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Self-affine polytopes. Applications to functional equations and matrix theory
Sbornik. Mathematics, Tome 202 (2011) no. 10, pp. 1413-1439 Cet article a éte moissonné depuis la source Math-Net.Ru

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A special kind of functional equation with compression of the argument — the affine self-similarity equation — is studied. The earlier known one-dimensional self-similarity equations are generalized to the multidimensional case of functions of several variables. A criterion for the existence and uniqueness of an $L_p$-solution is established. Description of such equations involves classification of finite-dimensional convex self-affine compact sets. In this work properties of such objects are thoroughly analysed; in particular, a counterexample to the well-known conjecture about the structure of such bodies, which was put forward in 1991, is given. Applications of the results obtained include some facts about the convergence of products of stochastic matrices; also, criteria for the convergence of some subdivision algorithms are suggested. Bibliography: 39 titles.
Keywords: convex polytope, functional equation, compression of the argument, stochastic matrix.
Mots-clés : partition 
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A. S. Voynov. Self-affine polytopes. Applications to functional equations and matrix theory. Sbornik. Mathematics, Tome 202 (2011) no. 10, pp. 1413-1439. http://geodesic.mathdoc.fr/item/SM_2011_202_10_a0/

[1] M. Barnsley, Fractals everywhere, Academic Press, Boston, MA, 1988 | MR | Zbl

[2] Ch. A. Micchelli, H. Prautzsch, “Uniform refinement of curves”, Linear Algebra Appl., 114–115 (1989), 841–870 | DOI | MR | Zbl

[3] G. de Rham, “Sur une courbe plane”, J. Math. Pures Appl. (9), 35 (1956), 25–42 | MR | Zbl

[4] I. Daubechies, J. C. Lagarias, “Two-scale difference equations. II. Local regularity, infinite products of matrices and fractals”, SIAM J. Math. Anal., 23:4 (1992), 1031–1079 | DOI | MR | Zbl

[5] D. Collela, Ch. Heil, “Characterizations of scaling functions: continuous solutions”, SIAM J. Matrix Anal. Appl., 15:2 (1994), 496–518 | DOI | MR | Zbl

[6] L. F. Villemoes, “Wavelet analysis of refinement equations”, SIAM J. Math. Anal., 25:5 (1994), 1433–1460 | DOI | MR | Zbl

[7] V. Yu. Protasov, “Fractal curves and wavelets”, Izv. Math., 70:5 (2006), 975–1013 | DOI | MR | Zbl

[8] I. Ya. Novikov, V. Yu. Protasov, M. A. Skopina, Wavelet theory, Amer. Math. Soc., Providence, RI, 2011 | MR | MR | Zbl | Zbl

[9] G. Deslauriers, S. Dubuc, “Symmetric iterative interpolation processes”, Constr. Approx., 5:1 (1989), 49–68 | DOI | MR | Zbl

[10] N. Dyn, D. Levin, “Interpolating subdivision schemes for the generation of curves and surfaces”, Multivariate approximation and interpolation (Duisburg, 1989), Internat. Ser. Numer. Math., 94, Birkhaüser, Basel, 1990, 91–106 | MR | Zbl

[11] Y. Wang, “Two-scale dilation equations and the mean spectral radius”, Random Comput. Dynam., 4:1 (1996), 49–72 | MR | Zbl

[12] R.-Q. Jia, “Subdivision schemes in $L_p$ spaces”, Adv. Comput. Math., 3:4 (1995), 309–341 | DOI | MR | Zbl

[13] K.-S. Lau, J. Wang, “Characterization of $L^p$-solutions for the two-scale dilation equations”, SIAM J. Math. Anal., 26:4 (1995), 1018–1046 | DOI | MR | Zbl

[14] G. Derfel, N. Dyn, D. Levin, “Generalized refinement equations and subdivision processes”, J. Approx. Theory, 80:2 (1995), 272–297 | DOI | MR | Zbl

[15] C. A. Cabrelli, Ch. Heil, U. M. Molter, Self-similarity and multiwavelets in higher dimensions, Mem. Amer. Math. Soc., 170, no. 807, Amer. Math. Soc., Providence, RI, 2004 | MR | Zbl

[16] V. Yu. Protasov, “The generalized joint spectral radius. A geometric approach”, Izv. Math., 61:5 (1997), 995–1030 | DOI | MR | Zbl

[17] M. Solomyak, E. Verbitsky, “On a spectral problem related to self-similar measures”, Bull. London Math. Soc., 27:3 (1995), 242–248 | DOI | MR | Zbl

[18] B. M. Hambly, J. Kigami, T. Kumagai, “Multifractal formalisms for the local spectral and walk dimensions”, Math. Proc. Cambridge Philos. Soc., 132:3 (2002), 555–571 | DOI | MR | Zbl

[19] V. Yu. Protasov, “Extremal $L_p$-norms of linear operators and self-similar functions”, Linear Algebra Appl., 428:10 (2008), 2339–2356 | DOI | MR | Zbl

[20] H. T. Croft, K. J. Falconer, R. K. Guy, Unsolved problems in geometry, Problem Books in Math., Springer-Verlag, New York, 1991 | MR | Zbl

[21] J. Tolke, J. M. Wills, Contributions to geometry (Siegen, 1978), Birkhaüser, Basel, 1979 | MR | Zbl

[22] E. Hertel C. Richter, “Self-affine convex polygons”, SIAM J. Math. Anal., 98:1–2 (2010), 79–89 | DOI | MR | Zbl

[23] D.-X. Zhou, “Self-similar lattice tilings and subdivision schemes”, SIAM J. Math. Anal., 33:1 (2001), 1–15 | DOI | MR | Zbl

[24] C. Richter, “Self-affine convex discs are polygons”, Beitr. Algebra Geom., Online First | DOI

[25] A. Voynov, “Self-affine polyhedra”, Proc. Steklov Inst. Math. (to appear)

[26] G.-C. Rota, W. G. Strang, “A note on the joint spectral radius”, Nederl. Akad. Wetensch. Proc. Ser. A, 63 (1960), 379–381 | MR | Zbl

[27] M. A. Berger, Y. Wang, “Bounded semigroups of matrices”, Linear Algebra Appl., 166 (1992), 21–27 | DOI | MR | Zbl

[28] V. Blondel, S. Gaubert, J. N. Tsitsiklis, “Approximating the spectral radius of sets of matrices in the max-algebra is NP-hard”, IEEE Trans. Automat. Control, 45:9 (2000), 1762–1765 | DOI | MR | Zbl

[29] V. Yu. Protasov, “Sovmestnyi spektralnyi radius i invariantnye mnozhestva lineinykh operatorov”, Fundament. i prikl. matem., 2:1 (1996), 205–231 | MR | Zbl

[30] G. Gripenberg, “Computing the joint spectral radius”, Linear Algebra Appl., 234 (1996), 43–60 | DOI | MR | Zbl

[31] V. Yu. Protasov, “On the regularity of de Rham curves”, Izv. Math., 68:3 (2004), 567–606 | DOI | MR | Zbl

[32] 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 | DOI | MR | Zbl

[33] I. Daubechies, J. C. Lagarias, “Corrigendum/addendum to: Sets of matrices all infinite products of which converge”, Linear Algebra Appl., 327:1–3 (2001), 69–83 | DOI | MR | Zbl

[34] R. A. Horn, Ch. R. Johnson, Matrix analysis, Cambridge Univ. Press, Cambridge, 1985 | MR | MR | Zbl | Zbl

[35] H. Furstenberg, “Noncommuting random products”, Trans. Amer. Math. Soc., 108 (1963), 377–428 | DOI | MR | Zbl

[36] H. Furstenberg, H. Kesten, “Products of random matrices”, Ann. Math. Statist., 31:2 (1960), 457–469 | DOI | MR | Zbl

[37] V. Yu. Protasov, “Semigroups of non-negative matrices”, Russian Math. Surveys, 65:6 (2010), 1186–1188 | DOI | MR | Zbl

[38] R.-Q. Jia, D.-X. Zhou, “Convergence of subdivision schemes associated with nonnegative masks”, SIAM J. Matrix Anal. Appl., 21:2 (1999), 418–430 | DOI | MR | Zbl

[39] V. Protasov, “Refinement equations with nonnegative coefficients”, J. Fourier Anal. Appl., 6:1 (2000), 55–78 | DOI | MR | Zbl