Algebraic polynomial bases of space $L_p$

Let $\{\varphi_n\}$ be a system, close to the orthonormal complete system $\{\chi_n\}$. An estimate is obtained for the deviation of the system $\{f_n\}$, obtained from $\{\varphi_n\}$ by Schmidt's method, from the system $\{\chi_n\}$. This estimate is used to show that, in any $L_p(-1,1)$, with $p\in(1,4/3]\cup[4,\infty)$, and for any $\lambda>\pi e/4=2,\!13\dots$, there exists an orthogonal algebraic system $\{P_n(x)\}_{n=0}^\infty$, forming a basis in $L_p$ and such that $\nu_n=\deg P_n(x)\le\lambda n$ for $n>n_0(p,\lambda)$.