This article is cited in 30 papers

On the Construction and Some Properties of Self-Similar Functions in the Spaces $L_p[0,1]$

I. A. Sheipak

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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.

Keywords: self-similar function, self-similar measure, fractal curve, monotonicity, function of bounded variation, Lebesgue classes.

UDC: 517.518

Received: 28.06.2006
Revised: 29.09.2006

DOI: 10.4213/mzm3741

 English version: Mathematical Notes, 2007, 81:6, 827–839

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