Geodesic
On the absolute convergence of Fourier--Haar series in the metric of $L^p(0,1)$, $0$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 46, Tome 467 (2018), pp. 34-54

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It is proved that for any $0\epsilon1$ there exists a measurable set $E\subset[0,1]$ with $|E|>1-\epsilon$ such that for any function $f(x)\in L^1[0,1]$ one can find a function $g(x)\in L^1[0,1]$ equal to $f(x)$ on $E$ such that its Fourier–Haar series converges absolutely in the metric of $L^p(0,1)$, $0$. 
@article{ZNSL_2018_467_a3,
     author = {M. G. Grigoryan},
     title = {On the absolute convergence of {Fourier--Haar} series in the metric of $L^p(0,1)$, $0<p<1$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {34--54},
     publisher = {mathdoc},
     volume = {467},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_467_a3/}
}
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M. G. Grigoryan. On the absolute convergence of Fourier--Haar series in the metric of $L^p(0,1)$, $0