Geodesic
A correction theorem and the dyadic space~$H(1,\infty)$
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 67-75

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It is proved that for every $L^\infty$-function $f$ and positive $\varepsilon$ there is a function $g$ whose partial sums of both Fourier and Walsh–Fourier series are uniformly bounded by $c(\log 1/\varepsilon)\|f\|_\infty$ and that satisfies $\operatorname{mes}\{f\ne g\}\varepsilon$. 
@article{ZNSL_1986_149_a4,
     author = {S. V. Kislyakov},
     title = {A correction theorem and the dyadic space~$H(1,\infty)$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--75},
     publisher = {mathdoc},
     volume = {149},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a4/}
}
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S. V. Kislyakov. A correction theorem and the dyadic space~$H(1,\infty)$. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 67-75. http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a4/