BIT Numerical Mathematics

, Volume 48, Issue 1, pp 5–27 | Cite as

Full rank positive matrix symbols: interpolation and orthogonality



We investigate full rank interpolatory vector subdivision schemes whose masks are positive definite on the unit circle except the point z=1. Such masks are known to give rise to convergent schemes with a cardinal limit function in the scalar case. In the full rank vector case, we show that there also exists a cardinal refinable function based on this mask, however, with respect to a different notion of refinability which nevertheless also leads to an iterative scheme for the computation of vector fields. Moreover, we show the existence of orthogonal scaling functions for multichannel wavelets and give a constructive method to obtain these scaling functions.

Key words

subdivision schemes refinement equation full rank schemes interpolatory matrix refinable function matrix spectral factorization 


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  1. 1.
    S. Bacchelli, M. Cotronei, and T. Sauer, Wavelets for multichannel signals, Adv. Appl. Math., 29 (2002), pp. 581–598.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    F. M. Callier, On polynomial matrix spectral factorization by symmetric extraction, IEEE Trans. Autom. Control, 30 (1985), pp. 453–464.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    C. Conti, M. Cotronei, and T. Sauer, Interpolatory vector subdivision schemes, in A. Cohen, J. L. Merrien, and L. L. Schumaker, eds., Curves and Surfaces, Avignon 2006, Nashboro Press, 2007, pp. 71–81.Google Scholar
  4. 4.
    C. Conti, M. Cotronei, and T. Sauer, Full rank interpolatory subdivision schemes: Kronecker, filters and multiresolution, submitted for publication.Google Scholar
  5. 5.
    M. Cotronei and T. Sauer, Full rank filters and polynomial reproduction, Commun. Pure Appl. Anal., 6 (2007), pp. 667–687.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, SIAM, 1992.Google Scholar
  7. 7.
    R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.MATHGoogle Scholar
  8. 8.
    INRIA, scilab – the open source platform for numerical computation,, 1989.Google Scholar
  9. 9.
    J. Ježek and V. Kučera, Efficient algorithm for matrix spectral factorization, Automatica, 21 (1985), pp. 663–669.CrossRefMATHGoogle Scholar
  10. 10.
    A. Klein, T. Sauer, A. Jedynak, and W. Skrandies, Conventional and wavelet coherence applied to human electrophysiological data, IEEE Trans. Biosignal Proc., 53 (2006), pp. 266–272.Google Scholar
  11. 11.
    M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Prindle, Weber & Schmidt, 1969, Paperback reprint, Dover Publications, 1992.Google Scholar
  12. 12.
    C. A. Micchelli, Interpolatory subdivision schemes and wavelets, J. Approximation Theory, 86 (1996), pp. 41–71.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. Comput. Math., 7(4) (1997), pp. 455–545.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    C. A. Micchelli and T. Sauer, On vector subdivision, Math. Z., 229 (1998), pp. 621–674.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  1. 1.Dipartimento di Energetica “Sergio Stecco”Università di FirenzeFirenzeItaly
  2. 2.DIMET – Dipartimento di Informatica, Matematica, Elettronica e TrasportiUniversità degli Studi “Mediterranea” di Reggio CalabriaReggio CalabriaItaly
  3. 3.Lehrstuhl für Numerische MathematikJustus-Liebig-Universität GießenGießenGermany

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