Geodesic
On convergence of Fourier series in complete orthonormal systems in the $L^1$-metric and almost everywhere
Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 445-466 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is proved that if $\{\varphi_n(x)\}$ is a complete orthonormal system of bounded functions and $\varepsilon>0$, then there exists a measurable set $E\subset[0,1]$ with $|E|>1-\varepsilon$ such that 1) for any function $f(x)\in L[0,1]$ there exists a function $g(x)\in L^1[0,1]$ with $g(x)=f(x)$ on $E$ and such that the Fourier series of $g(x)$ in the system $\{\varphi_n(x)\}$ converges in the $L^1$-metric; and 2) there exists a subsequence of natural numbers $m_k\nearrow\infty$ such that for any function $f(x)\in L^1[0,1]$ there exists a function $g(x)\in L^1[0,1]$ such that $g(x)=f(x)$ for $x\in E$, $\displaystyle\lim_{k\to\infty}\sum\limits_{n=1}^{m_k}\alpha_n(g)\varphi_n(x)=g(x)$ almost everywhere on $[0,1]$, and $\{\alpha_n(g)\}\in l_p$ for all $p>2$, where $\displaystyle\alpha_n(g)=\int_0^1g(x)\varphi_n(x)\,dx$, $n=1,2\dots$ . 
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     author = {M. G. Grigoryan},
     title = {On convergence of {Fourier} series in complete orthonormal systems in the $L^1$-metric and almost everywhere},
     journal = {Sbornik. Mathematics},
     pages = {445--466},
     year = {1991},
     volume = {70},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1991_70_2_a7/}
}
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M. G. Grigoryan. On convergence of Fourier series in complete orthonormal systems in the $L^1$-metric and almost everywhere. Sbornik. Mathematics, Tome 70 (1991) no. 2, pp. 445-466. http://geodesic.mathdoc.fr/item/SM_1991_70_2_a7/

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