Skip to main content
SpringerLink
Log in

Remarkable Haar spaces of multivariate piecewise polynomials

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Allasia, G.: Approssimazione della funzione di distribuzione normale mediante funzioni spline. Statistica XLI(2), 325–332 (1981)

    MathSciNet  Google Scholar 

  2. Allasia, G., Cavoretto, R., De Rossi, A.: Lobachevsky spline functions and interpolation to scattered data. Comp. Appl. Math. 32(1), 71–87 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  4. Bochner, S.: Vorlesungen über Fouriersche Integrale. Akademische Verlagsgesellschaft, Leipzig (1932)

    MATH  Google Scholar 

  5. Bochner, S.: Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math. Ann. 108, 378–410 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  8. Cheney, W., Light, W.: A Course in Approximation Theory. Brooks/Cole, Pacific Grove (2000)

    MATH  Google Scholar 

  9. Curtis, P.C.: n-parameter families and best approximation. Pacific J. Math. 9(4), 1013–1027 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fasshauer, G.E.: Meshfree Approximation Methods with Matlab. World Scientific Publishing, Singapore (2007)

    Book  MATH  Google Scholar 

  11. Fasshauer, G.E., McCourt, M.: Kernel–Based Approximation Methods Using Matlab. World Scientific Publishing, Singapore (2016)

    MATH  Google Scholar 

  12. Gnedenko, B.V.: The Theory of Probability. MIR, Moscow (1976)

  13. Haar, A.: Die Minkowskische Geometrie und die Annäherung an stetige Funktionen. Math. Ann. 18, 294–311 (1918)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  16. Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, vol. 2, 2nd edn. Wiley (1995)

  17. Lobachevsky, N.: Probabilité des résultats moyens tirés d’observations répetées. J. Reine Angew. Math. 24, 164–170 (1842)

    MathSciNet  Google Scholar 

  18. Lukacs, E.: Characteristic Functions, 2nd edn. Griffin, London (1970)

    Google Scholar 

  19. Mairhuber, J.C.: On Haar’s theorem concerning Chebyshev approximation problems having unique solutions. Proc. Am. Math. Soc. 7(4), 609–615 (1956)

    MathSciNet  MATH  Google Scholar 

  20. Rényi, A.: Calcul des Probabilités. Dunod, Paris (1966)

    MATH  Google Scholar 

  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)

  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)

    MATH  Google Scholar 

  23. Tricomi, F.G.: Ueber die Summe mehrerer zufälliger Veränderlichen mit konstanten Verteilungsgesetzen. Jahresb. Deutschen Math. Ver. 42, 174–179 (1932)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  25. Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Department of Mathematics “G. Peano”, University of Turin, via Carlo Alberto 10, 10123, Turin, Italy

    Giampietro Allasia

Authors
  1. Giampietro Allasia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giampietro Allasia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allasia, G. Remarkable Haar spaces of multivariate piecewise polynomials. Numer Algor 78, 661–672 (2018). https://doi.org/10.1007/s11075-017-0394-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-017-0394-x

Keywords

Mathematics subject classification 2010

Search

Navigation