Abstract
We present a construction of affinely self-similar functions. In terms of the parameters of self-similarity transformations, a condition is given for these functions to belong to the classes L p[0, 1] as well as to the space C[0, 1]. Some properties of these functions (monotonicity and bounded variation) are studied. A relationship between self-similar functions and self-similar measures is established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Hutchinson, “Fractals and self-similarity,” Indiana Univ. Math. J. 30, 713–747 (1981).
M. Barnsley, Fractals Everywhere (Academic Press, Boston, 1988).
M. Solomyak and E. Verbitsky, “On a spectral problem related to self-similar measures,” Bull. London Math. Soc. 27(3), 242–248 (1995).
R. S. Strihartz, “Self-similar measures and their Fourier transform, I,” Indiana Univ. Math. J. 39(3), 797–817 (1990).
A. Zygmund, Trigonometric Series (Cambridge University Press, New York, 1959; Mir, Moscow, 1965), Vol. 1 [in Russian].
G. de Rham, “Un peu de mathématique à propos d’une courbe plane,” Elem. Math. 2(4, 5), 73–76, 89–97 (1947).
G. de Rham, “Sur une courbe plane,” J. Math. Pures Appl. (9) 35, 25–42 (1956).
G. de Rham, “Sur les courbes limitées de polygones obtenus par trisection,” Enseign. Math. (2) 5, 29–43 (1959).
A. S. Cavaretta, W. Dahmen and C. A. Micchelli, “Stationary dubdivision,” Mem. Amer. Math. Soc. 93(453) (1991).
G. A. Derfel’, “A probabilistic method for studying a class of functional-differential equations,” Ukrain. Mat. Zh. 41(10), 1137–1141 (1989) [Ukrainian Math. J. 41 (1989) (10), 1137–1141 (1990)].
P. P. Nikitin, “The Hausdorff dimension of the harmonic measure on a de Rham curve,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283, 206–223 (2001) [J. Math. Sci. (N. Y.) 121 (3), 2409–2418 (2004)].
I. Daubechies and J. Lagarias, “Two scale difference equations. I. Existence and global regularity of solutions,” SIAM J. Math. Anal. 22(5), 1388–1410 (1991).
I. Daubechies and J. Lagarias, “Two scale difference equations. I. Local regularity, infinite products of matrices and fractals,” SIAM J. Math. Anal. 23(4), 1031–1079 (1992).
V. Protasov, “Refinement equations with nonnegative coefficients,” J. Fourier Anal. Appl. 6(1), 55–78 (2000).
V. Yu. Protasov, “Fractal curves and wavelets,” Izv. Ross. Akad. Nauk Ser. Mat. 70(5), 123–162 (2006) [Izvestiya Mathematics 70 (5), 975–1013 (2006)].
K.-S. Lau and J. Wang, “Characterization of L p-solutions for two-scale dilation equations,” SIAM J. Math. Anal. 26(4), 1018–1046 (1995).
I. A. Sheipak and A. A. Vladimirov, Self-similar functions in L 2[0, 1] and Sturm—Liouville Problem with singular indefinite weight, arxiv: math. FA/0405410.
I. S. Kac and M. G. Krein, “The spectral functions of a string,” Supplement to the Russian translation of F. Atkinson, Discrete and Continuous Boundary Problems (Academic Press, New York-London, 1964; Moscow, Mir, 1968), pp. 648–733 [in Russian].
I. P. Natanson, Theory of Functions of a Real Variable (Nauka, Moscow, 1974) [in Russian].
Author information
Authors and Affiliations
Additional information
Original Russian Text © I. A. Sheipak, 2007, published in Matematicheskie Zametki, 2007, Vol. 81, No. 6, pp. 924–938.
Rights and permissions
About this article
Cite this article
Sheipak, I.A. On the construction and some properties of self-similar functions in the spaces L p[0, 1]. Math Notes 81, 827–839 (2007). https://doi.org/10.1134/S0001434607050306
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1134/S0001434607050306