摘要
Let E be a Moran fractal and Hs(E) denote the s-dimensional Hausdorff measure of E. In this paper, we define a orthonormal and complete system. of functions in the Hilbert space L2(E,Hs) and prove that partial sums of the Fourier series,with respect to Φ, of each function f(x)∈L1(E,Hs) converge to f(x) at Hs-a.e. x∈E. Moreover, the Fourier series of f, for f∈Lp(E,Hs), p≥1, converges to f in Lp-norm. When Moran fractals degenerate into self-similar fractals, our results well agree with M. Reyes's results.
Let E be a Moran fractal and Hs(E) denote the s-dimensional Hausdorff measure of E. In this paper, we define a orthonormal and complete system. of functions in the Hilbert space L2(E,Hs) and prove that partial sums of the Fourier series,with respect to Φ, of each function f(x)∈L1(E,Hs) converge to f(x) at Hs-a.e. x∈E. Moreover, the Fourier series of f, for f∈Lp(E,Hs), p≥1, converges to f in Lp-norm. When Moran fractals degenerate into self-similar fractals, our results well agree with M. Reyes's results.
