On properties of an incomplete system of functions close to powerfunctions

On the segment $[0,1]$ we consider a system of functions $\{x^{\lambda_\nu}[1+\varepsilon_\nu(x)]\}$, where the $\varepsilon_\nu(x)$ are small in a definite sense, $\lambda_\nu>0$, $\sum\limits_1^\infty\frac1{\lambda_\nu}<\infty$. We study the functions $y(x)$ which are approximated by linear combinations of functions of this system in $L_p(0,1)$ or in $C[0,1]$ .