Numerical Algorithms

, Volume 78, Issue 2, pp 661–672 | Cite as

Remarkable Haar spaces of multivariate piecewise polynomials

Original Paper
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Abstract

Some families of Haar spaces in \(\mathbb {R}^{d},~ d\ge 1,\) whose basis functions are d-variate piecewise polynomials, are highlighted. The starting point is a sequence of univariate piecewise polynomials, called Lobachevsky splines, arised in probability theory and asymptotically related to the normal density function. Then, it is shown that d-variate Lobachevsky splines can be expressed as products of Lobachevsky splines. All these splines have simple analytic expressions and subsets of them are suitable for scattered data interpolation, allowing efficient computation and plain error analysis.

Keywords

Scattered data interpolation Lobachevsky splines Gaussians Positive definite functions 

Mathematics subject classification 2010

65D05 65D07 42A82 

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Notes

Acknowledgments

The author is very grateful to the anonymous referee for many accurate and helpful comments on a draft of this note.

References

  1. 1.
    Allasia, G.: Approssimazione della funzione di distribuzione normale mediante funzioni spline. Statistica XLI(2), 325–332 (1981)MathSciNetGoogle Scholar
  2. 2.
    Allasia, G., Cavoretto, R., De Rossi, A.: Lobachevsky spline functions and interpolation to scattered data. Comp. Appl. Math. 32(1), 71–87 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bates, G.E.: Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya scheme. Ann. Math. Stat. 26, 705–720 (1955)CrossRefMATHGoogle Scholar
  4. 4.
    Bochner, S.: Vorlesungen über Fouriersche Integrale. Akademische Verlagsgesellschaft, Leipzig (1932)MATHGoogle Scholar
  5. 5.
    Bochner, S.: Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math. Ann. 108, 378–410 (1933)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brinks, R.: On the convergence of derivatives of B-splines to derivatives of the Gaussian functions. Comp. Appl. Math. 27(1), 79–92 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Cheney, W., Light, W.: A Course in Approximation Theory. Brooks/Cole, Pacific Grove (2000)MATHGoogle Scholar
  9. 9.
    Curtis, P.C.: n-parameter families and best approximation. Pacific J. Math. 9(4), 1013–1027 (1959)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fasshauer, G.E.: Meshfree Approximation Methods with Matlab. World Scientific Publishing, Singapore (2007)CrossRefMATHGoogle Scholar
  11. 11.
    Fasshauer, G.E., McCourt, M.: Kernel–Based Approximation Methods Using Matlab. World Scientific Publishing, Singapore (2016)MATHGoogle Scholar
  12. 12.
    Gnedenko, B.V.: The Theory of Probability. MIR, Moscow (1976)Google Scholar
  13. 13.
    Haar, A.: Die Minkowskische Geometrie und die Annäherung an stetige Funktionen. Math. Ann. 18, 294–311 (1918)MATHGoogle Scholar
  14. 14.
    Hall, P.: The distribution of means for samples of size N drawn from a population in which the variate takes values between 0 and 1, all such values being equally probable. Biometrika 19(3/4), 240–245 (1927)CrossRefMATHGoogle Scholar
  15. 15.
    Irwin, J.O.: On the frequency distribution of the means of samples from a population having any law of frequency with finite moments, with special reference to Pearson’s Type II. Biometrika 19(3/4), 225–239 (1927)CrossRefMATHGoogle Scholar
  16. 16.
    Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 2, 2nd edn. Wiley (1995)Google Scholar
  17. 17.
    Lobachevsky, N.: Probabilité des résultats moyens tirés d’observations répetées. J. Reine Angew. Math. 24, 164–170 (1842)MathSciNetGoogle Scholar
  18. 18.
    Lukacs, E.: Characteristic Functions, 2nd edn. Griffin, London (1970)Google Scholar
  19. 19.
    Mairhuber, J.C.: On Haar’s theorem concerning Chebyshev approximation problems having unique solutions. Proc. Am. Math. Soc. 7(4), 609–615 (1956)MathSciNetMATHGoogle Scholar
  20. 20.
    Rényi, A.: Calcul des Probabilités. Dunod, Paris (1966)MATHGoogle Scholar
  21. 21.
    Tricomi, F.G.: Una quistione di probabilità. In: Atti Primo Congresso Nazionale di Scienza delle Assicurazioni (Torino, 20–23 Settembre 1928), vol. I, pp. 243–259. Chiantore, Torino (1928)Google Scholar
  22. 22.
    Tricomi, F.G.: Su di una variabile casuale connessa con un notevole tipo di partizioni di un numero intero. Giorn. Istit. Italiano Attuari 2, 455–468 (1931)MATHGoogle Scholar
  23. 23.
    Tricomi, F.G.: Ueber die Summe mehrerer zufälliger Veränderlichen mit konstanten Verteilungsgesetzen. Jahresb. Deutschen Math. Ver. 42, 174–179 (1932)MATHGoogle Scholar
  24. 24.
    Unser, M., Aldroubi, A., Eden, M.: On the asymptotic convergence of B-splines wavelets to Gabor functions. IEEE Trans. Inform. Theory 8(2), 864–872 (1991)MathSciNetMATHGoogle Scholar
  25. 25.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar

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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics “G. Peano”University of TurinTurinItaly

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