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A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines

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Abstract

This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.

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References

  1. Basu, S., Pollack, R., Roy, M.F.: Algorithms in real algebraic geometry. In: Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2006)

    Google Scholar 

  2. Cavaretta, A.S., Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Am. Math. Soc. 453, 1–185 (1991)

    MathSciNet  Google Scholar 

  3. Charina, M., Conti, C., Sauer, T.: Regularity of multivariate vector subdivision schemes. Numer. Algorithms 39, 97–113 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Charina, M., Conti, C.: Polynomial reproduction of multivariate scalar subdivision schemes. J. Comput. Appl. Math. (2012). doi:10.1016/j.cam.2012.06.013

    MATH  Google Scholar 

  5. Charina, M., Conti, C., Jetter, K., Zimmermann, G.: Scalar multivariate subdivision schemes and box splines. Comput. Aided Geom. Des. 28(5), 285–306 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Conti, C., Gemignani, L., Romani, L.: From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks. Linear Algebra Appl. 431, 1971–1987 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Conti, C., Gemignani, L., Romani, L.: From approximating to interpolatory non-stationary subdivision schemes with the same generation properties. Adv. Comput. Math. 35(2–4), 217–241 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Conti, C., Hormann, K.: Polynomial reproduction for univariate subdivision schemes of any arity. J. Approx. Theory 163(4), 413–437 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Conti, C., Romani, L.: Algebraic conditions on non-stationary subdivision symbols for exponential polynomial reproduction. J. Comput. Appl. Math. 236, 543–556 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dyn, N., Levin, D.: Subdivision schemes in geometric modelling. Acta Numer. 11, 73–144 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dyn, N., Gregory, J., Levin, D.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graphics 9, 160–169 (1990)

    Article  MATH  Google Scholar 

  12. Rabarison, A.F., de Villiers, J.: On bivariate interpolatory mask symbols. In: Proceedings in Mathematics Subdivision and Refinable Functions, Approximation Algorithms for Complex Systems, part 2, vol. 3, pp. 275–292. Springer, Berlin (2011)

    Chapter  Google Scholar 

  13. Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, New York (1982)

    MATH  Google Scholar 

  14. Ji, H., Riemenschneider, S.D., Shen, Z.: Multivariate compactly supported fundamental refinable functions, duals and biorthogonal wavelets. Stud. Appl. Math. 102, 173–204 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jia, R.Q.: Subdivision schemes in L p spaces. Adv. Comput. Math. 3, 309–341 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jia, R.Q.: Interpolatory subdivision schemes induced by box splines. Appl. Comput. Harmon. Anal. 8(3), 286–292 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Han, B., Jia, R.Q.: Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. Anal. 29, 1177–1199 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lawton, W., Lee, S.L., Shen, Z.: Stability and orthonormality of multivariate refinable functions. SIAM J. Math. Anal. 28, 999–1014 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Newman, M.: Integral matrices. In: Pure and Applied Mathematics, vol. 45. Academic Press, New York (1972)

    Google Scholar 

  20. Riemenschneider, S.D., Shen, Z.: Multidimensional interpolatory subdivision schemes. SIAM J. Numer. Anal. 34, 2357–2381 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sauer, T.: Multivariate refinable functions, differences and ideals: a simple tutorial. J. Comput. Appl. Math. 221(2), 447–459 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Energetica “Sergio Stecco”, Università di Firenze, Viale Morgagni 40/44, 50134, Firenze, Italy

    Costanza Conti

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Luca Gemignani

  3. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via R. Cozzi 53, 20125, Milano, Italy

    Lucia Romani

Authors
  1. Costanza Conti
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  2. Luca Gemignani
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  3. Lucia Romani
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Correspondence to Costanza Conti.

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Communicated by: T. Lyche.

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Conti, C., Gemignani, L. & Romani, L. A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines. Adv Comput Math 39, 395–424 (2013). https://doi.org/10.1007/s10444-012-9285-9

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  • DOI: https://doi.org/10.1007/s10444-012-9285-9

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