THE FOURIER SERIES EXPANSIONS OF FUNCTIONS DEFINED ON S-SETS

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THE FOURIER SERIES EXPANSIONS OF FUNCTIONS DEFINED ON S-SETS

Let $ E $ be a compact $s$-sets of $ R^n$. The authors define an orthonormal system $ \Phi $ of functions on $E$ and obtain that, for any $f(x)\in L^1(E, \Cal H^s)$, the Fourier series of $f$, with respect to $\Phi $, is equal to $f(x)$ at $\Cal H^s$-a.e. $x\in E.$ Moreover, for any $ f\in L^p(E,\Cal H^s) \ \ (p\ge 1),$ the partial sums of the Fourier series, with respect to $\Phi$, of $f$ converges to $f$ in $L^p-$norm.