. On minimax cardinal spline interpolation. Stat Inference Stoch Process 25, 17–41 (2022). https://doi.org/10.1007/s11203-021-09261-5

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  • DOI: https://doi.org/10.1007/s11203-021-09261-5

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