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On multiresolution analysis of multiplicityd

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Abstract

In this paper we study some basic properties of multiresolution analysis of multiplicityd in several variables and discuss some examples related to the spaces of cardinal splines with respect to the unidiagonal or the crisscross partition of the plane. Furthermore, in analogy with [8], we show that if the scaling functions are compactly supported, then it is possible to find compactly supported mother waveletsψ l ,l=1,...,2n dd, in such a way that the family {2jn/2 ψ l (2j xv)} is a semiorthogonal basis ofL 2 (ℝn).

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References

  1. Alpert BK (1992) Construction of simple multiscale bases for fast matrix operations. In:Ruskai et al. (eds) Wavelets and Their Applications, pp 211–226. Boston: Jones and Bartlett

    Google Scholar 

  2. Auscher P (1992) Wavelets with boundary conditions on the interval. In:Chui CK (ed) Wavelets: A Tutorial in Theory and Applications, pp 217–236. San Diego: Academic Press

    Google Scholar 

  3. Alfeld P (1986) On the dimension of multivariate piecewise polynomials. In:Griffiths DF, Watson GA (ed) Numerical Analysis, pp. 1–23. New York: Longmans

    Google Scholar 

  4. Alfield P, Schumaker LL, Sirvent M (1992) On dimension and existence of local bases for multivariate spline spaces. J Approx Theory70: 243–264

    Google Scholar 

  5. Chui CK (1992) An Introduction to Wavelets. San Diego: Academic Press

    Google Scholar 

  6. Chui CK (1988) Multivariate Splines. Philadelphia

  7. Cohen A, Schlenker JM (1993) Compactly supported bidimensional wavelets bases with hexagonal symmetry. Constr Approx9: 209–236

    Google Scholar 

  8. Chui CK, Stöckler J, Ward JD (1992) Compactly supported box-spline wavelets. Approx Theory Appl8: 77–99

    Google Scholar 

  9. Chui CK, Wang JZ (1992) On compactly supported spline wavelets and a duality principle. Trans Amer Math Soc330: 903–916

    Google Scholar 

  10. De Boor C (1987) B-form basics. In:Farin GE (ed) Geometry Modeling. Algorithms and New Trends, pp 131–148. Philadelphia: SIAM

    Google Scholar 

  11. Gröchenig K (1987) Analyse multiéchelle et bases d'ondelettes. CR Acad Sci Paris305: 13–15

    Google Scholar 

  12. Geronimo JS, Hardin DP, Massopust PR (1994) Fractal functions and wavelet expansions based on several scaling functions. J Approx Theory78: 373–401

    Google Scholar 

  13. Goodmann TNT, Lee SL, Tang WS (1993) Wavelets in wandering subspaces. Trans Amer Math Soc338: 639–654

    Google Scholar 

  14. Gröchenig K, Madych WR (1992) Multiresolution analysis, Haar basis and self-similar tilings of ℝn. IEEE Trans Information Theory38: 556–568

    Google Scholar 

  15. Grünbaum B, Shephard GC (1986) Tilings and Patterns. New York: WH Freeman

    Google Scholar 

  16. Hervé L (1994) Multi-resolution analysis of multiplicity d; applications to dyadic interpolation. Appl Comput Harmonic Anal1: 299–315

    Google Scholar 

  17. Iversen B (1990/91) Lectures on crystallographic groups. Aarhus Univ Lect Notes 60

  18. Jaffard S (1989) Construction et propriété des bases d'ondelettes. Remarque sur la controllabilité exacte. PhD Thesis Palaiseau: Ecole polytechnique

    Google Scholar 

  19. Jia RQ, Micchelli CA (1991) Using the refinement equations for the construction of prewavelets. In:Laurent PJ et al (ed) Curves and Surfaces, pp 209–246. San Diego: Academic Press

    Google Scholar 

  20. Lemarié-Rieusset PG (1993) Ondelettes generalisées et function d'échelle a support compact. Revista Mat Iberoamericana9: 333–37

    Google Scholar 

  21. Martin GE (1980) Transformation Geometry. Springer

  22. Meyer Y (1990) Ondelettes et operateurs. Paris: Hermann

    Google Scholar 

  23. Plonka F (1997) Approximation order provided by refinable function vector. Constr Approx13: 221–244

    Google Scholar 

  24. Plonka F, Strela V (1995) Construction of multiscaling functions with approximation and symmetry (preprint)

  25. Strela V (1996) Multiwavelets: Regularity, orthogonality and symmetry via two-scale similarity transforms. Studies in Appl Math

  26. Strang G, Strela V (1994) Orthogonal wavelets with vanishing moments. J Optical Eng33: 2104–2107

    Google Scholar 

  27. Stöckler J (1992) Multivariate wavelets. In:Chui CK (ed) Wavelets: A Tutorial in Theory and Applications, pp 325–356. San Diego: Academic Press

    Google Scholar 

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De Michele, L., Soardi, P.M. On multiresolution analysis of multiplicityd . Monatshefte für Mathematik 124, 255–272 (1997). https://doi.org/10.1007/BF01298247

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  • DOI: https://doi.org/10.1007/BF01298247

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