On the construction and some properties of self-similar functions in the spaces L p[0, 1]

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.