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

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

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English version:
Izvestiya: Mathematics, 2006, 70:5, 975–1013

Bibliographic databases:

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

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

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. V. Yu. Protasov, “Self-similarity equations and the $p$-radius of operators”, Russian Math. Surveys, 62:6 (2007), 1221–1223
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
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
5. Protasov V.Yu., “Extremal $L_p$-norms of linear operators and self-similar functions”, Linear Algebra Appl., 428:10 (2008), 2339–2356
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
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
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
9. Xu Jianhong, “On the trace characterization of the joint spectral radius”, Electron. J. Linear Algebra, 20 (2010), 367–375
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
11. V. Yu. Protasov, “Invariant functions for the Lyapunov exponents of random matrices”, Sb. Math., 202:1 (2011), 101–126
12. Xu Jianhong, Xiao Mingqing, “A characterization of the generalized spectral radius with Kronecker powers”, Automatica, 47 (2011), 1530–1533
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
14. A. S. Voynov, “Self-affine polytopes. Applications to functional equations and matrix theory”, Sb. Math., 202:10 (2011), 1413–1439
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
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
17. Serge Dubuc, “Palindromic Matrices of Order Two and Three-Point Subdivision Schemes”, Constr Approx, 2012
18. Chitour Ya., Mason P., Sigalotti M., “On the Marginal Instability of Linear Switched Systems”, Syst. Control Lett., 61:6 (2012), 747–757
19. Carl Dettmann, “Open circle maps: small hole asymptotics”, Nonlinearity, 26:1 (2013), 307
20. Liu J., Xiao M., “Rank-One Characterization of Joint Spectral Radius of Finite Matrix Family”, Linear Alg. Appl., 438:8 (2013), 3258–3277
21. V.Y.. Protasov, Raphaël.M.. Jungers, “Resonance and marginal instability of switching systems”, Nonlinear Analysis: Hybrid Systems, 17 (2015), 81
22. A. S. Voynov, V. Yu. Protasov, “Compact noncontraction semigroups of affine operators”, Sb. Math., 206:7 (2015), 921–940
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
24. Protasov V.Yu., Voynov A.S., “Matrix semigroups with constant spectral radius”, Linear Alg. Appl., 513 (2017), 376–408
25. Barany B., Kiss G., Kolossvary I., “Pointwise Regularity of Parameterized Affine Zipper Fractal Curves”, Nonlinearity, 31:4 (2018), 1705–1733
26. Cicone A., Guglielmi N., Protasov V.Yu., “Linear Switched Dynamical Systems on Graphs”, Nonlinear Anal.-Hybrid Syst., 29 (2018), 165–186
27. I. A. Sheipak, “O pokazatelyakh Geldera samopodobnykh funktsii”, Funkts. analiz i ego pril., 53:1 (2019), 67–78
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