Order of growth of the degrees of a polynomial basis of a space of continuous functions

The problem under consideration is the one posed independently by C. Foias and I. Singer and by P. L. Ul'yanov concerning the minimal growth of the degrees $\nu_n$ of a polynomial basis $\{t_n(x)\}_0^\infty$ of a space of continuous functions. It is shown that there exist an absolute constant $C$ and a polynomial basis $\{t_n(x)\}_0^\infty$ such that
$$ \nu_n\le C(n\ln^+\ln(n+1)+1),\quad n=0,1,2,\dots $$
The feasibility of the method employed is also considered.