Geodesic
Subdivision schemes on the dyadic half-line
Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 910-929.

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We consider subdivision schemes, which are used for the approximation of functions and generation of curves on the dyadic half-line. In the classical case of functions on the real line, the theory of subdivision schemes is widely known because of its applications in constructive approximation theory and signal processing as well as for generating fractal curves and surfaces. We define and study subdivision schemes on the dyadic half-line (the positive half-line endowed with the standard Lebesgue measure and the digit-wise binary addition operation), where the role of exponentials is played by Walsh functions. We obtain necessary and sufficient conditions for the convergence of subdivision schemes in terms of the spectral properties of matrices and in terms of the smoothness of solutions of the corresponding refinement equation. We also investigate the problem of convergence of subdivision schemes with non-negative coefficients. We obtain an explicit criterion for the convergence of algorithms with four coefficients. As an auxiliary result, we define fractal curves on the dyadic half-line and prove a formula for their smoothness. The paper contains various illustrative examples and numerical results.
Keywords: subdivision schemes, dyadic half-line, fractal curves, smoothness of fractal curves, spectral properties of matrices. 
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M. A. Karapetyants. Subdivision schemes on the dyadic half-line. Izvestiya. Mathematics , Tome 84 (2020) no. 5, pp. 910-929. http://geodesic.mathdoc.fr/item/IM2_2020_84_5_a3/

[1] B. I. Golubov, Elementy dvoichnogo analiza, 2-e izd., LKI, M., 2007, 208 pp.

[2] B. Golubov, A. Efimov, V. Skvortsov, Walsh series and transforms. Theory and applications, Math. Appl. (Soviet Ser.), 64, Kluwer Acad. Publ., Dordrecht, 1991, xiv+368 pp. | DOI | MR | MR | Zbl | Zbl

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

[4] G. de Rham, “Sur les courbes limités de polygones obtenus par trisection”, Enseign. Math. (2), 5 (1959), 29–43 | MR | Zbl

[5] G. M. Chaikin, “An algorithm for high-speed curve generation”, Comput. Graphics and Image Processing, 3:4 (1974), 346–349 | DOI

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

[7] A. S. Cavaretta, W. Dahmen, C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc., 93, no. 453, Amer. Math. Soc., Providence, RI, 1991, vi+186 pp. | DOI | MR | Zbl

[8] N. Dyn, “Linear and nonlinear subdivision schemes in geometric modeling”, Foundations of computational mathematics (Hong Kong, 2008), London Math. Soc. Lecture Note Ser., 363, Cambridge Univ. Press, Cambridge, 2009, 68–92 | MR | Zbl

[9] N. Dyn, D. Levin, J. A. Gregory, “A butterfly subdivision scheme for surface interpolation with tension control”, ACM Trans. Graph., 9:2 (1990), 160–169 | DOI | Zbl

[10] M. Marinov, N. Dyn, D. Levin, “Geometrically controlled 4-point interpolatory schemes”, Advances in multiresolution for geometric modelling, Math. Vis., Springer, Berlin, 2005, 301–315 | DOI | MR | Zbl

[11] I. Ya. Novikov, V. Yu. Protasov, M. A. Skopina, Wavelet theory, Transl. Math. Monogr., 239, Amer. Math. Soc., Providence, RI, 2011, xiv+506 pp. | MR | MR | Zbl | Zbl

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

[13] I. Daubechies, J. C. Lagarias, “Two-scale difference equations. I. Existence and global regularity of solutions”, SIAM. J. Math. Anal., 22:5 (1991), 1388–1410 | DOI | MR | Zbl

[14] E. A. Rodionov, Yu. A. Farkov, “Estimates of the smoothness of dyadic orthogonal wavelets of Daubechies type”, Math. Notes, 86:3 (2009), 407–421 | DOI | DOI | MR | Zbl

[15] G.-C. Rota, W. G. Strang, “A note on the joint spectral radius”, Nederl. Akad. Wetensch. Proc. Ser. A, 63, Indag. Math. 22 (1960), 379–381 | DOI | MR | Zbl

[16] N. Dyn, D. Levin, “Subdivision schemes in geometric modelling”, Acta Numer., 11 (2002), 73–144 | DOI | MR | Zbl

[17] A. A. Melkman, “Subdivision schemes with non-negative masks converge always – unless they obviously cannot?”, Ann. Numer. Math., 4:1-4 (1997), 451–460 | MR | Zbl

[18] V. Yu. Protasov, “Spectral factorization of 2-block Toeplitz matrices and refinement equations”, St. Petersburg Math. J., 18:4 (2007), 607–646 | DOI | MR | Zbl