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Multivariate Refinable Functions of High Approximation Order Via Quotient Ideals of Laurent Polynomials

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Abstract

We give an algebraic interpretation of the well-known “zero-condition” or “sum rule” for multivariate refinable functions with respect to an arbitrary scaling matrix. The main result is a characterization of these properties in terms of containment in a quotient ideal, however not in the ring of polynomials but in the ring of Laurent polynomials.

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References

  1. W.W. Adams and P. Loustaunau, An Introduction to Groebner Bases (Amer. Math. Soc., Providence, RI, 1994).

    Google Scholar 

  2. A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93(453) (1991).

  3. D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms, Undergraduate Texts in Mathematics (Springer, New York, 1996).

    Google Scholar 

  4. D. Cox, J. Little and D. O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, Vol. 185 (Springer, New York, 1998).

    Google Scholar 

  5. W. Dahmen and C.A. Micchelli, Biorthogonal wavelet expansion, Constr. Approx. 13 (1997) 294–328.

    Google Scholar 

  6. K. Jetter and G. Plonka, A survey on L 2-approximation order from shift-invariant spaces, in: Multivariate Approximation and Applications, eds. N. Dyn, D. Leviatan, D. Levin and A. Pinkus (2001) pp. 73–111.

  7. R.Q. Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998) 647–665.

    Google Scholar 

  8. V. Latour, J. Müller and W. Nickel, Stationary subdivision for general scaling matrices, Math. Z. 227 (1998) 645–661.

    Google Scholar 

  9. M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities (Weber & Schmidt, Prindle, 1969); paperback reprint (Dover, New York, 1992).

    Google Scholar 

  10. C.A. Micchelli and T. Sauer, On vector subdivision, Math. Z. 229 (1998) 621–674.

    Google Scholar 

  11. T. Sauer, Polynomial interpolation, ideals and approximation order of refinable functions, Proc. Amer. Math. Soc. 130 (2002) 3335–3347.

    Google Scholar 

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Authors and Affiliations

  1. FB Mathematik, Universität Dortmund, Vogelpothsweg 87, D-44221, Dortmund, Germany

    H. Michael Möller

  2. Lehrstuhl für Numerische Mathematik, Justus-Liebig-Universität Gießen, Heinrich-Buff-Ring 44, D-35392, Gießen, Germany

    Thomas Sauer

Authors
  1. H. Michael Möller
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  2. Thomas Sauer
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Möller, H.M., Sauer, T. Multivariate Refinable Functions of High Approximation Order Via Quotient Ideals of Laurent Polynomials. Advances in Computational Mathematics 20, 205–228 (2004). https://doi.org/10.1023/A:1025889132677

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