Nonuniform B-Splines and B-Wavelets

  • Ewald Quak
Part of the Mathematics and Visualization book series (MATHVISUAL)


The purpose of this tutorial is to give a basic introduction into the refinement of nonuniform B-splines, a finite-dimensional multiresolution analysis based on nonuniform B-splines and nonuniform B-wavelets as bases of the corresponding wavelet spaces.


Minimal Support Subdivision Scheme Riesz Basis Bernstein Polynomial Minimal Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    G.-P. Bonneau. BLaC wavelets and non-nested wavelets. This volume.Google Scholar
  2. 2.
    C. de Boor. A Practical Guide to Splines. Springer, New York, 1978.zbMATHCrossRefGoogle Scholar
  3. 3.
    C. K. Chui. An Introduction to Wavelets. Academic Press, Boston, 1992.zbMATHGoogle Scholar
  4. 4.
    C. K. Chui and E. Quak. Wavelets on a bounded interval. Numerical Methods in Approximation Theory, ISNM 105, D. Braess and L. L. Schumaker (eds.), Birkhäuser, Basel, 1992, 53–75.CrossRefGoogle Scholar
  5. 5.
    A. Cohen, I. Daubechies, and P. Vial. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmonic Anal. 1, 1993, 54–81.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    E. Cohen, T. Lyche, and R. Riesenfeld. Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics. Comp. Graphics and Image Proc. 14, 1980, 87–111.CrossRefGoogle Scholar
  7. 7.
    M. Conrad and J. Prestin. Multiresolution on the sphere. This volume.Google Scholar
  8. 8.
    I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992.zbMATHCrossRefGoogle Scholar
  9. 9.
    N. Dyn. Interpolatory subdivision schemes. This volume.Google Scholar
  10. 10.
    N. Dyn. Analysis of convergence and smoothness by the formalism of Laurent polynomials. This volume.Google Scholar
  11. 11.
    G. Faber. Über stetige Funktionen. Math. Ann. 66, 1909, 81–94.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    G. Farin. Curves and Surfaces for Computer-Aided Geometric Design, 4th edition, Academic Press, San Diego, 1998.Google Scholar
  13. 13.
    M. Floater, E. Quak, and M. Reimers. Filter bank algorithms for piecewise linear prewavelets on arbitrary triangulations. J. Comput. Appl. Math. 119, 2000, 185–207.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    J. Hoschek and D. Lasser. Fundamentals of Computer Aided Geometric Design. AKPeters, Wellesley, 1993.zbMATHGoogle Scholar
  15. 15.
    A. Iske. Scattered data modelling using radial basis functions. This volume.Google Scholar
  16. 16.
    T. Lyche and K. Mørken. Making the Oslo algorithm more efficient. SIAM J. Numer. Anal. 8, 1988, 185–208.zbMATHGoogle Scholar
  17. 17.
    T. Lyche and K. Mørken. Spline-wavelets of minimal support. Numerical Methods in Approximation Theory, ISNM 105, D. Braess and L. L. Schumaker (eds.), Birkhäuser, Basel, 1992, 177–194.CrossRefGoogle Scholar
  18. 18.
    T. Lyche, K. Mørken, and E. Quak. Theory and algorithms for nonuniform spline wavelets. Multivariate Approximation Theory and Applications, N. Dyn, D. Leviatan, D. Levin, and A. Pinkus (eds.), Cambridge University Press, 2001, 152–187.Google Scholar
  19. 19.
    J. Mikkelsen, P. Oja, and E. Quak. Stability of piecewise linear wavelets. Preprint, 2001.Google Scholar
  20. 20.
    P. Oja and E. Quak. An example concerning the L p-stability of piecewise linear B-wavelets. To appear in Algorithms for Approximation 4, I. Anderson and J. Levesley (eds.).Google Scholar
  21. 21.
    P. Oswald. Multilevel Finite Element Approximation: Theory and Applications. Teubner, Stuttgart, 1994.zbMATHCrossRefGoogle Scholar
  22. 22.
    E. Quak and N. Weyrich. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval. Appl. Comput. Harmonic Anal. 1, 1994, 217–231.MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    M. Sabin. Subdivision of box-splines. This volume.Google Scholar
  24. 24.
    M. Sabin. Eigenanalysis and artifacts of subdivision curves and surfaces. This volume.Google Scholar
  25. 25.
    I. J. Schoenberg. Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 4, 1946, Part A: 45–99, Part B: 112–141.MathSciNetGoogle Scholar
  26. 26.
    L. L. Schumaker. Spline Functions: Basic Theory. Wiley, New York, 1981.zbMATHGoogle Scholar
  27. 27.
    A. Sommerfeld. Eine besonders anschauliche Ableitung des Gaußschen Fehlergesetzes. Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstag. Verlag von Johann Ambrosius Barth, 1904, 848–859.Google Scholar
  28. 28.
    E. Stollnitz, T. DeRose, and D. Salesin. Wavelets in Computer Graphics, Morgan Kaufmann, San Francisco, 1996.Google Scholar
  29. 29.
    W. Sweldens and P. Schröder. Building your own wavelets at home. Wavelets in Computer Graphics, ACM SIGGRAPH Course notes, 1996.Google Scholar
  30. 30.
    F. Zeilfelder. Scattered data fitting with bivariate splines. This volume.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ewald Quak
    • 1
  1. 1.SINTEF Applied MathematicsOsloNorway

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