The study of the Fourier series of functions defined on Moran fractals (Q1367248)

Let \(E\subset \mathbb{R}^n\) be a Moran set. Let \(\Phi\) be an orthonormal system defined on \(E\) satisfying some conditions. The authors prove that for any \(f\in L^1(E, {\mathcal H}^s)\), the Fourier series of \(f\), with respect to \(\Phi\), is equal to \(s\), \({\mathcal H}^s\)- a.e; and for any \(f\in L^p(E, {\mathcal H}^s)\), \(p\geq 1\), the partial sum of the Fourier series of \(f\) converges to \(f\) in \(L^p\)-norm. These results generalize that of Reyes who considered the case of self similar sets, and have been generalized by \textit{J. Liang}, \textit{W. Li}, \textit{F. Su} and \textit{F. Ren} [Chin. Ann. Math., Ser. B 18, No. 2, 201-212 (1997; Zbl 0886.42006)] to the \(s\)-set.