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Izv. RAN. Ser. Mat., 2006, Volume 70, Issue 5, Pages 123–162 (Mi izv674)  

This article is cited in 27 scientific papers (total in 27 papers)

Fractal curves and wavelets

V. Yu. Protasov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We introduce the notion of a summable fractal curve generated by a finite family of affine operators. This generalizes well-known notions of affine fractals and continuous fractal curves to the case of non-contractive operators. We establish a criterion for the existence of a fractal curve for a given family of operators, obtain criteria for that curve to belong to various function spaces and derive formulae for the exponents of regularity in those spaces as well as asymptotically sharp estimates for the moduli of continuity. These results are applied to the study of well-known curves (Koch, de Rham, and so on), refinable functions and wavelets. We also study the local behaviour of continuous fractal curves. We obtain a formula for the exponent of local regularity of continuous fractal curves at a given point and characterize the sets of points with a fixed local regularity. It is shown that the values of the local regularity of any fractal curve fill out some closed interval. Nevertheless, the regularity is the same at almost all points (in the Lebesgue measure) and can be computed from the Lyapunov exponent of certain linear operators. We apply this technique to refinement equations and compactly supported wavelets. As an example, we compute the moduli of continuity and exponents of local regularity and $L_p$-regularity for several Daubechies wavelets.

DOI: https://doi.org/10.4213/im674

Full text: PDF file (810 kB)
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English version:
Izvestiya: Mathematics, 2006, 70:5, 975–1013

Bibliographic databases:

UDC: 517.51
MSC: 26A16, 28A80, 39B22
Received: 31.10.2005

Citation: V. Yu. Protasov, “Fractal curves and wavelets”, Izv. RAN. Ser. Mat., 70:5 (2006), 123–162; Izv. Math., 70:5 (2006), 975–1013

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. Yu. Protasov, “Spectral factorization of 2-block Toeplitz matrices and refinement equations”, St. Petersburg Math. J., 18:4 (2007), 607–646  mathnet  crossref  mathscinet  zmath  elib
    2. V. Yu. Protasov, “Self-similarity equations and the $p$-radius of operators”, Russian Math. Surveys, 62:6 (2007), 1221–1223  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. I. A. Sheipak, “On the Construction and Some Properties of Self-Similar Functions in the Spaces $L_p[0,1]$”, Math. Notes, 81:6 (2007), 827–839  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. Jungers R.M., Protasov V., Blondel V.D., “Efficient algorithms for deciding the type of growth of products of integer matrices”, Linear Algebra Appl., 428:10 (2008), 2296–2311  crossref  mathscinet  zmath  isi  elib  scopus
    5. Protasov V.Yu., “Extremal $L_p$-norms of linear operators and self-similar functions”, Linear Algebra Appl., 428:10 (2008), 2339–2356  crossref  mathscinet  zmath  isi  elib  scopus
    6. E. A. Rodionov, Yu. A. Farkov, “Estimates of the Smoothness of Dyadic Orthogonal Wavelets of Daubechies Type”, Math. Notes, 86:3 (2009), 407–421  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Protasov V.Y., Jungers R.M., Blondel V.D., “Joint spectral characteristics of matrices: a conic programming approach”, SIAM J. Matrix Anal. Appl., 31:4 (2010), 2146–2162  crossref  zmath  isi  scopus
    8. I. A. Sheipak, “Singular points of a self-similar function of spectral order zero: self-similar Stieltjes string”, Math. Notes, 88:2 (2010), 275–286  mathnet  crossref  crossref  mathscinet  isi
    9. Xu Jianhong, “On the trace characterization of the joint spectral radius”, Electron. J. Linear Algebra, 20 (2010), 367–375  mathscinet  zmath  isi  elib
    10. Jungers R.M., Protasov V.Y., “Weak stability of switching dynamical systems and fast computation of the $p$-radius of matrices”, 49th IEEE Conference on Decision and Control (CDC), 2010, 7328–7333  crossref  isi  scopus
    11. V. Yu. Protasov, “Invariant functions for the Lyapunov exponents of random matrices”, Sb. Math., 202:1 (2011), 101–126  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Xu Jianhong, Xiao Mingqing, “A characterization of the generalized spectral radius with Kronecker powers”, Automatica, 47 (2011), 1530–1533  crossref  mathscinet  zmath  isi  scopus
    13. Jungers R.M., Protasov V.Y., “Fast methods for computing the $p$-radius of matrices”, SIAM J. Sci. Comput., 33:3 (2011), 1246–1266  crossref  mathscinet  zmath  isi  elib  scopus
    14. A. S. Voynov, “Self-affine polytopes. Applications to functional equations and matrix theory”, Sb. Math., 202:10 (2011), 1413–1439  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    15. Farkov Yu.A., Maksimov A.Yu., Stroganov S.A., “On biorthogonal wavelets related to the Walsh functions”, Int. J. Wavelets Multiresolut. Inf. Process., 9:3 (2011), 485–499  crossref  mathscinet  zmath  isi  elib  scopus
    16. N. V. Gaganov, I. A. Sheipak, “A boundedness criterion for the variations of self-similar functions”, Siberian Math. J., 53:1 (2012), 55–71  mathnet  crossref  mathscinet  isi
    17. Serge Dubuc, “Palindromic Matrices of Order Two and Three-Point Subdivision Schemes”, Constr Approx, 2012  crossref  mathscinet  isi  scopus
    18. Chitour Ya., Mason P., Sigalotti M., “On the Marginal Instability of Linear Switched Systems”, Syst. Control Lett., 61:6 (2012), 747–757  crossref  mathscinet  zmath  isi  elib  scopus
    19. Carl Dettmann, “Open circle maps: small hole asymptotics”, Nonlinearity, 26:1 (2013), 307  crossref  mathscinet  zmath  isi  scopus
    20. Liu J., Xiao M., “Rank-One Characterization of Joint Spectral Radius of Finite Matrix Family”, Linear Alg. Appl., 438:8 (2013), 3258–3277  crossref  mathscinet  zmath  isi  elib  scopus
    21. V.Y.. Protasov, Raphaël.M.. Jungers, “Resonance and marginal instability of switching systems”, Nonlinear Analysis: Hybrid Systems, 17 (2015), 81  crossref  mathscinet  zmath  scopus
    22. A. S. Voynov, V. Yu. Protasov, “Compact noncontraction semigroups of affine operators”, Sb. Math., 206:7 (2015), 921–940  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    23. Krivoshein A., Protasov V., Skopina M., “Smoothness of Wavelets”: Krivoshein, A Protasov, V Skopina, M, Multivariate Wavelet Frames, Industrial and Applied Mathematics, Springer-Verlag Singapore Pte Ltd, 2016, 209–237  crossref  mathscinet  isi
    24. Protasov V.Yu., Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408  crossref  mathscinet  zmath  isi  elib  scopus
    25. Barany B., Kiss G., Kolossvary I., “Pointwise Regularity of Parameterized Affine Zipper Fractal Curves”, Nonlinearity, 31:4 (2018), 1705–1733  crossref  zmath  isi  scopus
    26. Cicone A., Guglielmi N., Protasov V.Yu., “Linear Switched Dynamical Systems on Graphs”, Nonlinear Anal.-Hybrid Syst., 29 (2018), 165–186  crossref  mathscinet  zmath  isi  scopus
    27. I. A. Sheipak, “O pokazatelyakh Geldera samopodobnykh funktsii”, Funkts. analiz i ego pril., 53:1 (2019), 67–78  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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