The study of the fourier series of functions defined on moran fractals

Abstract

LetE be a Moran fractal andH s (E) denote thes-dimensional Hausdorff measure ofE. In this paper, we define a orthonormal and complete system φ of functions in the Hilbert spaceL2(E,H s ) and prove that partial sums of the Fourier series, with respect to φ, of each functionf(x)∈L1(E,H s ) converge tof(x) atH s -a.e.x∈E. Moreover, the Fourier series off, forf∈L p (E,H s ),p≥1, converges off inL p -norm. When Moran fractals degenerate into self-similar fractals, our results well agree with M. Reyes's results.