A system of functions

A system of functions
$$ f_k(x)=\sum_{i=1}^r\alpha_i\varphi_i(x)^k+b_i\overline\varphi_i(x)^k,\quad k=1,2,\dots $$
is considered on the interval $[0,l]$.
Under certain conditions on the $\varphi_i(x)$, it is proved that the system $1\cup\{f_k(x)\}_{k=1}^\infty$ is complete in the space $L_p(0,l)$. In the case $r=1$ it is proved, under certain additional assumptions, that the system $\{f_k(x)\}_{k=0}^\infty$ is minimal.