A correction theorem and the dyadic space $H(1,\infty)$

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 $\operatorname{mes}\{f\ne g\}<\varepsilon$.