Estimates of best approximations of transformed Fourier series in $L^p$-norm and $p$-variational norm

We consider functions $F=F(\lambda,f)$ with transformed Fourier series $\sum\limits^\infty_{n=1}\lambda_nA_n(x)$, where $\smash[t]{\sum\limits^\infty_{n=1}A_n(x)}$ is the Fourier series of a function $f$. Let $C_p$ be the space of $2\pi$-periodic $p$-absolutely continuous functions with $p$-variational norm. The estimates of best approximations of $F$ in $L^p$ in terms of best approximations of $f$ in $C_p$ are given. Also the dual problem for $F$ in $C_p$ and $f$ in $L^p$ is treated. In the important case of fractional derivative, the sharpness of estimates is established.