A Haar Basis And Convergences Of Fractal Spaces

In this paper, we firstly give the construction of some typical fractal sets, the kth Cantor set in which k is an odd integer. Then discuss some topological properties and fractal characters of the kth Cantor sets; According to the metricδ_r, we discuss the symbol space (Σ_m~∞,δ_r) and analysis the special topological properties of the symbol space; Then according to the properties of the regular borel measure on the symbol space and the homeomorphic relation of the self-similar set and symbol space, which the self-similar set satisfy the strong separated condition, we gain the properties of a self-similar measure, which is defined in the self-similar set with satisfying to the strong separated condition, that is to say,it is a regular borel measure, on the basis of it, we especially discuss the functional space L~2(C,μ) and show that it is a separated Hilbert space; At last , applying to the properties of projected mapping, we have showed that there is a Haar basis in the space L~2(C,μ), then discuss some convergence with respect to the basis.