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Smoothness of Wavelets

  • Aleksandr Krivoshein
  • Vladimir Protasov
  • Maria SkopinaEmail author
Chapter
  • 656 Downloads
Part of the Industrial and Applied Mathematics book series (INAMA)

Abstract

The regularity of multivariate wavelet frames with an arbitrary dilation is studied by the matrix approach. The formulas for the Hölder exponents in spaces C and \(L_p\) are obtained in terms of the joint spectral radius of the corresponding transition matrices. Some results on higher order regularity, on the local regularity, and on the asymptotics of the moduli of continuity in various spaces are presented.

References

  1. 1.
    Jia, R.Q.: Characterization of smoothness of multivariate refinable functions in Sobolev spaces. Trans. Amer. Math. Soc. 351, 4089–4112 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Han, B.: Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix Anal. Appl. 24(3), 693–714 (2003)Google Scholar
  3. 3.
    Han, B.: Vector cascade algorithms and refinable function vectors in Sobolev spaces. J. Approx. Theory 124(1), 44–88 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Han, B.: Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix. Adv. Comput. Math. 24(1–4), 375–403 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cavaretta, A.S. Dahmen, W., Micchelli, C.A.: Stationary subdivision. Mem. Am. Math. Soc. 93(453), (1991)Google Scholar
  6. 6.
    Cabrelli, C.A., Heil, C., Molter, U.M.: Accuracy of lattice translates of several refinable multidimensional refinable functions. J. Approx. Theory 95, 5–52 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cabrelli, C.A., Heil, C., Molter, U.M., Self-similarity and multiwavelets in higher dimensions. Memoirs. Amer. Math. Soc. 170(807), (2004)Google Scholar
  8. 8.
    Cohen, A., Gröchenig, K., Villemoes, L.: Regularity of multivariate refinable functions. Constr. Approx. 15, 241–255 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Charina, M., Protasov, V.Y.: Matrix approach to analyse smoothness of multivariate wavelets, preprintGoogle Scholar
  10. 10.
    Novikov, I.Y., Protasov, V.Y. Skopina, M.A.: Wavelet Theory. AMS, Providence, RI, Translations Mathematical Monographs, V. 239 (2011)Google Scholar
  11. 11.
    Rota, G.C., Strang, G.: A note on the joint spectral radius. Kon. Nederl. Acad. Wet. Proc. 63, 379–381 (1960)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Protasov V.Y.: Spectral decomposition of 2-block Toeplitz matrices and refinement equations, St. Petersburg Math. J. 18(4), 607–646 (2007)Google Scholar
  13. 13.
    Collela, D., Heil, C.: Characterization of scaling functions. I. Continuous solutions. SIAM J. Matrix Anal. Appl. 15, 496–518 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Barabanov, N.E.: Lyapunov indicator for discrete inclusions, I-III. Autom. Remote Control 49(2), 152–157 (1988)zbMATHGoogle Scholar
  15. 15.
    Daubechies, I., Lagarias, J.: Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals. SIAM J. Math. Anal. 23, 1031–1079 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Villemoes, L.: Wavelet analysis of refinement equations. SIAM J. Math. Anal. 25(5), 1433–1460 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Protasov, VYu.: Fractal curves and wavelets. Izv. Math. 70(5), 123–162 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114, 185–204 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guglielmi, N., Protasov, V.Y.: Invariant polytopes of sets of matrices with applications to regularity of wavelets and subdivisions. SIAM J. Matrix Anal. Appl. 37(1), 18–52 (2016)Google Scholar
  20. 20.
    Chitour, Y., Mason, P., Sigalotti, M.: On the marginal instability of linear switched systems. Syst. Cont. Lett. 61, 747–757 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Protasov, V.Y., Jungers, R.: Resonance and marginal instability of switching systems. Nonlinear Anal.: Hybrid Syst. 17, 81–93 (2015)Google Scholar
  22. 22.
    Protasov, V.Y.: Extremal \(L_p\)-norms and self-similar functions. Linear Alg. Appl. 428(10), 2339–2357 (2008)Google Scholar
  23. 23.
    Protasov, V.Y.: The generalized spectral radius. A geometric approach, Izvestiya Math. 61, 995–1030 (1997)Google Scholar
  24. 24.
    Lawton, W., Lee, S.N., Shen, Z.: Convergence of multidimensional cascade algorithm. Numer. Math. 78(3), 427–438 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Guglielmi, N., Protasov, VY.: Exact computation of joint spectral characteristics of matrices. Found. Comput. Math. 13(1), 37–97 (2013)Google Scholar
  26. 26.
    Gripenberg, G.: A necessary and sufficient condition for the existence of father wavelet. Stud. Math. 114(3), 207–226 (1995)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Möller, C., Reif, U.: A tree-based approach to joint spectral radius determination. Linear Alg. Appl. 563, 154–170 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2016

Authors and Affiliations

  • Aleksandr Krivoshein
    • 1
  • Vladimir Protasov
    • 2
  • Maria Skopina
    • 1
    Email author
  1. 1.Department of Applied Mathematics and Control ProcessesSaint Petersburg State UniversitySaint PetersburgRussia
  2. 2.Department of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia

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