Singularities of carleman type for subsystems of a trigonometric system

We prove that for arbitrary $\varepsilon>0$ there exists a sequence of positive integers $\{n_k\}$ such that a) the system $\{\cos n_kX,\sin n_kX\}$ is a basis with respect to the $C[-\pi,\pi]$ norm in the closure of its linear hull, and b) a continuous function $f(x)$ belonging to the closure of the linear hull of the system can be found such that its Fourier coefficients $a_n$ and $b_n$ satisfy the relation
$$ \sum{n=1}^\infty|a_n|^{2-\varepsilon}+|b_n|^{2-\varepsilon}=\infty. $$