Universal functions in ‘correction’ problems guaranteeing the convergence of Fourier–Walsh series

We prove the existence of a function $g(x)\in L^1[0,1]$ with monotone decreasing Fourier–Walsh coefficients $\{c_k(g)\}_{k=0}^\infty\downarrow$ which is universal in $L^p[0,1]$, $p\geqslant1$, in the sense of modification with respect to the signs of the Fourier coefficients for the Walsh system. In other words, for every function $f\in L^p[0;1]$ and every $\varepsilon>0$ one can find a function $\widetilde f\in L^p[0;1]$ such that the measure $|\{x\in[0;1]\colon f(x)=\widetilde f(x)\}|$ is greater than $1-\varepsilon$, the Fourier series of $\widetilde f(x)$ in the Walsh system converges to $\widetilde f(x)$ in the $L^p[0,1]$-norm and $|c_k(\widetilde f)|=c_k(g)$, $k\in\operatorname{Spec}(\widetilde f)$. We also prove that for every $\varepsilon$, $0<\varepsilon<1$, one can find a measurable set $E\subset [0,1]$ of measure $|E|>1-\varepsilon$ and a function $g\in L^1[0;1]$ with $0<c_{k+1}(g)<c_k(g)$, $k=0,1,2,\dots$, such that for every function $f\in L^1[0,1]$ there is a function $\widetilde f\in L^1[0,1]$ with the following properties: $\widetilde f$ coincides with $f$ on $E$, the Fourier–Walsh series of $\widetilde f(x)$ converges to $\widetilde f(x)$ in the norm of $L^1[0,1]$ and the absolute values of all terms in the sequence of the Fourier–Walsh coefficients of the newly obtained function satisfy $|c_k(\widetilde f)|=c_k(g)$, $k=0,1,2,\dots$ .