Behavior of the Fourier–Walsh coefficients of a corrected function

We prove that, given a sequence $\{a_k\}^\infty_{k=1}$ with $a_k\downarrow0$ and $\{a_k\}^\infty_{k=1}\not\in l_2$, reals $0<\epsilon<1$ and $p\in[1,2]$, and $f\in L^p(0,1)$, we can find $\tilde f\in L^p(0,1)$ with $\operatorname{mes}\{f\ne\tilde f\}<\epsilon$ whose nonzero Fourier–Walsh coefficients $c_k(\tilde f)$ are such that $|c_k(\tilde f)|=a_k$ for $k\in\operatorname{spec}(\tilde f)$.