Abstract
We consider self-similar functions in the space B[0, 1] of bounded functions and prove a boundedness criterion in terms of self-similarity parameters for the variations of these functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mandelbrot B., Fractals: Form, Chance, and Dimension, Freeman, San Francisco (1977).
Hutchinson J., “Fractals and self-similarity,” Indiana Math. J., 30, No. 5, 713–747 (1981).
Barnsley M., Fractals Everywhere, Academic Press, New York (1988).
De Rham G., “Une peu de mathématique à propos d’une courbe plane,” Rev. Math. Elemetaires, 2, No. 4 and 5, 73–76 and 89–97 (1947).
De Rham G., “Sur une courbe plane,” J. Math. Pures Appl., 35, No. 9, 25–42 (1956).
De Rham G., “Sur les courbes limités de polygones obtenus par trisection,” Enseign. Math., 5, No. 2, 29–43 (1959).
Protasov V., “Refinement equations with nonnegative coefficients,” J. Fourier Anal. Appl., 6, No. 1, 55–78 (2000).
Protasov V. Yu., “Fractal curves and wavelets,” Izv.: Math., 70, No. 5, 975–1013 (2006).
Sheipak I. A., “On the construction and some properties of self-similar functions in the spaces L p[0, 1],” Math. Notes, 81, No. 6, 827–839 (2007).
Solomyak M. and Verbitsky E., “On a spectral problem related to self-similar measures,” Bull. London Math. Soc., 27, 242–248 (1995).
Vladimirov A. A. and Sheipak I. A., “Self-similar functions in (L 2[0, 1]) and the Sturm-Liouville problem with singular indefinite weight,” Sb.: Math., 197, No. 11, 15–69 (2006).
Vladimirov A. A. and Sheipak I. A., “The indefinite Sturm-Liouville problem for some classes with self-similar singular weights,” Tr. MIRAN, 255, 88–98 (2006).
Sheipak I. A., “Singular points of a self-similar function of spectral order zero: Self-similar Stieltjes string,” Math. Notes, 88, No. 2, 275–286 (2010).
Natanson I. P., Theory of Functions of Real Variable, Ungar, New York (1955).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text Copyright © 2012 Gaganov N. V. and Sheipak I. A.
The authors were supported by the Russian Foundation for Basic Research (Grants 10-01-00423-a and 11-01-12115-ofi-m-2011).
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 1, pp. 68–88, January–February, 2012.
Rights and permissions
About this article
Cite this article
Gaganov, N.V., Sheipak, I.A. A boundedness criterion for the variations of self-similar functions. Sib Math J 53, 55–71 (2012). https://doi.org/10.1134/S0037446612010053
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446612010053