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Convergence and error estimate of cascade algorithms with infinitely supported masks in L p (ℝs)

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Abstract

The cascade algorithm plays an important role in computer graphics and wavelet analysis. In this paper, we first investigate the convergence of cascade algorithms associated with a polynomially decaying mask and a general dilation matrix in L p (ℝs) (1 ⩾ p ⩾ ∞) spaces, and then we give an error estimate of the cascade algorithms associated with truncated masks. It is proved that under some appropriate conditions if the cascade algorithm associated with a polynomially decaying mask converges in the L p -norm, then the cascade algorithms associated with the truncated masks also converge in the L p -norm. Moreover, the error between the two resulting limit functions is estimated in terms of the masks.

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References

  1. Cavaretta A S, Dahmen W, Micchelli C A. Stationary subdivision. Mem Amer Math Soc, 1991, 93: 1–186

    MathSciNet  Google Scholar 

  2. Chen D R, Jia R Q, Riemenschneider S D. Convergence of vector subdivision schemes in Sobolev spaces. Appl Comp Harmon Anal, 2002, 12: 128–149

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen D R, Plonka G. Convergence of cascade algorithms in sobolev spaces for perturbed refinement masks. J Approx Theory, 2002, 119: 133–155

    Article  MathSciNet  MATH  Google Scholar 

  4. Daubechies I, Huang Y. A decay theorem for refinable functions. Appl Math Lett, 1994, 7: 1–4

    Article  MathSciNet  MATH  Google Scholar 

  5. Daubechies I, Huang Y. How does truncation of the mask affect a refinable function? Constr Approx, 1995, 11: 365–380

    Article  MathSciNet  MATH  Google Scholar 

  6. Goodman T N T, Micchelli C A, Ward J D. Spectral radius formulas for subdivision operators. In: Schumaker L L, Webb G, eds. Recent Advances in Wavelet Analysis. New York: Academic Press, 1994, 335–360

    Google Scholar 

  7. Han B. Error estimate of a subdivision scheme with a truncated refinement mask. Unpublished manuscript, 1997

  8. Han B. Subdivision schemes, biorthogonal wavelets and image compression. PhD Thesis. University of Alberta, 1998

  9. Han B. The initial functions in a subdivision scheme. In: Zhou D -X, ed. Wavelet Analysis, Twenty Years Developments. Singapore: World Scientific Press, 2002, 154–178

    Chapter  Google Scholar 

  10. Han B. Refinable functions and cascade algorithms in weighted spaces with hölder continuous masks. SIAM J Math Anal, 2008, 41: 70–102

    Article  Google Scholar 

  11. Han B, Hogan T A. How is a vector pyramid scheme affected by perturbation in the mask? In: Chui C K, Schumaker L L, eds. Approximation Theory IX. Nashville: Vanderbilt University Press, 1998, 97–104

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Han B, Jia R Q. Characterization of Riesz bases of wavelets generated from multiresolution analysis. Appl Comput Harmon Anal, 2007, 23: 321–345

    Article  MathSciNet  MATH  Google Scholar 

  14. Han B, Shen Z. Wavelets from the Loop scheme. J Fourier Anal Appl, 2005, 11: 615–637

    Article  MathSciNet  MATH  Google Scholar 

  15. Han B, Shen Z. Wavelets with short support. SIAM J Math Anal, 2006, 38: 530–556

    Article  MathSciNet  Google Scholar 

  16. Herley C, Vetterli M. Wavelets and recursive filter banks. IEEE Tran Signal Proc, 1993, 41: 2536–2556

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Jia R Q. Cascade algorithms in wavelet analysis. In: Zhou D X, ed. Wavelet Analysis, Twenty Years Developments. Singapore: World Scientific Press, 2002, 196–230

    Chapter  Google Scholar 

  19. Jia R Q, Li S. Refinable functions with exponential decay: An approach via cascade algorithms. J Fourier Anal Appl, 2011, 17: 1008–1034

    Article  MATH  Google Scholar 

  20. Jia R Q, Micchelli C A. Using the refinement equations for the construction of pre-wavelets II: Powers of two. In: Laurent P J, Laurent A, Le Méhauté A, et al. eds. Curves and Surfaces. New York: Academic Press, 1991, 209–246

    Google Scholar 

  21. Jia R Q, Riemenschneider S D, Zhou D X. Vector subdivision schemes and multiple wavelets. Math Comp, 1998, 67: 1533–1563

    Article  MathSciNet  MATH  Google Scholar 

  22. Lange K. Numerical Analysis for Statisticians. New York: Springer, 1999

    MATH  Google Scholar 

  23. Li S. Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms of nonhomogeneous refinement equations. Adv Comput Math, 2004, 20: 311–331

    Article  MathSciNet  MATH  Google Scholar 

  24. Li S, Pan Y L. Subdivisions with infinitely supported mask. J Comput Appl Math, 2008, 214: 288–303

    Article  MathSciNet  MATH  Google Scholar 

  25. Li S, Pan Y L. Subdivision schemes with polynomially decaying masks. Adv Comput Math, 2010, 32: 487–507

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen Z. Refinable function vectors. SIAM J Math Anal, 1998, 29: 235–250

    Article  MathSciNet  MATH  Google Scholar 

  27. Strang G, Fix G. A Fourier analysis of the finite-element variational method. In: Geymonat G, ed. Constructive Aspects of Functional Analysis. Berlin: Springer-Verlag, 1973, 793–840

    Google Scholar 

  28. Unser M, Blu T. Fractional splines and wavelets. SIAM Rev, 2000, 42: 43–67

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang J, Li S. Smoothness of multivariate refinable functions with infinitely supported masks. J Approx Theory, 2010, 162: 1279–1293

    Article  MathSciNet  MATH  Google Scholar 

  30. Yang J, Li S. Convergence rates of cascade algorithms with infinitely supported masks. Canad Math Bull, 2011, doi:10.4153/CMB-2011-081-6

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Hohai University, Nanjing, 210098, China

    JianBin Yang

  2. Department of Mathematics, College of Sciences, Zhejiang University, Hangzhou, 310027, China

    Song Li

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  1. JianBin Yang
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  2. Song Li
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Correspondence to Song Li.

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Yang, J., Li, S. Convergence and error estimate of cascade algorithms with infinitely supported masks in L p (ℝs). Sci. China Math. 55, 577–592 (2012). https://doi.org/10.1007/s11425-011-4320-8

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  • DOI: https://doi.org/10.1007/s11425-011-4320-8

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