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 spaceL 2(E,H s) and prove that partial sums of the Fourier series, with respect to φ, of each functionf(x)∈L 1(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.
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This work is supported in part by the National Natural Science Foundation of China.
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Liang, J., Ren, F. The study of the fourier series of functions defined on moran fractals. Acta Mathematicae Applicatae Sinica 13, 158–166 (1997). https://doi.org/10.1007/BF02015137
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DOI: https://doi.org/10.1007/BF02015137